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Consider the following configuration of 64 triangles: We wish to colour the interior of each triangle with one of three

The cube, 41063625 (345^3), can be permuted to produce two other cubes: 56623104 (384^3) and 66430125 (405^3).

For an n-tuple of integers t = (a1, dots, an), let (x1, dots, xn) be the solutions of the polynomial equation x^n + a1 x

k defects are randomly distributed amongst n integrated-circuit chips produced by a factory (any number of defects may b

Consider all permutations of 1, 2, ldots N, listed in lexicographic order.

A simple quadrilateral is a polygon that has four distinct vertices, has no straight angles and does not self-intersect.

How many triangles are there with integral sides, at least one integral angle (measured in degrees), and a perimeter tha

In the hexadecimal number system numbers are represented using 16 different digits: The hexadecimal number mathrm{AF} wh

In the context of formal languages, any finite sequence of letters of a given alphabet Sigma is called a word over Sigma

The function operatorname{mathbf{lcm}}(a,b) denotes the least common multiple of a and b.

Let Aq(n) be the number of subsets, B, of the set 1, 2, ..., q cdot n that satisfy two conditions: 1) B has exactly n el

For any prime p the number N(p, q) is defined by N(p, q) = sum{n = 0}^q Tn cdot p^n with Tn generated by the following r

How many integers 0 le n lt 10^{18} have the property that the sum of the digits of n equals the sum of digits of 137n?

Let T(m, n) be the number of the binomial coefficients ^iCn that are divisible by 10 for n le i lt m (i, m and n are pos

A particular school offers cash rewards to children with good attendance and punctuality.

Secret Santa is a process that allows n people to give each other presents, so that each person gives a single present a

Let Sm = (x1, x2, dots , xm) be the m-tuple of positive real numbers with x1 + x2 + cdots + xm = m for which Pm = x1 cdo

Let us call an integer sided triangle with sides a le b le c barely acute if the sides satisfy a^2 + b^2 = c^2 + 1.

For a positive integer n, let s(n) be the integer obtained by shifting the leftmost digit of the decimal representation

Let an be the largest real root of a polynomial g(x) = x^3 - 2^n cdot x^2 + n.

A k-input binary truth table is a map from k input bits (binary digits, 0 [false] or 1 [true]) to 1 output bit.

ABC is an integer sided triangle with incenter I and perimeter p.

It can be shown that the polynomial n^4 + 4n^3 + 2n^2 + 5n is a multiple of 6 for every integer n.

An electric circuit uses exclusively identical capacitors of the same value C.

A deck of cards numbered from 1 to n is shuffled randomly such that each permutation is equally likely.

Albert chooses a positive integer k, then two real numbers a, b are randomly chosen in the interval [0,1] with uniform d

Given a character string s, we define L(k,s) to be the length of the longest substring of s which appears at least k tim

A train is used to transport four carriages in the order: ABCD.

Consider the fraction, dfrac n d, where n and d are positive integers.

The number 145 is well known for the property that the sum of the factorial of its digits is equal to 145: Perhaps less

Given an integer sided triangle ABC: Let I be the incenter of ABC.

Let n be a positive integer.

By replacing the 1st digit of the 2-digit number 3, it turns out that six of the nine possible values: 13, 23, 43, 53, 7

Two players play a game with two piles of stones, alternating turns.

Consider the following algorithm for sorting a list: - 1.

Given a set of points on a plane, we define a convex hole to be a convex polygon having as vertices any of the given poi

A pack of cards contains 4n cards with four identical cards of each value.

The coefficients in the expansion of (x+1)^k are called binomial coefficients.

Let's call two numbers friend numbers if their representation in base 10 has at least one common digit.

Let a0, a1, dots be an integer sequence defined by: - a0 = 1; - for n ge 1, an is the sum of the digits of all preceding

Let us consider mixtures of three substances: A, B and C.

NOTE: This is a more difficult version of Problem 114.

Let C be the circle with radius r, x^2 + y^2 = r^2.

Members of a species of bacteria occur in two different types: alpha and beta.

Anton and Bertrand love to play three pile Nim.

Consider a wire of length 1 unit between two posts.

Bob plays a single-player game of chance using two standard 6-sided dice and twelve cards numbered 1 to 12.

The divisors of 12 are: 1,2,3,4,6 and 12.

When (1+sqrt 7) is raised to an integral power, n, we always get a number of the form (a+bsqrt 7).

Looking at the table below, it is easy to verify that the maximum possible sum of adjacent numbers in any direction (hor

A sequence is created by starting with a positive integer n and incrementing by (n+m) at the m^{th} step.

