Project Euler
Project Euler problem solutions.
Project Euler
Solutions to Project Euler problems.
Problems
| # | Problem | Time | Description |
|---|---|---|---|
| 1 | Problem 1 | — | If we list all the natural numbers below 10 that are multiples of 3 or 5, we … |
| 2 | Problem 2 | — | Each new term in the Fibonacci sequence is generated by adding the previous t… |
| 3 | Problem 3 | — | The prime factors of 13195 are 5, 7, 13 and 29. |
| 4 | Problem 4 | — | A palindromic number reads the same both ways. |
| 5 | Problem 5 | — | 2520 is the smallest number that can be divided by each of the numbers from 1… |
| 6 | Problem 6 | — | The sum of the squares of the first ten natural numbers is, The square of the… |
| 7 | Problem 7 | — | By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see th… |
| 8 | Problem 8 | — | The four adjacent digits in the 1000-digit number that have the greatest prod… |
| 9 | Problem 9 | — | A Pythagorean triplet is a set of three natural numbers, a lt b lt c, for whi… |
| 10 | Problem 10 | — | The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17. |
| 11 | Problem 11 | 2m 32s | In the 20 times 20 grid below, four numbers along a diagonal line have been m… |
| 12 | Problem 12 | 3m 26s | The sequence of triangle numbers is generated by adding the natural numbers. |
| 13 | Problem 13 | 2m 32s | Work out the first ten digits of the sum of the following one-hundred 50-digi… |
| 14 | Problem 14 | 44s | The following iterative sequence is defined for the set of positive integers:… |
| 15 | Problem 15 | 54s | Starting in the top left corner of a 2 times 2 grid, and only being able to m… |
| 16 | Problem 16 | 1m 27s | 2^{15} = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26. |
| 17 | Problem 17 | 1m 51s | If the numbers 1 to 5 are written out in words: one, two, three, four, five, … |
| 18 | Problem 18 | 2m 46s | By starting at the top of the triangle below and moving to adjacent numbers o… |
| 19 | Problem 19 | 34s | You are given the following information, but you may prefer to do some resear… |
| 20 | Problem 20 | 1m 28s | n! means n times (n - 1) times cdots times 3 times 2 times 1. |
| 21 | Problem 21 | 55s | Let d(n) be defined as the sum of proper divisors of n (numbers less than n w… |
| 22 | Problem 22 | 1m 37s | Using names.txt (right click and 'Save Link/Target As...'), a 46K text file c… |
| 23 | Problem 23 | 1m 37s | A perfect number is a number for which the sum of its proper divisors is exac… |
| 24 | Problem 24 | 30s | A permutation is an ordered arrangement of objects. |
| 25 | Problem 25 | 24s | The Fibonacci sequence is defined by the recurrence relation: Fn = F{n - 1} +… |
| 26 | Problem 26 | 3m 2s | A unit fraction contains 1 in the numerator. |
| 27 | Problem 27 | 45s | Euler discovered the remarkable quadratic formula: n^2 + n + 41 It turns out … |
| 28 | Problem 28 | 2m 19s | Starting with the number 1 and moving to the right in a clockwise direction a… |
| 29 | Problem 29 | 5m 35s | Consider all integer combinations of a^b for 2 le a le 5 and 2 le b le 5: If … |
| 30 | Problem 30 | 29s | Surprisingly there are only three numbers that can be written as the sum of f… |
| 31 | Problem 31 | 10m 28s | In the United Kingdom the currency is made up of pound (£) and pence (p). |
| 32 | Problem 32 | 12m 48s | We shall say that an n-digit number is pandigital if it makes use of all the … |
| 33 | Problem 33 | 10m 31s | The fraction 49/98 is a curious fraction, as an inexperienced mathematician i… |
| 34 | Problem 34 | 49s | 145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145. |
| 35 | Problem 35 | 46s | The number, 197, is called a circular prime because all rotations of the digi… |
| 36 | Problem 36 | 37s | The decimal number, 585 = 10010010012 (binary), is palindromic in both bases. |
| 37 | Problem 37 | 57s | The number 3797 has an interesting property. |
| 38 | Problem 38 | 36s | Take the number 192 and multiply it by each of 1, 2, and 3: By concatenating … |
| 39 | Problem 39 | 29s | If p is the perimeter of a right angle triangle with integral length sides, a… |
| 40 | Problem 40 | 1m 18s | An irrational decimal fraction is created by concatenating the positive integ… |
| 41 | Problem 41 | 32s | We shall say that an n-digit number is pandigital if it makes use of all the … |
| 42 | Problem 42 | 45s | The nth term of the sequence of triangle numbers is given by, tn = frac12n(n+… |
| 43 | Problem 43 | 33s | The number, 1406357289, is a 0 to 9 pandigital number because it is made up o… |
| 44 | Problem 44 | 1m 40s | Pentagonal numbers are generated by the formula, Pn=n(3n-1)/2. |
| 45 | Problem 45 | 37s | Triangle, pentagonal, and hexagonal numbers are generated by the following fo… |
| 46 | Problem 46 | 47s | It was proposed by Christian Goldbach that every odd composite number can be … |
| 47 | Problem 47 | 34s | The first two consecutive numbers to have two distinct prime factors are: The… |
| 48 | Problem 48 | 32s | The series, 1^1 + 2^2 + 3^3 + cdots + 10^{10} = 10405071317. |
| 49 | Problem 49 | 2m 42s | The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increas… |
| 50 | Problem 50 | 1m 47s | The prime 41, can be written as the sum of six consecutive primes: This is th… |
| 51 | Problem 51 | 2m 20s | By replacing the 1st digit of the 2-digit number 3, it turns out that six of … |
| 52 | Problem 52 | 2m 20s | It can be seen that the number, 125874, and its double, 251748, contain exact… |
| 53 | Problem 53 | 2m 19s | There are exactly ten ways of selecting three from five, 12345: 123, 124, 125… |
| 54 | Problem 54 | 50s | In the card game poker, a hand consists of five cards and are ranked, from lo… |
| 55 | Problem 55 | 37s | If we take 47, reverse and add, 47 + 74 = 121, which is palindromic. |
| 56 | Problem 56 | 48s | A googol (10^{100}) is a massive number: one followed by one-hundred zeros; 1… |
| 57 | Problem 57 | 35s | It is possible to show that the square root of two can be expressed as an inf… |
| 58 | Problem 58 | 43s | Starting with 1 and spiralling anticlockwise in the following way, a square s… |
| 59 | Problem 59 | 41s | Each character on a computer is assigned a unique code and the preferred stan… |
| 60 | Problem 60 | 42s | The primes 3, 7, 109, and 673, are quite remarkable. |
| 61 | Problem 61 | 2m 46s | Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers ar… |
| 62 | Problem 62 | 2m 45s | The cube, 41063625 (345^3), can be permuted to produce two other cubes: 56623… |
| 63 | Problem 63 | 1m 26s | The 5-digit number, 16807=7^5, is also a fifth power. |
| 64 | Problem 64 | 46s | All square roots are periodic when written as continued fractions and can be … |
| 65 | Problem 65 | 6m 37s | The square root of 2 can be written as an infinite continued fraction. |
| 66 | Problem 66 | 5m 40s | Consider quadratic Diophantine equations of the form: For example, when D=13,… |
| 67 | Problem 67 | 5m 40s | By starting at the top of the triangle below and moving to adjacent numbers o… |
| 68 | Problem 68 | 1m 48s | Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, an… |
| 69 | Problem 69 | 3m 33s | Euler's totient function, phi(n) [sometimes called the phi function], is defi… |
| 70 | Problem 70 | 51s | Euler's totient function, phi(n) [sometimes called the phi function], is used… |
| 71 | Problem 71 | 44s | Consider the fraction, dfrac n d, where n and d are positive integers. |
| 72 | Problem 72 | 32s | Consider the fraction, dfrac n d, where n and d are positive integers. |
| 73 | Problem 73 | 1m 12s | Consider the fraction, dfrac n d, where n and d are positive integers. |
| 74 | Problem 74 | 51s | The number 145 is well known for the property that the sum of the factorial o… |
| 75 | Problem 75 | 1m 54s | It turns out that pu{12 cm} is the smallest length of wire that can be bent t… |
| 76 | Problem 76 | 1m 53s | It is possible to write five as a sum in exactly six different ways: How many… |
| 77 | Problem 77 | 1m 36s | It is possible to write ten as the sum of primes in exactly five different wa… |
| 78 | Problem 78 | 2m 20s | Let p(n) represent the number of different ways in which n coins can be separ… |
| 79 | Problem 79 | 1m 57s | A common security method used for online banking is to ask the user for three… |
| 80 | Problem 80 | 46s | It is well known that if the square root of a natural number is not an intege… |
| 81 | Problem 81 | 34s | In the 5 by 5 matrix below, the minimal path sum from the top left to the bot… |
| 82 | Problem 82 | 2m 16s | NOTE: This problem is a more challenging version of Problem 81. |
| 83 | Problem 83 | 1m 11s | NOTE: This problem is a significantly more challenging version of Problem 81. |
| 84 | Problem 84 | 1m 9s | In the game, Monopoly, the standard board is set up in the following way: !00… |
| 85 | Problem 85 | 45s | By counting carefully it can be seen that a rectangular grid measuring 3 by 2… |
| 86 | Problem 86 | 1m 57s | A spider, S, sits in one corner of a cuboid room, measuring 6 by 5 by 3, and … |
| 87 | Problem 87 | 36s | The smallest number expressible as the sum of a prime square, prime cube, and… |
| 88 | Problem 88 | 47s | A natural number, N, that can be written as the sum and product of a given se… |
| 89 | Problem 89 | 40s | For a number written in Roman numerals to be considered valid there are basic… |
| 90 | Problem 90 | 39s | Each of the six faces on a cube has a different digit (0 to 9) written on it;… |
| 91 | Problem 91 | 2m 20s | The points P(x1, y1) and Q(x2, y2) are plotted at integer co-ordinates and ar… |
| 92 | Problem 92 | 2m 19s | A number chain is created by continuously adding the square of the digits in … |
| 93 | Problem 93 | 2m 19s | By using each of the digits from the set, 1, 2, 3, 4, exactly once, and makin… |
| 94 | Problem 94 | 1m 46s | It is easily proved that no equilateral triangle exists with integral length … |
| 95 | Problem 95 | 4m 22s | The proper divisors of a number are all the divisors excluding the number its… |
| 96 | Problem 96 | 2m 39s | Su Doku (Japanese meaning number place) is the name given to a popular puzzle… |
| 97 | Problem 97 | 26s | The first known prime found to exceed one million digits was discovered in 19… |
| 98 | Problem 98 | 41s | By replacing each of the letters in the word CARE with 1, 2, 9, and 6 respect… |
| 99 | Problem 99 | 1m 37s | Comparing two numbers written in index form like 2^{11} and 3^7 is not diffic… |
| 100 | Problem 100 | 58s | If a box contains twenty-one coloured discs, composed of fifteen blue discs a… |
| 101 | Problem 101 | 1m 37s | If we are presented with the first k terms of a sequence it is impossible to … |
| 102 | Problem 102 | 3m 34s | Three distinct points are plotted at random on a Cartesian plane, for which -… |
| 103 | Problem 103 | 3m 34s | Let S(A) represent the sum of elements in set A of size n. |
| 104 | Problem 104 | 39s | The Fibonacci sequence is defined by the recurrence relation: Fn = F{n - 1} +… |
| 105 | Problem 105 | 3m 13s | Let S(A) represent the sum of elements in set A of size n. |
| 106 | Problem 106 | 46s | Let S(A) represent the sum of elements in set A of size n. |
| 107 | Problem 107 | 59s | The following undirected network consists of seven vertices and twelve edges … |
| 108 | Problem 108 | 1m 7s | In the following equation x, y, and n are positive integers. |
| 109 | Problem 109 | 5m 55s | In the game of darts a player throws three darts at a target board which is s… |
| 110 | Problem 110 | 5m 48s | In the following equation x, y, and n are positive integers. |
| 111 | Problem 111 | 51s | Considering 4-digit primes containing repeated digits it is clear that they c… |
| 112 | Problem 112 | 2m 23s | Working from left-to-right if no digit is exceeded by the digit to its left i… |
| 113 | Problem 113 | 3m 49s | Working from left-to-right if no digit is exceeded by the digit to its left i… |
| 114 | Problem 114 | 32s | A row measuring seven units in length has red blocks with a minimum length of… |
| 115 | Problem 115 | 39s | NOTE: This is a more difficult version of Problem 114. |
| 116 | Problem 116 | 40s | A row of five grey square tiles is to have a number of its tiles replaced wit… |
| 117 | Problem 117 | 46s | Using a combination of grey square tiles and oblong tiles chosen from: red ti… |
| 118 | Problem 118 | 41s | Using all of the digits 1 through 9 and concatenating them freely to form dec… |
| 119 | Problem 119 | 34s | The number 512 is interesting because it is equal to the sum of its digits ra… |
| 120 | Problem 120 | 1m 55s | Let r be the remainder when (a - 1)^n + (a + 1)^n is divided by a^2. |
| 121 | Problem 121 | 43s | A bag contains one red disc and one blue disc. |
| 122 | Problem 122 | 44s | The most naive way of computing n^{15} requires fourteen multiplications: But… |
| 123 | Problem 123 | 1m 52s | Let pn be the nth prime: 2, 3, 5, 7, 11, dots, and let r be the remainder whe… |
| 124 | Problem 124 | 3m 45s | The radical of n, operatorname{rad}(n), is the product of the distinct prime … |
| 125 | Problem 125 | 4m 27s | The palindromic number 595 is interesting because it can be written as the su… |
| 126 | Problem 126 | 33s | The minimum number of cubes to cover every visible face on a cuboid measuring… |
| 127 | Problem 127 | 39s | The radical of n, operatorname{rad}(n), is the product of distinct prime fact… |
| 128 | Problem 128 | 44s | A hexagonal tile with number 1 is surrounded by a ring of six hexagonal tiles… |
| 129 | Problem 129 | 2m 59s | A number consisting entirely of ones is called a repunit. |
| 130 | Problem 130 | 2m 19s | A number consisting entirely of ones is called a repunit. |
| 131 | Problem 131 | 2m 5s | There are some prime values, p, for which there exists a positive integer, n,… |
| 132 | Problem 132 | 53s | A number consisting entirely of ones is called a repunit. |
| 133 | Problem 133 | 1m 21s | A number consisting entirely of ones is called a repunit. |
| 134 | Problem 134 | 41s | Consider the consecutive primes p1 = 19 and p2 = 23. |
| 135 | Problem 135 | 33s | Given the positive integers, x, y, and z, are consecutive terms of an arithme… |
| 136 | Problem 136 | 49s | The positive integers, x, y, and z, are consecutive terms of an arithmetic pr… |
| 137 | Problem 137 | 43s | Consider the infinite polynomial series AF(x) = x F1 + x^2 F2 + x^3 F3 + dots… |
| 138 | Problem 138 | 44s | Consider the isosceles triangle with base length, b = 16, and legs, L = 17. |
| 139 | Problem 139 | 3m 51s | Let (a, b, c) represent the three sides of a right angle triangle with integr… |
| 140 | Problem 140 | 2m 6s | Consider the infinite polynomial series AG(x) = x G1 + x^2 G2 + x^3 G3 + cdot… |
| 141 | Problem 141 | 49s | A positive integer, n, is divided by d and the quotient and remainder are q a… |
| 142 | Problem 142 | 42s | Find the smallest x + y + z with integers x gt y gt z gt 0 such that x + y, x… |
| 143 | Problem 143 | 39s | Let ABC be a triangle with all interior angles being less than 120 degrees. |
| 144 | Problem 144 | 41s | In laser physics, a "white cell" is a mirror system that acts as a delay line… |
| 145 | Problem 145 | 3m 22s | Some positive integers n have the property that the sum [n + operatorname{rev… |
| 146 | Problem 146 | 53s | The smallest positive integer n for which the numbers n^2 + 1, n^2 + 3, n^2 +… |
| 147 | Problem 147 | 2m 10s | In a 3 times 2 cross-hatched grid, a total of 37 different rectangles could b… |
| 148 | Problem 148 | 43s | We can easily verify that none of the entries in the first seven rows of Pasc… |
| 149 | Problem 149 | 49s | Looking at the table below, it is easy to verify that the maximum possible su… |
| 150 | Problem 150 | 1m 24s | In a triangular array of positive and negative integers, we wish to find a su… |
| 151 | Problem 151 | 52s | A printing shop runs 16 batches (jobs) every week and each batch requires a s… |
| 152 | Problem 152 | 2m 21s | There are several ways to write the number dfrac{1}{2} as a sum of square rec… |
| 153 | Problem 153 | 2m 58s | As we all know the equation x^2=-1 has no solutions for real x. |
| 154 | Problem 154 | 1m 39s | A triangular pyramid is constructed using spherical balls so that each ball r… |
| 155 | Problem 155 | 3m 28s | An electric circuit uses exclusively identical capacitors of the same value C. |
| 156 | Problem 156 | 54s | Starting from zero the natural numbers are written down in base 10 like this:… |
| 157 | Problem 157 | 3m 44s | Consider the diophantine equation frac 1 a + frac 1 b = frac p {10^n} with a,… |
| 158 | Problem 158 | 6m 3s | Taking three different letters from the 26 letters of the alphabet, character… |
| 159 | Problem 159 | 46s | A composite number can be factored many different ways. |
| 160 | Problem 160 | 56s | For any N, let f(N) be the last five digits before the trailing zeroes in N!. |
| 161 | Problem 161 | 1m 22s | A triomino is a shape consisting of three squares joined via the edges. |
| 162 | Problem 162 | 47s | In the hexadecimal number system numbers are represented using 16 different d… |
| 163 | Problem 163 | 2m 58s | Consider an equilateral triangle in which straight lines are drawn from each … |
| 164 | Problem 164 | 1m 8s | How many 20 digit numbers n (without any leading zero) exist such that no thr… |
| 165 | Problem 165 | 46s | A segment is uniquely defined by its two endpoints. |
| 166 | Problem 166 | 1m 10s | A 4 times 4 grid is filled with digits d, 0 le d le 9. |
| 167 | Problem 167 | 1m 27s | For two positive integers a and b, the Ulam sequence U(a,b) is defined by U(a… |
| 168 | Problem 168 | 42s | Consider the number 142857. |
| 169 | Problem 169 | 43s | Define f(0)=1 and f(n) to be the number of different ways n can be expressed … |
| 170 | Problem 170 | 42s | Take the number 6 and multiply it by each of 1273 and 9854: By concatenating … |
| 171 | Problem 171 | 6m 30s | For a positive integer n, let f(n) be the sum of the squares of the digits (i… |
| 172 | Problem 172 | 37s | How many 18-digit numbers n (without leading zeros) are there such that no di… |
| 173 | Problem 173 | 32s | We shall define a square lamina to be a square outline with a square "hole" s… |
| 174 | Problem 174 | 36s | We shall define a square lamina to be a square outline with a square "hole" s… |
| 175 | Problem 175 | 2m 39s | Define f(0)=1 and f(n) to be the number of ways to write n as a sum of powers… |
| 176 | Problem 176 | 1m 4s | The four right-angled triangles with sides (9,12,15), (12,16,20), (5,12,13) a… |
| 177 | Problem 177 | 58s | Let ABCD be a convex quadrilateral, with diagonals AC and BD. |
| 178 | Problem 178 | 2m 2s | Consider the number 45656. |
| 179 | Problem 179 | 58s | Find the number of integers 1 lt n lt 10^7, for which n and n + 1 have the sa… |
| 180 | Problem 180 | 47s | For any integer n, consider the three functions and their combination We call… |
| 181 | Problem 181 | 1m 20s | Having three black objects B and one white object W they can be grouped in 7 … |
| 182 | Problem 182 | 1m 58s | The RSA encryption is based on the following procedure: Generate two distinct… |
| 183 | Problem 183 | 46s | Let N be a positive integer and let N be split into k equal parts, r = N/k, s… |
| 184 | Problem 184 | 4m 17s | Consider the set Ir of points (x,y) with integer co-ordinates in the interior… |
| 185 | Problem 185 | 42s | The game Number Mind is a variant of the well known game Master Mind. |
| 186 | Problem 186 | 1m 48s | Here are the records from a busy telephone system with one million users: |
| 187 | Problem 187 | 1m 8s | A composite is a number containing at least two prime factors. |
| 188 | Problem 188 | 38s | The hyperexponentiation or tetration of a number a by a positive integer b, d… |
| 189 | Problem 189 | 3m 13s | Consider the following configuration of 64 triangles: We wish to colour the i… |
| 190 | Problem 190 | 3m 53s | Let Sm = (x1, x2, dots , xm) be the m-tuple of positive real numbers with x1 … |
| 191 | Problem 191 | 2m 19s | A particular school offers cash rewards to children with good attendance and … |
| 192 | Problem 192 | 3m 39s | Let x be a real number. |
| 193 | Problem 193 | 1m 18s | A positive integer n is called squarefree, if no square of a prime divides n,… |
| 194 | Problem 194 | 1m 7s | Consider graphs built with the units A: and B: , where the units are glued al… |
| 195 | Problem 195 | 3m | Let's call an integer sided triangle with exactly one angle of 60 degrees a 6… |
| 196 | Problem 196 | 3m | Build a triangle from all positive integers in the following way: 1 2 3 4 5 6… |
| 197 | Problem 197 | 2m 40s | Given is the function f(x) = lfloor 2^{30.403243784 - x^2}rfloor times 10^{-9… |
| 198 | Problem 198 | 48s | A best approximation to a real number x for the denominator bound d is a rati… |
| 199 | Problem 199 | 1m 2s | Three circles of equal radius are placed inside a larger circle such that eac… |
| 200 | Problem 200 | 1m 24s | We shall define a sqube to be a number of the form, p^2 q^3, where p and q ar… |
| 201 | Problem 201 | 2m 57s | For any set A of numbers, let operatorname{sum}(A) be the sum of the elements… |
| 202 | Problem 202 | 1m 59s | Three mirrors are arranged in the shape of an equilateral triangle, with thei… |
| 203 | Problem 203 | 37s | The binomial coefficients displaystyle binom n k can be arranged in triangula… |
| 204 | Problem 204 | 44s | A Hamming number is a positive number which has no prime factor larger than 5. |
| 205 | Problem 205 | 1m 7s | Peter has nine four-sided (pyramidal) dice, each with faces numbered 1, 2, 3, 4. |
| 206 | Problem 206 | 38s | Find the unique positive integer whose square has the form 1234567890, where … |
| 207 | Problem 207 | 2m 20s | For some positive integers k, there exists an integer partition of the form 4… |
| 208 | Problem 208 | 1m 4s | A robot moves in a series of one-fifth circular arcs (72^circ), with a free c… |
| 209 | Problem 209 | 1m 3s | A k-input binary truth table is a map from k input bits (binary digits, 0 [fa… |
| 210 | Problem 210 | 41s | Consider the set S(r) of points (x,y) with integer coordinates satisfying |
| 211 | Problem 211 | 37s | For a positive integer n, let sigma2(n) be the sum of the squares of its divi… |
| 212 | Problem 212 | 1m 47s | An axis-aligned cuboid, specified by parameters (x0, y0, z0), (dx, dy, dz), c… |
| 213 | Problem 213 | 6m 24s | A 30 times 30 grid of squares contains 900 fleas, initially one flea per square. |
| 214 | Problem 214 | 36s | Let phi be Euler's totient function, i.e. |
| 215 | Problem 215 | 36s | Consider the problem of building a wall out of 2 times 1 and 3 times 1 bricks… |
| 216 | Problem 216 | 1m 18s | Consider numbers t(n) of the form t(n) = 2n^2 - 1 with n gt 1. |
| 217 | Problem 217 | 1m 31s | A positive integer with k (decimal) digits is called balanced if its first lc… |
| 218 | Problem 218 | 1m 4s | Consider the right angled triangle with sides a=7, b=24 and c=25. |
| 219 | Problem 219 | 1m 15s | Let A and B be bit strings (sequences of 0's and 1's). |
| 220 | Problem 220 | 1m 12s | Let D0 be the two-letter string "Fa". |
| 221 | Problem 221 | 4m 49s | We shall call a positive integer A an "Alexandrian integer", if there exist i… |
| 222 | Problem 222 | 2m 46s | What is the length of the shortest pipe, of internal radius pu{50 mm}, that c… |
| 223 | Problem 223 | 6m 23s | Let us call an integer sided triangle with sides a le b le c barely acute if … |
| 224 | Problem 224 | 4m 15s | Let us call an integer sided triangle with sides a le b le c barely obtuse if… |
| 225 | Problem 225 | 45s | The sequence 1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355, 653, 1201, dots is … |
| 226 | Problem 226 | 5m 31s | The blancmange curve is the set of points (x, y) such that 0 le x le 1 and y … |
| 227 | Problem 227 | 5m 31s | The Chase is a game played with two dice and an even number of players. |
| 228 | Problem 228 | 55s | Let Sn be the regular n-sided polygon – or shape – whose vertices vk (k = 1, … |
| 229 | Problem 229 | 2m 41s | Consider the number 3600. |
| 230 | Problem 230 | 3m 10s | For any two strings of digits, A and B, we define F{A, B} to be the sequence … |
| 231 | Problem 231 | 46s | The binomial coefficient displaystyle binom {10} 3 = 120. |
| 232 | Problem 232 | 1m 21s | Two players share an unbiased coin and take it in turns to play The Race. |
| 233 | Problem 233 | 4m 20s | Let f(N) be the number of points with integer coordinates that are on a circl… |
| 234 | Problem 234 | 55s | For an integer n ge 4, we define the lower prime square root of n, denoted by… |
| 235 | Problem 235 | 2m 28s | Given is the arithmetic-geometric sequence u(k) = (900-3k)r^{k - 1}. |
| 236 | Problem 236 | 15m 38s | Suppliers 'A' and 'B' provided the following numbers of products for the luxu… |
| 237 | Problem 237 | 56s | Let T(n) be the number of tours over a 4 times n playing board such that: - T… |
| 238 | Problem 238 | 3m 25s | Create a sequence of numbers using the "Blum Blum Shub" pseudo-random number … |
| 239 | Problem 239 | 38s | A set of disks numbered 1 through 100 are placed in a line in random order. |
| 240 | Problem 240 | 1m 4s | There are 1111 ways in which five 6-sided dice (sides numbered 1 to 6) can be… |
| 241 | Problem 241 | 2m 47s | For a positive integer n, let sigma(n) be the sum of all divisors of n. |
| 242 | Problem 242 | 2m 57s | Given the set 1,2,dots,n, we define f(n, k) as the number of its k-element su… |
| 243 | Problem 243 | 45s | A positive fraction whose numerator is less than its denominator is called a … |
| 244 | Problem 244 | 1m 3s | You probably know the game Fifteen Puzzle. |
| 245 | Problem 245 | 7m 4s | We shall call a fraction that cannot be cancelled down a resilient fraction. |
| 246 | Problem 246 | 10m 42s | A definition for an ellipse is: Given a circle c with centre M and radius r a… |
| 247 | Problem 247 | 1m 16s | Consider the region constrained by 1 le x and 0 le y le 1/x. |
| 248 | Problem 248 | 1m 27s | The first number n for which phi(n)=13! is 6227180929. |
| 249 | Problem 249 | 6m 18s | Let S = 2, 3, 5, dots, 4999 be the set of prime numbers less than 5000. |