There is a method that is used by Bell ringers to generate all variations of the order that bells are rung.

Let sk be the number of 1’s when writing the numbers from 0 to k in binary.

An infinitely long cylinder has its curved surface fully covered with different coloured but otherwise identical rectang

For any N, let f(N) be the last five digits before the trailing zeroes in N!.

A positive number is pandigital in base b if it contains all digits from 0 to b - 1 at least once when written in base b

Define f(0)=1 and f(n) to be the number of different ways n can be expressed as a sum of integer powers of 2 using each

The smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is 28.

For every integer n1, the family of functions f{n,a,b} is defined by f{n,a,b}(x)equiv a x + b mod n for a,b,x integer an

Consider the equation 17^pa+19^pb+23^pc = n where a, b, c and p are positive integers, i.e.

Alice walks on a lattice grid.

Dave is doing his homework on the balcony and, preparing a presentation about Pythagorean triangles, has just cut out a

Consider the numbers 15, 16 and 18: 15=3times 5 and 3+5=8.

Jeff eats a pie in an unusual way.

Let g(m) be the integer defined by the following double sum of products of binomial coefficients: You are given that g(1

Find the unique positive integer whose square has the form 1234567890, where each “” is a single digit.

Let S(A) represent the sum of elements in set A of size n.

Gauss famously proved that every positive integer can be expressed as the sum of three triangular numbers (including 0 a

Starting from zero the natural numbers are written down in base 10 like this: Consider the digit d=1.

For a positive integer n, let f(n) be the sum of the squares of the digits (in base 10) of n, e.g.

The following undirected network consists of seven vertices and twelve edges with a total weight of 243.

A Gaussian integer is a number z = a + bi where a, b are integers and i^2 = -1.

The number 7 is special, because 7 is 111 written in base 2, and 11 written in base 6 (i.e.

A cubic Bézier curve is defined by four points: P0, P1, P2, and P3.

Two players share an unbiased coin and take it in turns to play The Race.

Let S(A) represent the sum of elements in set A of size n.

Let an be a sequence recursively defined by:quad a1=1,quaddisplaystyle an=biggl(sum{k=1}^{n-1}kcdot akbiggr)bmod n.

Considering 4-digit primes containing repeated digits it is clear that they cannot all be the same: 1111 is divisible by

There are 16 positive integers that do not have a zero in their digits and that have a digital sum equal to 5, namely: 5

A bracelet is made by connecting at least three numbered beads in a circle.

Let P{a,b} denote a path in a atimes b lattice grid with following properties: - The path begins at (0,0) and ends at (a

The sequence 1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355, 653, 1201, dots is defined by T1 = T2 = T3 = 1 and Tn = T{n -

Two friends A and B are great fans of Chess.

A group of chefs (numbered 1, 2, etc) participate in a turn-based strategic cooking competition.

An axis-aligned cuboid, specified by parameters (x0, y0, z0), (dx, dy, dz), consists of all points (X,Y,Z) such that x0

Let ak, bk, and ck represent the three solutions (real or complex numbers) to the equation frac 1 x = (frac k x)^2(k+x^2

A cyclic number with n digits has a very interesting property: When it is multiplied by 1, 2, 3, 4, dots, n, all the pro

A set, S, of integers is called 123-separable if S, 2S and 3S are disjoint.

Given a non-square integer d, any real x can be approximated arbitrarily close by quadratic integers a+bsqrt{d}, where a

On the parabola y = x^2/k, three points A(a, a^2/k), B(b, b^2/k) and C(c, c^2/k) are chosen.

The function f is defined for all positive integers as follows: It can be proven that f(n) is integer for all values of

We define a simber to be a positive integer in which any odd digit, if present, occurs an odd number of times, and any e

We use xoplus y for the bitwise XOR of x and y.

A triangle is cut into four pieces by two straight lines, each starting at one vertex and ending on the opposite edge.

A standard 52 card deck comprises thirteen ranks in four suits.

Given an integer n, n geq 3, let B=mathrm{false},mathrm{true} and let B^n be the set of sequences of n values from B.

Consider the set Ir of points (x,y) with integer co-ordinates in the interior of the circle with radius r, centered at t

Define G(N) = sumS operatorname{lcm}(S) where S ranges through all subsets of 1, dots, N and operatorname{lcm} denotes t

Let g(n) be a sequence defined as follows: g(4) = 13, g(n) = g(n-1) + gcd(n, g(n-1)) for n gt 4.

An infinite sequence of real numbers a(n) is defined for all integers n as follows: For example, a(0) = dfrac{1}{1!} + d