| 250 | Problem 250 | 53s | Find the number of non-empty subsets of 1^1, 2^2, 3^3,dots, 250250^{250250}, … |
| 251 | Problem 251 | 4m 32s | A triplet of positive integers (a, b, c) is called a Cardano Triplet if it sa… |
| 252 | Problem 252 | 5m 24s | Given a set of points on a plane, we define a convex hole to be a convex poly… |
| 253 | Problem 253 | 5m 55s | A small child has a “number caterpillar” consisting of forty jigsaw pieces, e… |
| 254 | Problem 254 | 11m 31s | Define f(n) as the sum of the factorials of the digits of n. |
| 255 | Problem 255 | 3m 26s | We define the rounded-square-root of a positive integer n as the square root … |
| 256 | Problem 256 | 31m 17s | Tatami are rectangular mats, used to completely cover the floor of a room, wi… |
| 257 | Problem 257 | 2m 26s | Given is an integer sided triangle ABC with sides a le b le c. |
| 258 | Problem 258 | 1m 25s | A sequence is defined as: - gk = 1, for 0 le k le 1999 - gk = g{k-2000} + g{k… |
| 259 | Problem 259 | 1m 32s | A positive integer will be called reachable if it can result from an arithmet… |
| 260 | Problem 260 | 31m 18s | A game is played with three piles of stones and two players. |
| 261 | Problem 261 | 59s | Let us call a positive integer k a square-pivot, if there is a pair of intege… |
| 262 | Problem 262 | 6m 30s | The following equation represents the continuous topography of a mountainous … |
| 263 | Problem 263 | 2m | Consider the number 6. |
| 264 | Problem 264 | 51s | Consider all the triangles having: - All their vertices on lattice pointsInte… |
| 265 | Problem 265 | 1m | 2^N binary digits can be placed in a circle so that all the N-digit clockwise… |
| 266 | Problem 266 | 45s | The divisors of 12 are: 1,2,3,4,6 and 12. |
| 267 | Problem 267 | 1m 11s | You are given a unique investment opportunity. |
| 268 | Problem 268 | 1m 7s | It can be verified that there are 23 positive integers less than 1000 that ar… |
| 269 | Problem 269 | 1m 32s | A root or zero of a polynomial P(x) is a solution to the equation P(x) = 0. |
| 270 | Problem 270 | 1m 37s | A square piece of paper with integer dimensions N times N is placed with a co… |
| 271 | Problem 271 | 59s | For a positive number n, define S(n) as the sum of the integers x, for which … |
| 272 | Problem 272 | 1m 3s | For a positive number n, define C(n) as the number of the integers x, for whi… |
| 273 | Problem 273 | 1m 32s | Consider equations of the form: a^2 + b^2 = N, 0 le a le b, a, b and N integer. |
| 274 | Problem 274 | 1m 11s | For each integer p gt 1 coprime to 10 there is a positive divisibility multip… |
| 275 | Problem 275 | 1m 4s | Let us define a balanced sculpture of order n as follows: - A polyominoAn arr… |
| 276 | Problem 276 | 1m 42s | Consider the triangles with integer sides a, b and c with a le b le c. |
| 277 | Problem 277 | 1m 20s | A modified Collatz sequence of integers is obtained from a starting value a1 … |
| 278 | Problem 278 | 47s | Given the values of integers 1 < a1 < a2 < dots < an, consider the linear com… |
| 279 | Problem 279 | 3m 48s | How many triangles are there with integral sides, at least one integral angle… |
| 280 | Problem 280 | 7m 20s | A laborious ant walks randomly on a 5 times 5 grid. |
| 281 | Problem 281 | 1m 1s | You are given a pizza (perfect circle) that has been cut into m cdot n equal … |
| 282 | Problem 282 | 1m 48s | defhtmltext1{style{font-family:inherit;}{text{1}}} For non-negative integers … |
| 283 | Problem 283 | 2m 35s | Consider the triangle with sides 6, 8, and 10. |
| 284 | Problem 284 | 4m 11s | The 3-digit number 376 in the decimal numbering system is an example of numbe… |
| 285 | Problem 285 | 1m 43s | Albert chooses a positive integer k, then two real numbers a, b are randomly … |
| 286 | Problem 286 | 51s | Barbara is a mathematician and a basketball player. |
| 287 | Problem 287 | 1m 40s | The quadtree encoding allows us to describe a 2^N times 2^N black and white i… |
| 288 | Problem 288 | 1m 2s | For any prime p the number N(p, q) is defined by N(p, q) = sum{n = 0}^q Tn cd… |
| 289 | Problem 289 | 1m 14s | Let C(x, y) be a circle passing through the points (x, y), (x, y + 1), (x + 1… |
| 290 | Problem 290 | 1m 40s | How many integers 0 le n lt 10^{18} have the property that the sum of the dig… |
| 291 | Problem 291 | 4m 42s | A prime number p is called a Panaitopol prime if p = dfrac{x^4 - y^4}{x^3 + y… |
| 292 | Problem 292 | 1m 15s | We shall define a pythagorean polygon to be a convex polygon with the followi… |
| 293 | Problem 293 | 1m 2s | An even positive integer N will be called admissible, if it is a power of 2 o… |
| 294 | Problem 294 | 1m 30s | For a positive integer k, define d(k) as the sum of the digits of k in its us… |
| 295 | Problem 295 | 7m 38s | We call the convex area enclosed by two circles a lenticular hole if: - The c… |
| 296 | Problem 296 | 31m 17s | Given is an integer sided triangle ABC with BC le AC le AB. |
| 297 | Problem 297 | 1m 10s | Each new term in the Fibonacci sequence is generated by adding the previous t… |
| 298 | Problem 298 | 4m 19s | Larry and Robin play a memory game involving a sequence of random numbers bet… |
| 299 | Problem 299 | 3m 26s | Four points with integer coordinates are selected: A(a, 0), B(b, 0), C(0, c) … |
| 300 | Problem 300 | 3m 35s | In a very simplified form, we can consider proteins as strings consisting of … |
| 301 | Problem 301 | 45s | Nim is a game played with heaps of stones, where two players take it in turn … |
| 302 | Problem 302 | 16m 15s | A positive integer n is powerful if p^2 is a divisor of n for every prime fac… |
| 303 | Problem 303 | 16m 16s | For a positive integer n, define f(n) as the least positive multiple of n tha… |
| 304 | Problem 304 | 10m 24s | For any positive integer n the function operatorname{nextprime}(n) returns th… |
| 305 | Problem 305 | 7m 43s | Let's call S the (infinite) string that is made by concatenating the consecut… |
| 306 | Problem 306 | 31m 18s | The following game is a classic example of Combinatorial Game Theory: Two pla… |
| 307 | Problem 307 | 37s | k defects are randomly distributed amongst n integrated-circuit chips produce… |
| 308 | Problem 308 | 24s | A program written in the programming language Fractran consists of a list of … |
| 309 | Problem 309 | 2m 20s | In the classic "Crossing Ladders" problem, we are given the lengths x and y o… |
| 310 | Problem 310 | 1m 11s | Alice and Bob play the game Nim Square. |
| 311 | Problem 311 | 45s | ABCD is a convex, integer sided quadrilateral with 1 le AB lt BC lt CD lt AD. |
| 312 | Problem 312 | 37m 12s | - A Sierpiński graph of order-1 (S1) is an equilateral triangle. |
| 313 | Problem 313 | 2m 3s | In a sliding game a counter may slide horizontally or vertically into an empt… |
| 314 | Problem 314 | 1m 5s | The moon has been opened up, and land can be obtained for free, but there is … |
| 315 | Problem 315 | 2m 29s | !0315clocks.gif Sam and Max are asked to transform two digital clocks into tw… |
| 316 | Problem 316 | 3m 49s | Let p = p1 p2 p3 cdots be an infinite sequence of random digits, selected fro… |
| 317 | Problem 317 | 1m 35s | A firecracker explodes at a height of pu{100 m} above level ground. |
| 318 | Problem 318 | 55s | Consider the real number sqrt 2 + sqrt 3. |
| 319 | Problem 319 | 2m 30s | Let x1, x2, dots, xn be a sequence of length n such that: - x1 = 2 - for all … |
| 320 | Problem 320 | 2m 44s | Let N(i) be the smallest integer n such that n! is divisible by (i!)^{1234567… |
| 321 | Problem 321 | 45s | A horizontal row comprising of 2n + 1 squares has n red counters placed at on… |
| 322 | Problem 322 | 6m 18s | Let T(m, n) be the number of the binomial coefficients ^iCn that are divisibl… |
| 323 | Problem 323 | 33s | Let y0, y1, y2, dots be a sequence of random unsigned 32-bit integers (i.e. |
| 324 | Problem 324 | 35s | Let f(n) represent the number of ways one can fill a 3 times 3 times n tower … |
| 325 | Problem 325 | 3m 29s | A game is played with two piles of stones and two players. |
| 326 | Problem 326 | 2m 19s | Let an be a sequence recursively defined by:quad a1=1,quaddisplaystyle an=big… |
| 327 | Problem 327 | 1m 21s | A series of three rooms are connected to each other by automatic doors. |
| 328 | Problem 328 | 1m 33s | We are trying to find a hidden number selected from the set of integers 1, 2,… |
| 329 | Problem 329 | 2m 26s | Susan has a prime frog. |
| 330 | Problem 330 | 4m 42s | An infinite sequence of real numbers a(n) is defined for all integers n as fo… |
| 331 | Problem 331 | 2m 59s | N times N disks are placed on a square game board. |
| 332 | Problem 332 | 4m 15s | A spherical triangle is a figure formed on the surface of a sphere by three g… |
| 333 | Problem 333 | 2m 38s | All positive integers can be partitioned in such a way that each and every te… |
| 334 | Problem 334 | 1m 16s | In Plato's heaven, there exist an infinite number of bowls in a straight line. |
| 335 | Problem 335 | 1m 21s | Whenever Peter feels bored, he places some bowls, containing one bean each, i… |
| 336 | Problem 336 | 6m 40s | A train is used to transport four carriages in the order: ABCD. |
| 337 | Problem 337 | 2m 23s | Let a1, a2, dots, an be an integer sequence of length n such that: - a1 = 6 -… |
| 338 | Problem 338 | 1m 27s | A rectangular sheet of grid paper with integer dimensions w times h is given. |
| 339 | Problem 339 | 1m 24s | "And he came towards a valley, through which ran a river; and the borders of … |
| 340 | Problem 340 | 1m 54s | For fixed integers a, b, c, define the crazy function F(n) as follows: F(n) =… |
| 341 | Problem 341 | 5m 9s | The Golomb's self-describing sequence (G(n)) is the only nondecreasing sequen… |
| 342 | Problem 342 | 12m 12s | Consider the number 50. |
| 343 | Problem 343 | 2m 4s | For any positive integer k, a finite sequence ai of fractions xi/yi is define… |
| 344 | Problem 344 | 2m 30s | One variant of N.G. de Bruijn's silver dollar game can be described as follow… |
| 345 | Problem 345 | 46s | We define the Matrix Sum of a matrix as the maximum possible sum of matrix el… |
| 346 | Problem 346 | 1m 45s | The number 7 is special, because 7 is 111 written in base 2, and 11 written i… |
| 347 | Problem 347 | 1m 6s | The largest integer le 100 that is only divisible by both the primes 2 and 3 … |
| 348 | Problem 348 | 1m 17s | Many numbers can be expressed as the sum of a square and a cube. |
| 349 | Problem 349 | 47s | An ant moves on a regular grid of squares that are coloured either black or w… |
| 350 | Problem 350 | 1m 11s | A list of size n is a sequence of n natural numbers. |
| 351 | Problem 351 | 6m 10s | A hexagonal orchard of order n is a triangular lattice made up of points with… |
| 352 | Problem 352 | 3m 8s | Each one of the 25 sheep in a flock must be tested for a rare virus, known to… |
| 353 | Problem 353 | 1m 26s | A moon could be described by the sphere C(r) with centre (0,0,0) and radius r. |
| 354 | Problem 354 | 19m 29s | Consider a honey bee's honeycomb where each cell is a perfect regular hexagon… |
| 355 | Problem 355 | 1m 46s | Define operatorname{Co}(n) to be the maximal possible sum of a set of mutuall… |
| 356 | Problem 356 | 1m 7s | Let an be the largest real root of a polynomial g(x) = x^3 - 2^n cdot x^2 + n. |
| 357 | Problem 357 | 46s | Consider the divisors of 30: 1,2,3,5,6,10,15,30. |
| 358 | Problem 358 | 1m 11s | A cyclic number with n digits has a very interesting property: When it is mul… |
| 359 | Problem 359 | 2m 5s | An infinite number of people (numbered 1, 2, 3, etc.) are lined up to get a r… |
| 360 | Problem 360 | 46s | Given two points (x1, y1, z1) and (x2, y2, z2) in three dimensional space, th… |
| 361 | Problem 361 | 1m 58s | The Thue-Morse sequence Tn is a binary sequence satisfying: - T0 = 0 - T{2n} … |
| 362 | Problem 362 | 2m 11s | Consider the number 54. |
| 363 | Problem 363 | 1m 20s | A cubic Bézier curve is defined by four points: P0, P1, P2, and P3. |
| 364 | Problem 364 | 2m 52s | There are N seats in a row. |
| 365 | Problem 365 | 44s | The binomial coefficient displaystyle{binom{10^{18}}{10^9}} is a number with … |
| 366 | Problem 366 | 56s | Two players, Anton and Bernhard, are playing the following game. |
| 367 | Problem 367 | 3m 55s | Bozo sort, not to be confused with the slightly less efficient bogo sort, con… |
| 368 | Problem 368 | 4m 24s | The harmonic series 1 + frac 1 2 + frac 1 3 + frac 1 4 + cdots is well known … |
| 369 | Problem 369 | 4m 28s | In a standard 52 card deck of playing cards, a set of 4 cards is a Badugi if … |
| 370 | Problem 370 | 53s | Let us define a geometric triangle as an integer sided triangle with sides a … |
| 371 | Problem 371 | 1m 10s | Oregon licence plates consist of three letters followed by a three digit numb… |
| 372 | Problem 372 | 2m 6s | Let R(M, N) be the number of lattice points (x, y) which satisfy MltxleN, Mlt… |
| 373 | Problem 373 | 2m 33s | Every triangle has a circumscribed circle that goes through the three vertices. |
| 374 | Problem 374 | 9m | An integer partition of a number n is a way of writing n as a sum of positive… |
| 375 | Problem 375 | 5m 2s | Let Sn be an integer sequence produced with the following pseudo-random numbe… |
| 376 | Problem 376 | 3m 48s | Consider the following set of dice with nonstandard pips: Die A: 1 4 4 4 4 4 … |
| 377 | Problem 377 | 1m 39s | There are 16 positive integers that do not have a zero in their digits and th… |
| 378 | Problem 378 | 45s | Let T(n) be the nth triangle number, so T(n) = dfrac{n(n + 1)}{2}. |
| 379 | Problem 379 | 1m 12s | Let f(n) be the number of couples (x, y) with x and y positive integers, x le… |
| 380 | Problem 380 | 1m 2s | An m times n maze is an m times n rectangular grid with walls placed between … |
| 381 | Problem 381 | 1m 3s | For a prime p let S(p) = (sum (p-k)!) bmod (p) for 1 le k le 5. |
| 382 | Problem 382 | 4m 21s | A polygon is a flat shape consisting of straight line segments that are joine… |
| 383 | Problem 383 | 2m 4s | Let f5(n) be the largest integer x for which 5^x divides n. |
| 384 | Problem 384 | 1m 30s | Define the sequence a(n) as the number of adjacent pairs of ones in the binar… |
| 385 | Problem 385 | 3m 46s | For any triangle T in the plane, it can be shown that there is a unique ellip… |
| 386 | Problem 386 | 3m 30s | Let n be an integer and S(n) be the set of factors of n. |
| 387 | Problem 387 | 50s | A Harshad or Niven number is a number that is divisible by the sum of its dig… |
| 388 | Problem 388 | 2m 5s | Consider all lattice points (a,b,c) with 0 le a,b,c le N. |
| 389 | Problem 389 | 1m 15s | An unbiased single 4-sided die is thrown and its value, T, is noted. |
| 390 | Problem 390 | 2m 27s | Consider the triangle with sides sqrt 5, sqrt {65} and sqrt {68}. |
| 391 | Problem 391 | 2m 6s | Let sk be the number of 1’s when writing the numbers from 0 to k in binary. |
| 392 | Problem 392 | 5m 24s | A rectilinear grid is an orthogonal grid where the spacing between the gridli… |
| 393 | Problem 393 | 44s | An n times n grid of squares contains n^2 ants, one ant per square. |
| 394 | Problem 394 | 1m 8s | Jeff eats a pie in an unusual way. |
| 395 | Problem 395 | 5m 18s | The Pythagorean tree is a fractal generated by the following procedure: Start… |
| 396 | Problem 396 | 3m 23s | For any positive integer n, the nth weak Goodstein sequence g1, g2, g3, dots … |
| 397 | Problem 397 | 46s | On the parabola y = x^2/k, three points A(a, a^2/k), B(b, b^2/k) and C(c, c^2… |
| 398 | Problem 398 | 2m 56s | Inside a rope of length n, n - 1 points are placed with distance 1 from each … |
| 399 | Problem 399 | 1m 7s | The first 15 Fibonacci numbers are: 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610. |
| 400 | Problem 400 | 8m 33s | A Fibonacci tree is a binary tree recursively defined as: - T(0) is the empty… |
| 401 | Problem 401 | 3m 52s | The divisors of 6 are 1,2,3 and 6. |
| 402 | Problem 402 | 16m 16s | It can be shown that the polynomial n^4 + 4n^3 + 2n^2 + 5n is a multiple of 6… |
| 403 | Problem 403 | 12m 23s | For integers a and b, we define D(a, b) as the domain enclosed by the parabol… |
| 404 | Problem 404 | 16m 16s | Ea is an ellipse with an equation of the form x^2 + 4y^2 = 4a^2. |
| 405 | Problem 405 | 10m 27s | We wish to tile a rectangle whose length is twice its width. |
| 406 | Problem 406 | 7m 51s | We are trying to find a hidden number selected from the set of integers 1, 2,… |
| 407 | Problem 407 | 52s | If we calculate a^2 bmod 6 for 0 leq a leq 5 we get: 0,1,4,3,4,1. |
| 408 | Problem 408 | 2m 3s | Let's call a lattice point (x, y) inadmissible if x, y and x+y are all positi… |
| 409 | Problem 409 | 4m 47s | Let n be a positive integer. |
| 410 | Problem 410 | 3m 4s | Let C be the circle with radius r, x^2 + y^2 = r^2. |
| 411 | Problem 411 | 4m 46s | Let n be a positive integer. |
| 412 | Problem 412 | 3m 22s | For integers m, n (0 leq n lt m), let L(m, n) be an m times m grid with the t… |
| 413 | Problem 413 | 5m 54s | We say that a d-digit positive number (no leading zeros) is a one-child numbe… |
| 414 | Problem 414 | 8m 40s | 6174 is a remarkable number; if we sort its digits in increasing order and su… |
| 415 | Problem 415 | 24m 14s | A set of lattice points S is called a titanic set if there exists a line pass… |
| 416 | Problem 416 | 4m 38s | A row of n squares contains a frog in the leftmost square. |
| 417 | Problem 417 | 47s | A unit fraction contains 1 in the numerator. |
| 418 | Problem 418 | 15m 7s | Let n be a positive integer. |
| 419 | Problem 419 | 46s | The look and say sequence goes 1, 11, 21, 1211, 111221, 312211, 13112221, 111… |
| 420 | Problem 420 | 3m 42s | A positive integer matrix is a matrix whose elements are all positive integers. |
| 421 | Problem 421 | 5m 6s | Numbers of the form n^{15}+1 are composite for every integer n gt 1. |
| 422 | Problem 422 | 3m 30s | Let H be the hyperbola defined by the equation 12x^2 + 7xy - 12y^2 = 625. |
| 423 | Problem 423 | 4m 22s | Let n be a positive integer. |
| 424 | Problem 424 | 2m | The above is an example of a cryptic kakuro (also known as cross sums, or eve… |
| 425 | Problem 425 | 58s | Two positive numbers A and B are said to be connected (denoted by "A leftrigh… |
| 426 | Problem 426 | 1m 4s | Consider an infinite row of boxes. |
| 427 | Problem 427 | 1m 38s | A sequence of integers S = si is called an n-sequence if it has n elements an… |
| 428 | Problem 428 | 53s | Let a, b and c be positive numbers. |
| 429 | Problem 429 | 1m 4s | A unitary divisor d of a number n is a divisor of n that has the property gcd… |
| 430 | Problem 430 | 1m 30s | N disks are placed in a row, indexed 1 to N from left to right. |
| 431 | Problem 431 | 2m 39s | Fred the farmer arranges to have a new storage silo installed on his farm and… |
| 432 | Problem 432 | 2m 32s | Let S(n,m) = sumphi(n times i) for 1 leq i leq m. |
| 433 | Problem 433 | 46s | Let E(x0, y0) be the number of steps it takes to determine the greatest commo… |
| 434 | Problem 434 | 53s | Recall that a graph is a collection of vertices and edges connecting the vert… |
| 435 | Problem 435 | 1m 22s | The Fibonacci numbers fn, n ge 0 are defined recursively as fn = f{n-1} + f{n… |
| 436 | Problem 436 | 1m 19s | Julie proposes the following wager to her sister Louise. |
| 437 | Problem 437 | 6m 48s | When we calculate 8^n modulo 11 for n=0 to 9 we get: 1, 8, 9, 6, 4, 10, 3, 2,… |
| 438 | Problem 438 | 11m 43s | For an n-tuple of integers t = (a1, dots, an), let (x1, dots, xn) be the solu… |
| 439 | Problem 439 | 4m 23s | Let d(k) be the sum of all divisors of k. |
| 440 | Problem 440 | 1m 51s | We want to tile a board of length n and height 1 completely, with either 1 ti… |
| 441 | Problem 441 | 1m 39s | For an integer M, we define R(M) as the sum of 1/(p cdot q) for all the integ… |
| 442 | Problem 442 | 48s | An integer is called eleven-free if its decimal expansion does not contain an… |
| 443 | Problem 443 | 2m 5s | Let g(n) be a sequence defined as follows: g(4) = 13, g(n) = g(n-1) + gcd(n, … |
| 444 | Problem 444 | 39s | A group of p people decide to sit down at a round table and play a lottery-ti… |
| 445 | Problem 445 | 5m 30s | For every integer n1, the family of functions f{n,a,b} is defined by f{n,a,b}… |
| 446 | Problem 446 | 1m 18s | For every integer n1, the family of functions f{n,a,b} is defined by f{n,a,b}… |
| 447 | Problem 447 | 3m | For every integer n 1, the family of functions f{n,a,b} is defined by for int… |
| 448 | Problem 448 | 1m 1s | The function operatorname{mathbf{lcm}}(a,b) denotes the least common multiple… |
| 449 | Problem 449 | 5m 21s | Phil the confectioner is making a new batch of chocolate covered candy. |
| 450 | Problem 450 | 44s | A hypocycloid is the curve drawn by a point on a small circle rolling inside … |
| 451 | Problem 451 | 1m 32s | Consider the number 15. |
| 452 | Problem 452 | 16m 13s | Define F(m,n) as the number of n-tuples of positive integers for which the pr… |
| 453 | Problem 453 | 16m 13s | A simple quadrilateral is a polygon that has four distinct vertices, has no s… |
| 454 | Problem 454 | 16m 13s | In the following equation x, y, and n are positive integers. |
| 455 | Problem 455 | 11m 24s | Let f(n) be the largest positive integer x less than 10^9 such that the last … |
| 456 | Problem 456 | 11m 24s | Define: xn = (1248^n bmod 32323) - 16161 yn = (8421^n bmod 30103) - 15051 Pn … |
| 457 | Problem 457 | 10m 51s | Let f(n) = n^2 - 3n - 1. |
| 458 | Problem 458 | 1m 3s | Consider the alphabet A made out of the letters of the word "text{project}": … |
| 459 | Problem 459 | 6m 58s | The flipping game is a two player game played on an N by N square board. |
| 460 | Problem 460 | 4m 48s | On the Euclidean plane, an ant travels from point A(0, 1) to point B(d, 1) fo… |
| 461 | Problem 461 | 19m 24s | Let fn(k) = e^{k/n} - 1, for all non-negative integers k. |
| 462 | Problem 462 | 1m 54s | A 3-smooth number is an integer which has no prime factor larger than 3. |
| 463 | Problem 463 | 3m 2s | The function f is defined for all positive integers as follows: - f(1)=1 - f(… |
| 464 | Problem 464 | 4m 5s | The Möbius function, denoted mu(n), is defined as: - mu(n) = (-1)^{omega(n)} … |
| 465 | Problem 465 | 35s | The kernel of a polygon is defined by the set of points from which the entire… |
| 466 | Problem 466 | 10m 56s | Let P(m,n) be the number of distinct terms in an mtimes n multiplication table. |
| 467 | Problem 467 | 1m 41s | An integer s is called a superinteger of another integer n if the digits of n… |
| 468 | Problem 468 | 38s | An integer is called B-smooth if none of its prime factors is greater than B. |
| 469 | Problem 469 | 1m 8s | In a room N chairs are placed around a round table. |
| 470 | Problem 470 | 3m 6s | Consider a single game of Ramvok: Let t represent the maximum number of turns… |
| 471 | Problem 471 | 2m 3s | The triangle triangle ABC is inscribed in an ellipse with equation frac {x^2}… |
| 472 | Problem 472 | 1m 47s | There are N seats in a row. |
| 473 | Problem 473 | 3m 20s | Let varphi be the golden ratio: varphi=frac{1+sqrt{5}}{2}. |
| 474 | Problem 474 | 36m 30s | For a positive integer n and digits d, we define F(n, d) as the number of the… |
| 475 | Problem 475 | 2m 47s | 12n musicians participate at a music festival. |
| 476 | Problem 476 | 1m 37s | Let R(a, b, c) be the maximum area covered by three non-overlapping circles i… |
| 477 | Problem 477 | 4m 45s | The number sequence game starts with a sequence S of N numbers written on a l… |
| 478 | Problem 478 | 1m 35s | Let us consider mixtures of three substances: A, B and C. |
| 479 | Problem 479 | 1m | Let ak, bk, and ck represent the three solutions (real or complex numbers) to… |
| 480 | Problem 480 | 15m 4s | Consider all the words which can be formed by selecting letters, in any order… |
| 481 | Problem 481 | 6m 4s | A group of chefs (numbered 1, 2, etc) participate in a turn-based strategic c… |
| 482 | Problem 482 | 10m 38s | ABC is an integer sided triangle with incenter I and perimeter p. |
| 483 | Problem 483 | 2m 23s | We define a permutation as an operation that rearranges the order of the elem… |
| 484 | Problem 484 | 1m 8s | The arithmetic derivative is defined by - p^prime = 1 for any prime p - (ab)^… |
| 485 | Problem 485 | 1m 19s | Let d(n) be the number of divisors of n. |
| 486 | Problem 486 | 3m 23s | Let F5(n) be the number of strings s such that: - s consists only of '0's and… |
| 487 | Problem 487 | 1m 42s | Let fk(n) be the sum of the kth powers of the first n positive integers. |
| 488 | Problem 488 | 2m 8s | Alice and Bob have enjoyed playing Nim every day. |
| 489 | Problem 489 | 4m 48s | Let G(a, b) be the smallest non-negative integer n for which operatorname{mat… |
| 490 | Problem 490 | 1m 17s | There are n stones in a pond, numbered 1 to n. |
| 491 | Problem 491 | 48s | We call a positive integer double pandigital if it uses all the digits 0 to 9… |
| 492 | Problem 492 | 2m 51s | Define the sequence a1, a2, a3, dots as: - a1 = 1 - a{n+1} = 6an^2 + 10an + 3… |
| 493 | Problem 493 | 46s | 70 coloured balls are placed in an urn, 10 for each of the seven rainbow colo… |
| 494 | Problem 494 | 1m 16s | The Collatz sequence is defined as: a{i+1} = left large{frac {ai} 2 atop 3 ai… |
| 495 | Problem 495 | 3m 27s | Let W(n,k) be the number of ways in which n can be written as the product of … |
| 496 | Problem 496 | 2m 46s | Given an integer sided triangle ABC: Let I be the incenter of ABC. |
| 497 | Problem 497 | 1m 54s | Bob is very familiar with the famous mathematical puzzle/game, "Tower of Hano… |
| 498 | Problem 498 | 58s | For positive integers n and m, we define two polynomials Fn(x) = x^n and Gm(x… |
| 499 | Problem 499 | 20m 18s | A gambler decides to participate in a special lottery. |
| 500 | Problem 500 | 46s | The number of divisors of 120 is 16. |
| 501 | Problem 501 | 48s | The eight divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. |
| 502 | Problem 502 | 44s | We define a block to be a rectangle with a height of 1 and an integer-valued … |
| 503 | Problem 503 | 2m 20s | Alice is playing a game with n cards numbered 1 to n. |
| 504 | Problem 504 | 2m 20s | Let ABCD be a quadrilateral whose vertices are lattice points lying on the co… |
| 505 | Problem 505 | 5m 9s | Let: begin{array}{ll} x(0)&=0 x(1)&=1 x(2k)&=(3x(k)+2x(lfloor frac k 2 rfloor… |
| 506 | Problem 506 | 3m 50s | Consider the infinite repeating sequence of digits: 1234321234321234321... |
| 507 | Problem 507 | 57s | Let tn be the tribonacci numbers defined as: t0 = t1 = 0; t2 = 1; tn = t{n-1}… |
| 508 | Problem 508 | 3m 50s | Consider the Gaussian integer i-1. |
| 509 | Problem 509 | 1m 36s | Anton and Bertrand love to play three pile Nim. |
| 510 | Problem 510 | 1m | Circles A and B are tangent to each other and to line L at three distinct poi… |
| 511 | Problem 511 | 2m 40s | Let Seq(n,k) be the number of positive-integer sequences ai{1 le i le n} of l… |
| 512 | Problem 512 | 1m 30s | Let varphi(n) be Euler's totient function. |
| 513 | Problem 513 | 7m 57s | ABC is an integral sided triangle with sides a le b le c. |
| 514 | Problem 514 | 3m 30s | A geoboard (of order N) is a square board with equally-spaced pins protruding… |
| 515 | Problem 515 | 1m 18s | Let d(p, n, 0) be the multiplicative inverse of n modulo prime p, defined as … |
| 516 | Problem 516 | 1m 31s | 5-smooth numbers are numbers whose largest prime factor doesn't exceed 5. |
| 517 | Problem 517 | 2m 4s | For every real number a gt 1 is given the sequence ga by: g{a}(x)=1 for x lt … |
| 518 | Problem 518 | 1m 30s | Let S(n) = sum a + b + c over all triples (a, b, c) such that: - a, b and c a… |
| 519 | Problem 519 | 6m 4s | An arrangement of coins in one or more rows with the bottom row being a block… |
| 520 | Problem 520 | 2m | We define a simber to be a positive integer in which any odd digit, if presen… |
| 521 | Problem 521 | 31m 6s | Let operatorname{smpf}(n) be the smallest prime factor of n. |
| 522 | Problem 522 | 4m 12s | Despite the popularity of Hilbert's infinite hotel, Hilbert decided to try ma… |
| 523 | Problem 523 | 1m 43s | Consider the following algorithm for sorting a list: - 1. |
| 524 | Problem 524 | 3m 54s | Consider the following algorithm for sorting a list: - 1. |
| 525 | Problem 525 | 2m 19s | An ellipse E(a, b) is given at its initial position by equation: frac {x^2} {… |
| 526 | Problem 526 | 4m 54s | Let f(n) be the largest prime factor of n. |
| 527 | Problem 527 | 3m 31s | A secret integer t is selected at random within the range 1 le t le n. |
| 528 | Problem 528 | 2m 2s | Let S(n, k, b) represent the number of valid solutions to x1 + x2 + cdots + x… |
| 529 | Problem 529 | 18m 21s | A 10-substring of a number is a substring of its digits that sum to 10. |
| 530 | Problem 530 | 2m 47s | Every divisor d of a number n has a complementary divisor n/d. |
| 531 | Problem 531 | 3m 35s | Let g(a, n, b, m) be the smallest non-negative solution x to the system: x = … |
| 532 | Problem 532 | 1m 19s | Bob is a manufacturer of nanobots and wants to impress his customers by givin… |
| 533 | Problem 533 | 6m 9s | The Carmichael function lambda(n) is defined as the smallest positive integer… |
| 534 | Problem 534 | 31m 3s | The classical eight queens puzzle is the well known problem of placing eight … |
| 535 | Problem 535 | 53s | Consider the infinite integer sequence S starting with: S = 1, 1, 2, 1, 3, 2,… |
| 536 | Problem 536 | 4m 1s | Let S(n) be the sum of all positive integers m not exceeding n having the fol… |
| 537 | Problem 537 | 1m 22s | Let pi(x) be the prime counting function, i.e. |
| 538 | Problem 538 | 5m 50s | Consider a positive integer sequence S = (s1, s2, dots, sn). |
| 539 | Problem 539 | 1m 27s | Start from an ordered list of all integers from 1 to n. |
| 540 | Problem 540 | 48s | A Pythagorean triple consists of three positive integers a, b and c satisfyin… |
| 541 | Problem 541 | 1m 17s | The nth harmonic number Hn is defined as the sum of the multiplicative invers… |
| 542 | Problem 542 | 3m 43s | Let S(k) be the sum of three or more distinct positive integers having the fo… |
| 543 | Problem 543 | 1m 49s | Define function P(n, k) = 1 if n can be written as the sum of k prime numbers… |
| 544 | Problem 544 | 3m 38s | Let F(r, c, n) be the number of ways to colour a rectangular grid with r rows… |
| 545 | Problem 545 | 7m 52s | The sum of the kth powers of the first n positive integers can be expressed a… |
| 546 | Problem 546 | 6m 44s | Define fk(n) = sum{i=0}^n fk(lfloorfrac i k rfloor) where fk(0) = 1 and lfloo… |
| 547 | Problem 547 | 4m 48s | Assuming that two points are chosen randomly (with uniform distribution) with… |
| 548 | Problem 548 | 3m 57s | A gozinta chain for n is a sequence 1,a,b,dots,n where each element properly … |
| 549 | Problem 549 | 1m 22s | The smallest number m such that 10 divides m! is m=5. |
| 550 | Problem 550 | 1m 55s | Two players are playing a game, alternating turns. |
| 551 | Problem 551 | 1m 40s | Let a0, a1, dots be an integer sequence defined by: - a0 = 1; - for n ge 1, a… |
| 552 | Problem 552 | 2m 40s | Let An be the smallest positive integer satisfying An bmod pi = i for all 1 l… |
| 553 | Problem 553 | 3m 48s | Let P(n) be the set of the first n positive integers 1, 2, dots, n. |
| 554 | Problem 554 | 10m 56s | On a chess board, a centaur moves like a king or a knight. |
| 555 | Problem 555 | 6m 8s | The McCarthy 91 function is defined as follows: We can generalize this defini… |
| 556 | Problem 556 | 3m 43s | A Gaussian integer is a number z = a + bi where a, b are integers and i^2 = -1. |
| 557 | Problem 557 | 3m 18s | A triangle is cut into four pieces by two straight lines, each starting at on… |
| 558 | Problem 558 | 4m 8s | Let r be the real root of the equation x^3 = x^2 + 1. |
| 559 | Problem 559 | 30m 25s | An ascent of a column j in a matrix occurs if the value of column j is smalle… |
| 560 | Problem 560 | 2m 19s | Coprime Nim is just like ordinary normal play Nim, but the players may only r… |
| 561 | Problem 561 | 1m 16s | Let S(n) be the number of pairs (a,b) of distinct divisors of n such that a d… |
| 562 | Problem 562 | 1m 41s | Construct triangle ABC such that: - Vertices A, B and C are lattice points in… |
| 563 | Problem 563 | 3m 47s | A company specialises in producing large rectangular metal sheets, starting f… |
| 564 | Problem 564 | 1m 34s | A line segment of length 2n-3 is randomly split into n segments of integer le… |
| 565 | Problem 565 | 31m 3s | Let sigma(n) be the sum of the divisors of n. |
| 566 | Problem 566 | 49s | Adam plays the following game with his birthday cake. |
| 567 | Problem 567 | 7m 59s | Tom has built a random generator that is connected to a row of n light bulbs. |
| 568 | Problem 568 | 2m 30s | Tom has built a random generator that is connected to a row of n light bulbs. |
| 569 | Problem 569 | 1m 46s | A mountain range consists of a line of mountains with slopes of exactly 45^ci… |
| 570 | Problem 570 | 2m 2s | A snowflake of order n is formed by overlaying an equilateral triangle (rotat… |
| 571 | Problem 571 | 7m 31s | A positive number is pandigital in base b if it contains all digits from 0 to… |
| 572 | Problem 572 | 7m 38s | A matrix M is called idempotent if M^2 = M. |
| 573 | Problem 573 | 5m 5s | n runners in very different training states want to compete in a race. |
| 574 | Problem 574 | 13m 23s | Let q be a prime and A ge B 0 be two integers with the following properties: … |
| 575 | Problem 575 | 1m 1s | It was quite an ordinary day when a mysterious alien vessel appeared as if fr… |
| 576 | Problem 576 | 30m 18s | A bouncing point moves counterclockwise along a circle with circumference 1 w… |
| 577 | Problem 577 | 3m 26s | An equilateral triangle with integer side length n ge 3 is divided into n^2 e… |
| 578 | Problem 578 | 3m 23s | Any positive integer can be written as a product of prime powers: p1^{a1} tim… |
| 579 | Problem 579 | 3m 22s | A lattice cube is a cube in which all vertices have integer coordinates. |
| 580 | Problem 580 | 1m 20s | A Hilbert number is any positive integer of the form 4k+1 for integer kgeq 0. |
| 581 | Problem 581 | 10m 47s | A number is p-smooth if it has no prime factors larger than p. |
| 582 | Problem 582 | 5m 27s | Let a, b and c be the sides of an integer sided triangle with one angle of 12… |
| 583 | Problem 583 | 9m 14s | A standard envelope shape is a convex figure consisting of an isosceles trian… |
| 584 | Problem 584 | 4m 7s | A long long time ago in a galaxy far far away, the Wimwians, inhabitants of p… |
| 585 | Problem 585 | 35s | Consider the term small sqrt{x+sqrt{y}+sqrt{z}} that is representing a nested… |
| 586 | Problem 586 | 50s | The number 209 can be expressed as a^2 + 3ab + b^2 in two distinct ways: qqua… |
| 587 | Problem 587 | 44s | A square is drawn around a circle as shown in the diagram below on the left. |
| 588 | Problem 588 | 9m 20s | The coefficients in the expansion of (x+1)^k are called binomial coefficients. |
| 589 | Problem 589 | 11m 52s | Christopher Robin and Pooh Bear love the game of Poohsticks so much that they… |
| 590 | Problem 590 | 6m 20s | Let H(n) denote the number of sets of positive integers such that the least c… |
| 591 | Problem 591 | 53s | Given a non-square integer d, any real x can be approximated arbitrarily clos… |
| 592 | Problem 592 | 5m 36s | For any N, let f(N) be the last twelve hexadecimal digits before the trailing… |
| 593 | Problem 593 | 1m 40s | We define two sequences S = S(1), S(2), ..., S(n) and S2 = S2(1), S2(2), ...,… |
| 594 | Problem 594 | 1m 40s | For a polygon P, let t(P) be the number of ways in which P can be tiled using… |
| 595 | Problem 595 | 2m 13s | A deck of cards numbered from 1 to n is shuffled randomly such that each perm… |
| 596 | Problem 596 | 2m 50s | Let T(r) be the number of integer quadruplets x, y, z, t such that x^2 + y^2 … |
| 597 | Problem 597 | 2m 15s | The Torpids are rowing races held annually in Oxford, following some curious … |
| 598 | Problem 598 | 3m 35s | Consider the number 48. |
| 599 | Problem 599 | 6m 25s | The well-known Rubik's Cube puzzle has many fascinating mathematical properties. |
| 600 | Problem 600 | 5m 55s | Let H(n) be the number of distinct integer sided equiangular convex hexagons … |
| 601 | Problem 601 | 1m 33s | For every positive number n we define the function mathop{streak}(n)=k as the… |
| 602 | Problem 602 | 1m 12s | Alice enlists the help of some friends to generate a random number, using a s… |
| 603 | Problem 603 | 1m 41s | Let S(n) be the sum of all contiguous integer-substrings that can be formed f… |
| 604 | Problem 604 | 4m 35s | Let F(N) be the maximum number of lattice points in an axis-aligned Ntimes N … |
| 605 | Problem 605 | 2m 5s | Consider an n-player game played in consecutive pairs: Round 1 takes place be… |
| 606 | Problem 606 | 9m 30s | A gozinta chain for n is a sequence 1,a,b,dots,n where each element properly … |
| 607 | Problem 607 | 1m 53s | Frodo and Sam need to travel 100 leagues due East from point A to point B. |
| 608 | Problem 608 | 4m 56s | Let D(m,n)=displaystylesum{dmid m}sum{k=1}^nsigma0(kd) where d runs through a… |
| 609 | Problem 609 | 7m 36s | For every n ge 1 the prime-counting function pi(n) is equal to the number of … |
| 610 | Problem 610 | 10m 53s | A random generator produces a sequence of symbols drawn from the set {I, V, X… |
| 611 | Problem 611 | 6m 34s | Peter moves in a hallway with N + 1 doors consecutively numbered from 0 throu… |
| 612 | Problem 612 | 53s | Let's call two numbers friend numbers if their representation in base 10 has … |
| 613 | Problem 613 | 1m 15s | Dave is doing his homework on the balcony and, preparing a presentation about… |
| 614 | Problem 614 | 1m 8s | An integer partition of a number n is a way of writing n as a sum of positive… |
| 615 | Problem 615 | 1m 14s | Consider the natural numbers having at least 5 prime factors, which don't hav… |
| 616 | Problem 616 | 3m 11s | Alice plays the following game, she starts with a list of integers L and on e… |
| 617 | Problem 617 | 56s | For two integers n,e gt 1, we define an (n,e)-MPS (Mirror Power Sequence) to … |
| 618 | Problem 618 | 2m 52s | Consider the numbers 15, 16 and 18: 15=3times 5 and 3+5=8. |
| 619 | Problem 619 | 1m 41s | For a set of positive integers a, a+1, a+2, dots , b, let C(a,b) be the numbe… |
| 620 | Problem 620 | 2m 5s | A circle C of circumference c centimetres has a smaller circle S of circumfer… |
| 621 | Problem 621 | 4m 27s | Gauss famously proved that every positive integer can be expressed as the sum… |
| 622 | Problem 622 | 1m 28s | A riffle shuffle is executed as follows: a deck of cards is split into two eq… |
| 623 | Problem 623 | 7m 42s | The lambda-calculus is a universal model of computation at the core of functi… |
| 624 | Problem 624 | 2m 16s | An unbiased coin is tossed repeatedly until two consecutive heads are obtained. |
| 625 | Problem 625 | 3m 27s | G(N)=sum{j=1}^Nsum{i=1}^j gcd(i,j). |
| 626 | Problem 626 | 4m 35s | A binary matrix is a matrix consisting entirely of 0s and 1s. |
| 627 | Problem 627 | 9m 54s | Consider the set S of all possible products of n positive integers not exceed… |
| 628 | Problem 628 | 2m 19s | A position in chess is an (orientated) arrangement of chess pieces placed on … |
| 629 | Problem 629 | 3m 7s | Alice and Bob are playing a modified game of Nim called Scatterstone Nim, wit… |
| 630 | Problem 630 | 1m | Given a set, L, of unique lines, let M(L) be the number of lines in the set a… |
| 631 | Problem 631 | 8m 5s | Let (p1 p2 ldots pk) denote the permutation of the set {1, ..., k} that maps … |
| 632 | Problem 632 | 1m 15s | For an integer n, we define the square prime factors of n to be the primes wh… |
| 633 | Problem 633 | 4m 2s | For an integer n, we define the square prime factors of n to be the primes wh… |
| 634 | Problem 634 | 9m 34s | Define F(n) to be the number of integers x≤n that can be written in the form … |
| 635 | Problem 635 | 3m 26s | Let Aq(n) be the number of subsets, B, of the set 1, 2, ..., q cdot n that sa… |
| 636 | Problem 636 | 1m 40s | Consider writing a natural number as product of powers of natural numbers wit… |
| 637 | Problem 637 | 18m 10s | Given any positive integer n, we can construct a new integer by inserting plu… |
| 638 | Problem 638 | 1m 29s | Let P{a,b} denote a path in a atimes b lattice grid with following properties… |
| 639 | Problem 639 | 31m 5s | A multiplicative function f(x) is a function over positive integers satisfyin… |
| 640 | Problem 640 | 2m 14s | Bob plays a single-player game of chance using two standard 6-sided dice and … |
| 641 | Problem 641 | 5m 29s | Consider a row of n dice all showing 1. |
| 642 | Problem 642 | 49s | Let f(n) be the largest prime factor of n and displaystyle F(n) = sum{i=2}^n … |
| 643 | Problem 643 | 2m 35s | Two positive integers a and b are 2-friendly when gcd(a,b) = 2^t, t gt 0. |
| 644 | Problem 644 | 34s | Sam and Tom are trying a game of (partially) covering a given line segment of… |
| 645 | Problem 645 | 45s | On planet J, a year lasts for D days. |
| 646 | Problem 646 | 1m 48s | Let n be a natural number and p1^{alpha1}cdot p2^{alpha2}cdots pk^{alphak} it… |
| 647 | Problem 647 | 1m 2s | It is possible to find positive integers A and B such that given any triangul… |
| 648 | Problem 648 | 45s | For some fixed rho in [0, 1], we begin a sum s at 0 and repeatedly apply a pr… |
| 649 | Problem 649 | 1m 3s | Alice and Bob are taking turns playing a game consisting of c different coins… |
| 650 | Problem 650 | 3m 33s | Let B(n) = displaystyle prod{k=0}^n {n choose k}, a product of binomial coeff… |
| 651 | Problem 651 | 2m 46s | An infinitely long cylinder has its curved surface fully covered with differe… |
| 652 | Problem 652 | 30s | Consider the values of log2(8), log4(64) and log3(27). |
| 653 | Problem 653 | 2m 4s | Consider a horizontal frictionless tube with length L millimetres, and a diam… |
| 654 | Problem 654 | 6m 34s | Let T(n, m) be the number of m-tuples of positive integers such that the sum … |
| 655 | Problem 655 | 10m 18s | The numbers 545, 5995 and 15151 are the three smallest palindromes divisible … |
| 656 | Problem 656 | 2m | Given an irrational number alpha, let Salpha(n) be the sequence Salpha(n)=lfl… |
| 657 | Problem 657 | 4m 4s | In the context of formal languages, any finite sequence of letters of a given… |
| 658 | Problem 658 | 2m 17s | In the context of formal languages, any finite sequence of letters of a given… |
| 659 | Problem 659 | 1m 18s | Consider the sequence n^2+3 with n ge 1. |
| 660 | Problem 660 | 1m 50s | We call an integer sided triangle n-pandigital if it contains one angle of 12… |
| 661 | Problem 661 | 5m 13s | Two friends A and B are great fans of Chess. |
| 662 | Problem 662 | 5m 11s | Alice walks on a lattice grid. |
| 663 | Problem 663 | 1m 39s | Let tk be the tribonacci numbers defined as: quad t0 = t1 = 0; quad t2 = 1; q… |
| 664 | Problem 664 | 3m 16s | Peter is playing a solitaire game on an infinite checkerboard, each square of… |
| 665 | Problem 665 | 3m 3s | Two players play a game with two piles of stones, alternating turns. |
| 666 | Problem 666 | 1m 27s | Members of a species of bacteria occur in two different types: alpha and beta. |
| 667 | Problem 667 | 1m 27s | After buying a Gerver Sofa from the Moving Sofa Company, Jack wants to buy a … |
| 668 | Problem 668 | 2m | A positive integer is called square root smooth if all of its prime factors a… |
| 669 | Problem 669 | 53s | The Knights of the Order of Fibonacci are preparing a grand feast for their k… |
| 670 | Problem 670 | 3m 17s | A certain type of tile comes in three different sizes - 1 times 1, 1 times 2,… |
| 671 | Problem 671 | 1m 1s | A certain type of flexible tile comes in three different sizes - 1 times 1, 1… |
| 672 | Problem 672 | 3m 7s | Consider the following process that can be applied recursively to any positiv… |
| 673 | Problem 673 | 1m 14s | At Euler University, each of the n students (numbered from 1 to n) occupies a… |
| 674 | Problem 674 | 2m 30s | We define the mathcal{I} operator as the function and mathcal{I}-expressions … |
| 675 | Problem 675 | 4m 58s | Let omega(n) denote the number of distinct prime divisors of a positive integ… |
| 676 | Problem 676 | 46s | Let d(i,b) be the digit sum of the number i in base b. |
| 677 | Problem 677 | 1m 13s | Let g(n) be the number of undirected graphs with n nodes satisfying the follo… |
| 678 | Problem 678 | 44s | If a triple of positive integers (a, b, c) satisfies a^2+b^2=c^2, it is calle… |
| 679 | Problem 679 | 49s | Let S be the set consisting of the four letters texttt{A'},texttt{E'},texttt{… |
| 680 | Problem 680 | 5m 56s | Let N and K be two positive integers. |
| 681 | Problem 681 | 6m 54s | Given positive integers a le b le c le d, it may be possible to form quadrila… |
| 682 | Problem 682 | 4m 1s | 5-smooth numbers are numbers whose largest prime factor doesn't exceed 5. |
| 683 | Problem 683 | 21m 3s | Consider the following variant of "The Chase" game. |
| 684 | Problem 684 | 1m 1s | Define s(n) to be the smallest number that has a digit sum of n. |
| 685 | Problem 685 | 13m 42s | Writing down the numbers which have a digit sum of 10 in ascending order, we … |
| 686 | Problem 686 | 1m 10s | 2^7=128 is the first power of two whose leading digits are "12". |
| 687 | Problem 687 | 2m 29s | A standard deck of 52 playing cards, which consists of thirteen ranks (Ace, T… |
| 688 | Problem 688 | 3m 35s | We stack n plates into k non-empty piles where each pile is a different size. |
| 689 | Problem 689 | 4m 13s | For 0 le x lt 1, define di(x) to be the ith digit after the binary point of t… |
| 690 | Problem 690 | 2m 52s | Tom (the cat) and Jerry (the mouse) are playing on a simple graph G. |
| 691 | Problem 691 | 2m 2s | Given a character string s, we define L(k,s) to be the length of the longest … |
| 692 | Problem 692 | 1m 17s | Siegbert and Jo take turns playing a game with a heap of N pebbles: 1. |
| 693 | Problem 693 | 3m 7s | Two positive integers x and y (x y) can generate a sequence in the following … |
| 694 | Problem 694 | 1m 57s | A positive integer n is considered cube-full, if for every prime p that divid… |
| 695 | Problem 695 | 4m | Three points, P1, P2 and P3, are randomly selected within a unit square. |
| 696 | Problem 696 | 9m 4s | The game of Mahjong is played with tiles belonging to s suits. |
| 697 | Problem 697 | 1m 15s | Given a fixed real number c, define a random sequence (Xn){nge 0} by the foll… |
| 698 | Problem 698 | 3m 6s | We define 123-numbers as follows: - 1 is the smallest 123-number. |
| 699 | Problem 699 | 1m 40s | Let sigma(n) be the sum of all the divisors of the positive integer n, for ex… |
| 700 | Problem 700 | 44s | Leonhard Euler was born on 15 April 1707. |
| 701 | Problem 701 | 2m 25s | Consider a rectangle made up of W times H square cells each with area 1. |
| 702 | Problem 702 | 1m 3s | A regular hexagon table of side length N is divided into equilateral triangle… |
| 703 | Problem 703 | 58s | Given an integer n, n geq 3, let B=mathrm{false},mathrm{true} and let B^n be … |
| 704 | Problem 704 | 1m 46s | Define g(n, m) to be the largest integer k such that 2^k divides binom{n}m. |
| 705 | Problem 705 | 2m 16s | The inversion count of a sequence of digits is the smallest number of adjacen… |
| 706 | Problem 706 | 2m 11s | For a positive integer n, define f(n) to be the number of non-empty substring… |
| 707 | Problem 707 | 2m 30s | Consider a wtimes h grid. |
| 708 | Problem 708 | 2m 57s | A positive integer, n, is factorised into prime factors. |
| 709 | Problem 709 | 3m 47s | Every day for the past n days Even Stevens brings home his groceries in a pla… |
| 710 | Problem 710 | 1m 28s | Solution to Project Euler Problem 710. |
| 711 | Problem 711 | 3m 58s | Oscar and Eric play the following game. |
| 712 | Problem 712 | 21m 59s | For any integer n0 and prime number p, define nup(n) as the greatest integer … |
| 713 | Problem 713 | 2m 18s | Turan has the electrical water heating system outside his house in a shed. |
| 714 | Problem 714 | 4m 55s | We call a natural number a duodigit if its decimal representation uses no mor… |
| 715 | Problem 715 | 45s | Let f(n) be the number of 6-tuples (x1,x2,x3,x4,x5,x6) such that: - All xi ar… |
| 716 | Problem 716 | 2m 10s | Consider a directed graph made from an orthogonal lattice of Htimes W nodes. |
| 717 | Problem 717 | 4m 38s | For an odd prime p, define f(p) = leftlfloorfrac{2^{(2^p)}}{p}rightrfloorbmod… |
| 718 | Problem 718 | 3m 11s | Consider the equation 17^pa+19^pb+23^pc = n where a, b, c and p are positive … |
| 719 | Problem 719 | 3m 46s | We define an S-number to be a natural number, n, that is a perfect square and… |
| 720 | Problem 720 | 1m 56s | Consider all permutations of 1, 2, ldots N, listed in lexicographic order. |
| 721 | Problem 721 | 1m 9s | Given is the function f(a,n)=lfloor (lceil sqrt a rceil + sqrt a)^n rfloor. |
| 722 | Problem 722 | 1m 10s | For a non-negative integer k, define where sigmak(n) = sum{d mid n} d^k is th… |
| 723 | Problem 723 | 1m 14s | A pythagorean triangle with catheti a and b and hypotenuse c is characterized… |
| 724 | Problem 724 | 1m 55s | A depot uses n drones to disperse packages containing essential supplies alon… |
| 725 | Problem 725 | 2m 57s | A number where one digit is the sum of the other digits is called a digit sum… |
| 726 | Problem 726 | 36m 25s | Consider a stack of bottles of wine. |
| 727 | Problem 727 | 59s | Let ra, rb and rc be the radii of three circles that are mutually and externa… |
| 728 | Problem 728 | 4m 7s | Consider n coins arranged in a circle where each coin shows heads or tails. |
| 729 | Problem 729 | 3m 29s | Consider the sequence of real numbers an defined by the starting value a0 and… |
| 730 | Problem 730 | 4m 30s | For a non-negative integer k, the triple (p,q,r) of positive integers is call… |
| 731 | Problem 731 | 1m 14s | Define A(n) to be the 10 decimal digits from the nth digit onward. |
| 732 | Problem 732 | 1m 58s | N trolls are in a hole that is DN cm deep. |
| 733 | Problem 733 | 3m 45s | Let ai be the sequence defined by ai=153^i bmod 10000019 for i ge 1. |
| 734 | Problem 734 | 1m 23s | The logical-OR of two bits is 0 if both bits are 0, otherwise it is 1. |
| 735 | Problem 735 | 7m 21s | Let f(n) be the number of divisors of 2n^2 that are no greater than n. |
| 736 | Problem 736 | 3m 40s | Define two functions on lattice points: r(x,y) = (x+1,2y) s(x,y) = (2x,y+1) A… |
| 737 | Problem 737 | 6m 54s | A game is played with many identical, round coins on a flat table. |
| 738 | Problem 738 | 2m 45s | Define d(n,k) to be the number of ways to write n as a product of k ordered i… |
| 739 | Problem 739 | 6m | Take a sequence of length n. |
| 740 | Problem 740 | 1m 51s | Secret Santa is a process that allows n people to give each other presents, s… |
| 741 | Problem 741 | 4m 6s | Let f(n) be the number of ways an ntimes n square grid can be coloured, each … |
| 742 | Problem 742 | 4m 3s | A symmetrical convex grid polygon is a polygon such that: - All its vertices … |
| 743 | Problem 743 | 5m 9s | A window into a matrix is a contiguous sub matrix. |
| 744 | Problem 744 | 1m 34s | "What? Where? When?" is a TV game show in which a team of experts attempt to … |
| 745 | Problem 745 | 1m 37s | For a positive integer, n, define g(n) to be the maximum perfect square that … |
| 746 | Problem 746 | 3m 56s | n families, each with four members, a father, a mother, a son and a daughter,… |
| 747 | Problem 747 | 2m 19s | Mamma Triangolo baked a triangular pizza. |
| 748 | Problem 748 | 1m 9s | Upside Down is a modification of the famous Pythagorean equation: A solution … |
| 749 | Problem 749 | 4m 2s | A positive integer, n, is a near power sum if there exists a positive integer… |
| 750 | Problem 750 | 3m 42s | Card Stacking is a game on a computer starting with an array of N cards label… |
| 751 | Problem 751 | 51s | A non-decreasing sequence of integers an can be generated from any positive r… |
| 752 | Problem 752 | 55s | When (1+sqrt 7) is raised to an integral power, n, we always get a number of … |
| 753 | Problem 753 | 3m 21s | Fermat's Last Theorem states that no three positive integers a, b, c satisfy … |
| 754 | Problem 754 | 51s | The Gauss Factorial of a number n is defined as the product of all positive n… |
| 755 | Problem 755 | 6m 23s | Consider the Fibonacci sequence 1,2,3,5,8,13,21,ldots. |
| 756 | Problem 756 | 2m 1s | Consider a function f(k) defined for all positive integers k0. |
| 757 | Problem 757 | 2m 39s | A positive integer N is stealthy, if there exist positive integers a, b, c, d… |
| 758 | Problem 758 | 2m 48s | There are 3 buckets labelled S (small) of 3 litres, M (medium) of 5 litres an… |
| 759 | Problem 759 | 3m 13s | The function f is defined for all positive integers as follows: It can be pro… |
| 760 | Problem 760 | 3m 21s | Define where oplus, vee, wedge are the bitwise XOR, OR and AND operator respe… |
| 761 | Problem 761 | 1m 35s | Two friends, a runner and a swimmer, are playing a sporting game: The swimmer… |
| 762 | Problem 762 | 5m 32s | Consider a two dimensional grid of squares. |
| 763 | Problem 763 | 1m 43s | Consider a three dimensional grid of cubes. |
| 764 | Problem 764 | 2m 24s | Consider the following Diophantine equation: where x, y and z are positive in… |
| 765 | Problem 765 | 6m 23s | Starting with 1 gram of gold you play a game. |
| 766 | Problem 766 | 44s | A sliding block puzzle is a puzzle where pieces are confined to a grid and by… |
| 767 | Problem 767 | 3m 26s | A window into a matrix is a contiguous sub matrix. |
| 768 | Problem 768 | 10m 17s | A certain type of chandelier contains a circular ring of n evenly spaced cand… |
| 769 | Problem 769 | 5m 56s | Consider the following binary quadratic form: A positive integer q has a prim… |
| 770 | Problem 770 | 3m 50s | A and B play a game. A has originally 1 gram of gold and B has an unlimited a… |
| 771 | Problem 771 | 45s | We define a pseudo-geometric sequence to be a finite sequence a0, a1, dotsc, … |
| 772 | Problem 772 | 1m 8s | A k-bounded partition of a positive integer N is a way of writing N as a sum … |
| 773 | Problem 773 | 3m 35s | Let Sk be the set containing 2 and 5 and the first k primes that end in 7. |
| 774 | Problem 774 | 48m 48s | Let '' denote the bitwise AND operation. |
| 775 | Problem 775 | 45s | When wrapping several cubes in paper, it is more efficient to wrap them all t… |
| 776 | Problem 776 | 54s | For a positive integer n, d(n) is defined to be the sum of the digits of n. |
| 777 | Problem 777 | 36m 25s | For coprime positive integers a and b, let C{a,b} be the curve defined by: wh… |
| 778 | Problem 778 | 1m 31s | If a,b are two nonnegative integers with decimal representations a=(dots a2a1… |
| 779 | Problem 779 | 5m 33s | For a positive integer n gt 1, let p(n) be the smallest prime dividing n, and… |
| 780 | Problem 780 | 2m 4s | For positive real numbers a,b, an atimes b torus is a rectangle of width a an… |
| 781 | Problem 781 | 45s | Let F(n) be the number of connected graphs with blue edges (directed) and red… |
| 782 | Problem 782 | 4m 26s | The complexity of an ntimes n binary matrix is the number of distinct rows an… |
| 783 | Problem 783 | 50s | Given n and k two positive integers we begin with an urn that contains kn whi… |
| 784 | Problem 784 | 36m 26s | Let's call a pair of positive integers p, q (p lt q) reciprocal, if there is … |
| 785 | Problem 785 | 1m 39s | Consider the following Diophantine equation: where x, y and z are positive in… |
| 786 | Problem 786 | 49s | The following diagram shows a billiard table of a special quadrilateral shape. |
| 787 | Problem 787 | 3m 36s | Two players play a game with two piles of stones. |
| 788 | Problem 788 | 45s | A dominating number is a positive integer that has more than half of its digi… |
| 789 | Problem 789 | 2m 32s | Given an odd prime p, put the numbers 1,...,p-1 into frac{p-1}{2} pairs such … |
| 790 | Problem 790 | 2m 55s | There is a grid of length and width 50515093 points. |
| 791 | Problem 791 | 3m 34s | Denote the average of k numbers x1, ..., xk by bar{x} = frac{1}{k} sumi xi. |
| 792 | Problem 792 | 36m 26s | We define nu2(n) to be the largest integer r such that 2^r divides n. |
| 793 | Problem 793 | 3m 36s | Let Si be an integer sequence produced with the following pseudo-random numbe… |
| 794 | Problem 794 | 6m 5s | This problem uses half open interval notation where [a,b) represents a le x < b. |
| 795 | Problem 795 | 4m 17s | For a positive integer n, the function g(n) is defined as For example, g(4) =… |
| 796 | Problem 796 | 3m 12s | A standard 52 card deck comprises thirteen ranks in four suits. |
| 797 | Problem 797 | 5m 1s | A monic polynomial is a single-variable polynomial in which the coefficient o… |
| 798 | Problem 798 | 1m 15s | Two players play a game with a deck of cards which contains s suits with each… |
| 799 | Problem 799 | 3m 35s | Pentagonal numbers are generated by the formula: Pn = tfrac 12n(3n-1) giving … |
| 800 | Problem 800 | 1m 3s | An integer of the form p^q q^p with prime numbers p neq q is called a hybrid-… |
| 801 | Problem 801 | 1m 55s | The positive integral solutions of the equation x^y=y^x are (2,4), (4,2) and … |
| 802 | Problem 802 | 2m 22s | Let Bbb R^2 be the set of pairs of real numbers (x, y). |
| 803 | Problem 803 | 6m 45s | Rand48 is a pseudorandom number generator used by some programming languages. |
| 804 | Problem 804 | 1m 9s | Let g(n) denote the number of ways a positive integer n can be represented in… |
| 805 | Problem 805 | 4m 9s | For a positive integer n, let s(n) be the integer obtained by shifting the le… |
| 806 | Problem 806 | 3m 57s | This problem combines the game of Nim with the Towers of Hanoi. |
| 807 | Problem 807 | 45s | Given a circle C and an integer n 1, we perform the following operations. |
| 808 | Problem 808 | 4m 45s | Both 169 and 961 are the square of a prime. |
| 809 | Problem 809 | 3m 5s | The following is a function defined for all positive rational values of x. |
| 810 | Problem 810 | 3m 4s | We use xoplus y for the bitwise XOR of x and y. |
| 811 | Problem 811 | 4m 13s | Let b(n) be the largest power of 2 that divides n. |
| 812 | Problem 812 | 7m 54s | A dynamical polynomial is a monicleading coefficient is 1 polynomial f(x) wit… |
| 813 | Problem 813 | 1m 2s | We use xoplus y to be the bitwise XOR of x and y. |
| 814 | Problem 814 | 6m 23s | 4n people stand in a circle with their heads down. |
| 815 | Problem 815 | 3m 21s | A pack of cards contains 4n cards with four identical cards of each value. |
| 816 | Problem 816 | 46s | We create an array of points Pn in a two dimensional plane using the followin… |
| 817 | Problem 817 | 4m 56s | Define m = M(n, d) to be the smallest positive integer such that when m^2 is … |
| 818 | Problem 818 | 2m 28s | The SET® card game is played with a pack of 81 distinct cards. |
| 819 | Problem 819 | 1m 47s | Given an n-tuple of numbers another n-tuple is created where each element of … |
| 820 | Problem 820 | 1m 13s | Let dn(x) be the nth decimal digit of the fractional part of x, or 0 if the f… |
| 821 | Problem 821 | 4m 54s | A set, S, of integers is called 123-separable if S, 2S and 3S are disjoint. |
| 822 | Problem 822 | 2m 30s | A list initially contains the numbers 2, 3, dots, n. |
| 823 | Problem 823 | 2m 1s | A list initially contains the numbers 2, 3, dots, n. |
| 824 | Problem 824 | 1m 10s | A Slider is a chess piece that can move one square left or right. |
| 825 | Problem 825 | 3m 25s | Two cars are on a circular track of total length 2n, facing the same directio… |
| 826 | Problem 826 | 1m 7s | Consider a wire of length 1 unit between two posts. |
| 827 | Problem 827 | 3m 8s | Define Q(n) to be the smallest number that occurs in exactly n Pythagorean tr… |
| 828 | Problem 828 | 51s | It is a common recreational problem to make a target number using a selection… |
| 829 | Problem 829 | 31m 13s | Given any integer n gt 1 a binary factor tree T(n) is defined to be: - A tree… |
| 830 | Problem 830 | 6m 1s | Let displaystyle S(n)=sumlimits{k=0}^{n}binom{n}{k}k^n. |
| 831 | Problem 831 | 6m 26s | Let g(m) be the integer defined by the following double sum of products of bi… |
| 832 | Problem 832 | 3m 36s | In this problem oplus is used to represent the bitwise exclusive or of two nu… |
| 833 | Problem 833 | 1m 44s | Triangle numbers Tk are integers of the form frac{k(k+1)} 2. |
| 834 | Problem 834 | 2m 20s | A sequence is created by starting with a positive integer n and incrementing … |
| 835 | Problem 835 | 4m 31s | A Pythagorean triangle is called supernatural if two of its three sides are c… |
| 836 | Problem 836 | 1m 26s | Let A be an affine plane over a radically integral local field F with residua… |
| 837 | Problem 837 | 18m 16s | Amidakuji (Japanese: 阿弥陀籤) is a method for producing a random permutation of … |
| 838 | Problem 838 | 1m 33s | Let f(N) be the smallest positive integer that is not coprime to any positive… |
| 839 | Problem 839 | 45s | The sequence Sn is defined by S0 = 290797 and Sn = S{n - 1}^2 bmod 50515093 f… |
| 840 | Problem 840 | 3m 22s | A partition of n is a set of positive integers for which the sum equals n. |
| 841 | Problem 841 | 5m 17s | The regular star polygon p/q, for coprime integers p,q with p gt 2q gt 0, is … |
| 842 | Problem 842 | 7m 24s | Given n equally spaced points on a circle, we define an n-star polygon as an … |
| 843 | Problem 843 | 8m 52s | This problem involves an iterative procedure that begins with a circle of nge… |
| 844 | Problem 844 | 48s | Consider positive integer solutions to a^2+b^2+c^2 = 3abc For example, (1,5,1… |
| 845 | Problem 845 | 56s | Let D(n) be the n-th positive integer that has the sum of its digits a prime. |
| 846 | Problem 846 | 46s | A bracelet is made by connecting at least three numbered beads in a circle. |
| 847 | Problem 847 | 54s | Jack has three plates in front of him. |
| 848 | Problem 848 | 5m 30s | Two players play a game. |
| 849 | Problem 849 | 5m 22s | In a tournament there are n teams and each team plays each other team twice. |
| 850 | Problem 850 | 12m 17s | Any positive real number x can be decomposed into integer and fractional part… |
| 851 | Problem 851 | 6m 24s | Let n be a positive integer and let En be the set of n-tuples of strictly pos… |
| 852 | Problem 852 | 5m 5s | This game has a box of N unfair coins and N fair coins. |
| 853 | Problem 853 | 2m 31s | For every positive integer n the Fibonacci sequence modulo n is periodic. |
| 854 | Problem 854 | 3m 29s | For every positive integer n the Fibonacci sequence modulo n is periodic. |
| 855 | Problem 855 | 4m 33s | Given two positive integers a,b, Alex and Bianca play a game in ab rounds. |
| 856 | Problem 856 | 1m 15s | A standard 52-card deck comprises 13 ranks in four suits. |
| 857 | Problem 857 | 2m 17s | A graph is made up of vertices and coloured edges. |
| 858 | Problem 858 | 3m 29s | Define G(N) = sumS operatorname{lcm}(S) where S ranges through all subsets of… |
| 859 | Problem 859 | 2m 20s | Odd and Even are playing a game with N cookies. |
| 860 | Problem 860 | 2m 52s | Gary and Sally play a game using gold and silver coins arranged into a number… |
| 861 | Problem 861 | 21m 9s | A unitary divisor of a positive integer n is a divisor d of n such that gcdle… |
| 862 | Problem 862 | 46s | For a positive integer n define T(n) to be the number of strictly larger inte… |
| 863 | Problem 863 | 1m 20s | Using only a six-sided fair dice and a five-sided fair dice, we would like to… |
| 864 | Problem 864 | 5m 7s | Let C(n) be the number of squarefree integers of the form x^2 + 1 such that 1… |
| 865 | Problem 865 | 6m 56s | A triplicate number is a positive integer such that, after repeatedly removin… |
| 866 | Problem 866 | 46s | A small child has a “number caterpillar” consisting of N jigsaw pieces, each … |
| 867 | Problem 867 | 46s | There are 5 ways to tile a regular dodecagon of side 1 with regular polygons … |
| 868 | Problem 868 | 1m 3s | There is a method that is used by Bell ringers to generate all variations of … |
| 869 | Problem 869 | 4m 44s | A prime is drawn uniformly from all primes not exceeding N. |
| 870 | Problem 870 | 1m 25s | Two players play a game with a single pile of stones of initial size n. |
| 871 | Problem 871 | 3m 59s | Let f be a function from a finite set S to itself. |
| 872 | Problem 872 | 1m 35s | A sequence of rooted trees Tn is constructed such that Tn has n nodes numbere… |
| 873 | Problem 873 | 2m 42s | Let W(p,q,r) be the number of words that can be formed using the letter A p t… |
| 874 | Problem 874 | 3m 47s | Let p(t) denote the (t+1)th prime number. |
| 875 | Problem 875 | 4m 24s | For a positive integer n we define q(n) to be the number of solutions to: whe… |
| 876 | Problem 876 | 11m 59s | Starting with three numbers a, b, c, at each step do one of the three operati… |
| 877 | Problem 877 | 2m 24s | We use xoplus y for the bitwise XOR of x and y. |
| 878 | Problem 878 | 10m 21s | We use xoplus y for the bitwise XOR of x and y. |
| 879 | Problem 879 | 1m 27s | A touch-screen device can be unlocked with a "password" consisting of a seque… |
| 880 | Problem 880 | 2m 3s | (x,y) is called a nested radical pair if x and y are non-zero integers such t… |
| 881 | Problem 881 | 1m 15s | For a positive integer n create a graph using its divisors as vertices. |
| 882 | Problem 882 | 5m 5s | Dr. One and Dr. Zero are playing the following partisan game. The game begins… |
| 883 | Problem 883 | 1m 23s | In this problem we consider triangles drawn on a hexagonal lattice, where eac… |
| 884 | Problem 884 | 2m 7s | Starting from a positive integer n, at each step we subtract from n the large… |
| 885 | Problem 885 | 1m 21s | For a positive integer d, let f(d) be the number created by sorting the digit… |
| 886 | Problem 886 | 15m 44s | A permutation of 2,3,ldots,n is a rearrangement of these numbers. |
| 887 | Problem 887 | 8m 3s | Consider the problem of determining a secret number from a set 1, ..., N by r… |
| 888 | Problem 888 | 6m 24s | Two players play a game with a number of piles of stones, alternating turns. |
| 889 | Problem 889 | 9m 29s | Recall the blancmange function from Problem 226: T(x) = sumlimits{n = 0}^inft… |
| 890 | Problem 890 | 11m 53s | Let p(n) be the number of ways to write n as the sum of powers of two, ignori… |
| 891 | Problem 891 | 7m 23s | A round clock only has three hands: hour, minute, second. |
| 892 | Problem 892 | 4m 6s | Consider a circle where 2n distinct points have been marked on its circumfere… |
| 893 | Problem 893 | 9m 18s | Define M(n) to be the minimum number of matchsticks needed to represent the n… |
| 894 | Problem 894 | 1m 31s | Consider a unit circlecircle with radius 1 C0 on the plane that does not encl… |
| 895 | Problem 895 | 40s | Gary and Sally play a game using gold and silver coins arranged into a number… |
| 896 | Problem 896 | 12m 15s | A contiguous range of positive integers is called a divisible range if all th… |
| 897 | Problem 897 | 10m 29s | Let G(n) denote the largest possible area of an n-gona polygon with n sides c… |
| 898 | Problem 898 | 2m 47s | Claire Voyant is a teacher playing a game with a class of students. |
| 899 | Problem 899 | 3m 52s | Two players play a game with two piles of stones. |
| 900 | Problem 900 | 3m 34s | Two players play a game with at least two piles of stones. |
| 901 | Problem 901 | 9m 51s | A driller drills for water. |
| 902 | Problem 902 | 3m 31s | A permutation pi of 1, dots, n can be represented in one-line notation as pi(… |
| 903 | Problem 903 | 1m 58s | A permutation pi of 1, dots, n can be represented in one-line notation as pi(… |
| 904 | Problem 904 | 2m 14s | Given a right-angled triangle with integer sides, the smaller angle formed by… |
| 905 | Problem 905 | 10m 47s | Three epistemologists, known as A, B, and C, are in a room, each wearing a ha… |
| 906 | Problem 906 | 6m 40s | Three friends attempt to collectively choose one of n options, labeled 1,dots… |
| 907 | Problem 907 | 2m 41s | An infant's toy consists of n cups, labelled C1,dots,Cn in increasing order o… |
| 908 | Problem 908 | 1m 56s | A clock sequence is a periodic sequence of positive integers that can be brok… |
| 909 | Problem 909 | 4m 50s | An L-expression is defined as any one of the following: - a natural number; -… |
| 910 | Problem 910 | 5m 28s | An L-expression is defined as any one of the following: - a natural number; -… |
| 911 | Problem 911 | 11m 21s | An irrational number x can be uniquely expressed as a continued fraction [a0;… |
| 912 | Problem 912 | 3m 11s | Let sn be the n-th positive integer that does not contain three consecutive o… |
| 913 | Problem 913 | 3m 30s | The numbers from 1 to 12 can be arranged into a 3 times 4 matrix in either ro… |
| 914 | Problem 914 | 4m 28s | For a given integer R consider all primitive Pythagorean triangles that can f… |
| 915 | Problem 915 | 10m 49s | The function s(n) is defined recursively for positive integers by s(1) = 1 an… |
| 916 | Problem 916 | 2m 53s | Let P(n) be the number of permutations of 1,2,3,ldots,2n such that: 1. |
| 917 | Problem 917 | 15m 43s | The sequence sn is defined by s1 = 102022661 and sn = s{n-1}^2 bmod {99838888… |
| 918 | Problem 918 | 52s | The sequence an is defined by a1=1, and then recursively for ngeq1: The first… |
| 919 | Problem 919 | 3m 57s | We call a triangle fortunate if it has integral sides and at least one of its… |
| 920 | Problem 920 | 2m 2s | For a positive integer n we define tau(n) to be the count of the divisors of n. |
| 921 | Problem 921 | 14m 39s | Consider the following recurrence relation: Note that a0 is the golden ratio. |
| 922 | Problem 922 | 4m 31s | A Young diagram is a finite collection of (equally-sized) squares in a grid-l… |
| 923 | Problem 923 | 3m 16s | A Young diagram is a finite collection of (equally-sized) squares in a grid-l… |
| 924 | Problem 924 | 3m 47s | Let B(n) be the smallest number larger than n that can be formed by rearrangi… |
| 925 | Problem 925 | 6m 39s | Let B(n) be the smallest number larger than n that can be formed by rearrangi… |
| 926 | Problem 926 | 2m 3s | A round number is a number that ends with one or more zeros in a given base. |
| 927 | Problem 927 | 11m 21s | A full k-ary tree is a tree with a single root node, such that every node is … |
| 928 | Problem 928 | 34m 27s | This problem is based on (but not identical to) the scoring for the card game… |
| 929 | Problem 929 | 3m 2s | A composition of n is a sequence of positive integers which sum to n. |
| 930 | Problem 930 | 11m 12s | Given nge 2 bowls arranged in a circle, mge 2 balls are distributed amongst t… |
| 931 | Problem 931 | 4m 9s | For a positive integer n construct a graph using all the divisors of n as the… |
| 932 | Problem 932 | 2m 14s | For the year 2025 Given positive integers a and b, the concatenation ab we ca… |
| 933 | Problem 933 | 15m 41s | Starting with one piece of integer-sized rectangle paper, two players make mo… |
| 934 | Problem 934 | 2m 41s | We define the unlucky prime of a number n, denoted u(n), as the smallest prim… |
| 935 | Problem 935 | 1m 4s | A square of side length b<1 is rolling around the inside of a larger square o… |
| 936 | Problem 936 | 46s | A peerless tree is a tree with no edge between two vertices of the same degree. |
| 937 | Problem 937 | 3m 19s | Let theta=sqrt{-2}. Define T to be the set of numbers of the form a+btheta, w… |
| 938 | Problem 938 | 1m 21s | A deck of cards contains R red cards and B black cards. |
| 939 | Problem 939 | 4m 30s | Two players A and B are playing a variant of Nim. |
| 940 | Problem 940 | 1m 18s | The Fibonacci sequence (fi) is the unique sequence such that - f0=0 - f1=1 - … |
| 941 | Problem 941 | 1m 28s | de Bruijn has a digital combination lock with k buttons numbered 0 to k-1 whe… |
| 942 | Problem 942 | 5m 7s | Given a natural number q, let p = 2^q - 1 be the q-th Mersenne number. |
| 943 | Problem 943 | 5m 40s | Given two unequal positive integers a and b, we define a self-describing sequ… |
| 944 | Problem 944 | 4m 35s | Given a set E of positive integers, an element x of E is called an element di… |
| 945 | Problem 945 | 9m 41s | We use xoplus y for the bitwise XOR of x and y. |
| 946 | Problem 946 | 7m 38s | Given the representation of a continued fraction alpha is a real number with … |
| 947 | Problem 947 | 2m 13s | The (a,b,m)-sequence, where 0 leq a,b lt m, is defined as $begin{align} g(0)&… |
| 948 | Problem 948 | 2m 7s | Left and Right play a game with a word consisting of L's and R's, alternating… |
| 949 | Problem 949 | 3m 31s | Left and Right play a game with a number of words, each consisting of L's and… |
| 950 | Problem 950 | 1m | A band of pirates has come into a hoard of treasure, and must decide how to d… |
| 951 | Problem 951 | 3m 5s | Two players play a game using a deck of 2n cards: n red and n black. |
| 952 | Problem 952 | 2m 51s | Given a prime p and a positive integer n lt p, let R(p, n) be the multiplicat… |
| 953 | Problem 953 | 38s | In the classical game of Nim two players take turns removing stones from piles. |
| 954 | Problem 954 | 5m 33s | A positive integer is called heptaphobic if it is not divisible by seven and … |
| 955 | Problem 955 | 2m 29s | A sequence (an){n ge 0} starts with a0 = 3 and for each n ge 0, - if an is a … |
| 956 | Problem 956 | 4m 42s | The total number of prime factors of n, counted with multiplicity, is denoted… |
| 957 | Problem 957 | — | There is a plane on which all points are initially white, except three red po… |
| 958 | Problem 958 | 4m 31s | The Euclidean algorithm can be used to find the greatest common divisor of tw… |
| 959 | Problem 959 | 1m 49s | A frog is placed on the number line. |
| 960 | Problem 960 | 1m 43s | nThere are n distinct piles of stones, each of size n-1. |
| 961 | Problem 961 | 2m 4s | nThis game starts with a positive integer. |
| 962 | Problem 962 | 47s | Given is an integer sided triangle ABC with BC le AC le AB.nk is the angular … |
| 963 | Problem 963 | 1m 48s | NOTE: This problem is related to Problem 882. |
| 964 | Problem 964 | 3m 20s | A group of k(k-1) / 2 + 1 children play a game of k rounds.nAt the beginning,… |
| 965 | Problem 965 | 2m 46s | Let {x} denote the fractional part of a real number x.nnDefine fN(x) to be th… |
| 966 | Problem 966 | 2m 9s | nLet I(a, b, c) be the largest possible area of intersection between a triang… |
| 967 | Problem 967 | 3m 19s | nA positive integer n is considered B-trivisible if the sum of all different … |
| 968 | Problem 968 | 2m 23s | nDefinennas the sum of 2^a3^b5^c7^d11^e over all quintuples of non-negative i… |
| 969 | Problem 969 | 1m 48s | nStarting at zero, a kangaroo hops along the real number line in the positive… |
| 970 | Problem 970 | 3m 31s | nStarting at zero, a kangaroo hops along the real number line in the positive… |
| 971 | Problem 971 | 3m 5s | Let p be a prime of the form 5k-4 and define fp(x) = left(x^k+xright) bmod p.… |
| 972 | Problem 972 | 9m 29s | nThe hyperbolic plane can be represented by the open unit disc, namely the se… |
| 973 | Problem 973 | 3m 45s | Solution to Project Euler Problem 973. |
| 974 | Problem 974 | 2m 41s | Solution to Project Euler Problem 974. |
| 975 | Problem 975 | 4m 3s | Solution to Project Euler Problem 975. |
| 976 | Problem 976 | 5m 25s | Solution to Project Euler Problem 976. |
| 977 | Problem 977 | 9m 7s | Solution to Project Euler Problem 977. |
| 978 | Problem 978 | 1m 56s | Solution to Project Euler Problem 978. |
| 979 | Problem 979 | 1m 50s | Solution to Project Euler Problem 979. |
| 980 | Problem 980 | 3m 54s | Solution to Project Euler Problem 980. |
| 981 | Problem 981 | 2m | Solution to Project Euler Problem 981. |
| 982 | Problem 982 | 3m 25s | Solution to Project Euler Problem 982. |
| 983 | Problem 983 | 2m 45s | Solution to Project Euler Problem 983. |
| 984 | Problem 984 | 2m 11s | Solution to Project Euler Problem 984. |
| 985 | Problem 985 | 2m 24s | Solution to Project Euler Problem 985. |
| 986 | Problem 986 | 6m 17s | Solution to Project Euler Problem 986. |
| 987 | Problem 987 | 3m 43s | Solution to Project Euler Problem 987. |
| 988 | Problem 988 | 6m 1s | Solution to Project Euler Problem 988. |
| 989 | Problem 989 | 2m 54s | Write Fn for the n-th Fibonacci number, with F1 = F2 = 1 and F{n+1} = Fn + F{… |
| 990 | Problem 990 | 4m 23s | Solution to Project Euler Problem 990. |
| 991 | Problem 991 | 5m 32s | Solution to Project Euler Problem 991. |
| 992 | Problem 992 | 4m 19s | Solution to Project Euler Problem 992. |
| 993 | Problem 993 | 2m 1s | Solution to Project Euler Problem 993. |
| 994 | Problem 994 | 5m 5s | Solution to Project Euler Problem 994. |
| 995 | Problem 995 | 14m 29s | For each prime p and each positive integer n define two polynomials: Let S(p)… |
| 996 | Problem 996 | 2m 38s | Solution to Project Euler Problem 996. |
| 997 | Problem 997 | 4m 48s | There are xyz dice arranged in an x times y times z box such that touching fa… |
| 998 | Problem 998 | 15m 56s | The minimum bounding square of a triangle is the smallest square that can be … |
Project Euler Problem 506
Consider the infinite repeating sequence of digits: 1234321234321234321...
Project Euler Problem 607
Frodo and Sam need to travel 100 leagues due East from point A to point B.
Project Euler Problem 713
Turan has the electrical water heating system outside his house in a shed.
Project Euler Problem 720
Consider all permutations of 1, 2, ldots N, listed in lexicographic order.
Project Euler Problem 773
Let Sk be the set containing 2 and 5 and the first k primes that end in 7.
Project Euler Problem 809
The following is a function defined for all positive rational values of x.
Project Euler Problem 815
A pack of cards contains 4n cards with four identical cards of each value.
Project Euler Problem 840
A partition of n is a set of positive integers for which the sum equals n.
Project Euler Problem 855
Given two positive integers a,b, Alex and Bianca play a game in ab rounds.