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Given is an integer sided triangle ABC with sides a le b le c.

Gary and Sally play a game using gold and silver coins arranged into a number of vertical stacks, alternating turns.

Consider positive integer solutions to a^2+b^2+c^2 = 3abc For example, (1,5,13) is a solution.

A binary matrix is a matrix consisting entirely of 0s and 1s.

There are several ways to write the number dfrac{1}{2} as a sum of square reciprocals using distinct integers.

Let n be a positive integer and let En be the set of n-tuples of strictly positive integers.

Let R(M, N) be the number of lattice points (x, y) which satisfy MltxleN, MltyleN and largeleftlfloorfrac{y^2}{x^2}right

For a positive number n, define C(n) as the number of the integers x, for which 1 lt x lt n and x^3 equiv 1 bmod n.

There are some prime values, p, for which there exists a positive integer, n, such that the expression n^3 + n^2p is a p

For a positive integer n, let sigma2(n) be the sum of the squares of its divisors.

A small child has a “number caterpillar” consisting of forty jigsaw pieces, each with one number on it, which, when conn

Using a combination of grey square tiles and oblong tiles chosen from: red tiles (measuring two units), green tiles (mea

The sequence Sn is defined by S0 = 290797 and Sn = S{n - 1}^2 bmod 50515093 for n 0.

Each new term in the Fibonacci sequence is generated by adding the previous two terms.

There are 3 buckets labelled S (small) of 3 litres, M (medium) of 5 litres and L (large) of 8 litres.

The Möbius function, denoted mu(n), is defined as: - mu(n) = (-1)^{omega(n)} if n is squarefree (where omega(n) is the n

An integer is called eleven-free if its decimal expansion does not contain any substring representing a power of 11 exce

The moon has been opened up, and land can be obtained for free, but there is a catch.

A Slider is a chess piece that can move one square left or right.

A riffle shuffle is executed as follows: a deck of cards is split into two equal halves, with the top half taken in the

This problem uses half open interval notation where [a,b) represents a le x < b.

A permutation of 2,3,ldots,n is a rearrangement of these numbers.

A Hamming number is a positive number which has no prime factor larger than 5.

Find the smallest x + y + z with integers x gt y gt z gt 0 such that x + y, x - y, x + z, x - z, y + z, y - z are all pe

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

A triplet of positive integers (a, b, c) is called a Cardano Triplet if it satisfies the condition: For example, (2,1,5)

A Pythagorean triangle is called supernatural if two of its three sides are consecutive integers.

Let H(n) be the number of distinct integer sided equiangular convex hexagons with perimeter not exceeding n.

Define operatorname{Co}(n) to be the maximal possible sum of a set of mutually co-prime elements from 1,2,dots,n.

Tom (the cat) and Jerry (the mouse) are playing on a simple graph G.

A positive integer n is called squarefree, if no square of a prime divides n, thus 1, 2, 3, 5, 6, 7, 10, 11 are squarefr

Define s(n) to be the smallest number that has a digit sum of n.

Let tn be the tribonacci numbers defined as: t0 = t1 = 0; t2 = 1; tn = t{n-1} + t{n-2} + t{n-3} for n ge 3 and let rn =

Peter has nine four-sided (pyramidal) dice, each with faces numbered 1, 2, 3, 4.

Anton and Bertrand love to play three pile Nim.

Consider a wtimes h grid.

A 30 times 30 grid of squares contains 900 fleas, initially one flea per square.

We call an integer sided triangle n-pandigital if it contains one angle of 120 degrees and, when the sides of the triang

If we calculate a^2 bmod 6 for 0 leq a leq 5 we get: 0,1,4,3,4,1.

Let Seq(n,k) be the number of positive-integer sequences ai{1 le i le n} of length n such that: - n is divisible by ai f

Let a, b and c be positive numbers.

Define Q(n) to be the smallest number that occurs in exactly n Pythagorean triples (a,b,c) where a lt b lt c.

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

The inversion count of a sequence of digits is the smallest number of adjacent pairs that must be swapped to sort the se

For a number written in Roman numerals to be considered valid there are basic rules which must be followed.

Define g(n, m) to be the largest integer k such that 2^k divides binom{n}m.

Two positive integers a and b are 2-friendly when gcd(a,b) = 2^t, t gt 0.

Let An be the smallest positive integer satisfying An bmod pi = i for all 1 le i le n, where pi is the i-th prime.

For a positive integer n, define f(n) as the least positive multiple of n that, written in base 10, uses only digits le

G(N)=sum{j=1}^Nsum{i=1}^j gcd(i,j).

Given an n-tuple of numbers another n-tuple is created where each element of the new n-tuple is chosen randomly from the

The divisors of 6 are 1,2,3 and 6.

For every positive number n we define the function mathop{streak}(n)=k as the smallest positive integer k such that n+k

We can easily verify that none of the entries in the first seven rows of Pascal's triangle are divisible by 7: | | | | |

A moon could be described by the sphere C(r) with centre (0,0,0) and radius r.

The Gauss Factorial of a number n is defined as the product of all positive numbers leq n that are relatively prime to n

Define fk(n) = sum{i=0}^n fk(lfloorfrac i k rfloor) where fk(0) = 1 and lfloor x rfloor denotes the floor function.

A k-bounded partition of a positive integer N is a way of writing N as a sum of positive integers not exceeding k.

Consider the infinite polynomial series AF(x) = x F1 + x^2 F2 + x^3 F3 + dots, where Fk is the kth term in the Fibonacci

defhtmltext1{style{font-family:inherit;}{text{1}}} For non-negative integers m, n, the Ackermann function A(m,n) is defi

Two players play a game with a single pile of stones of initial size n.

The 3-digit number 376 in the decimal numbering system is an example of numbers with the special property that its squar

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

A positive integer n is considered cube-full, if for every prime p that divides n, so does p^3.

The flipping game is a two player game played on an N by N square board.

Two friends, a runner and a swimmer, are playing a sporting game: The swimmer is swimming within a circular pool while t

(x,y) is called a nested radical pair if x and y are non-zero integers such that dfrac{x}{y} is not a cube of a rational

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

Consider the sequence n^2+3 with n ge 1.

We shall call a positive integer A an "Alexandrian integer", if there exist integers p, q, r such that: and For example,

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

Let varphi be the golden ratio: varphi=frac{1+sqrt{5}}{2}.

Jeff eats a pie in an unusual way.

The Thue-Morse sequence Tn is a binary sequence satisfying: - T0 = 0 - T{2n} = Tn - T{2n + 1} = 1 - Tn The first several

By counting carefully it can be seen that a rectangular grid measuring 3 by 2 contains eighteen rectangles: Although the

An integer partition of a number n is a way of writing n as a sum of positive integers.

Let S = 2, 3, 5, dots, 4999 be the set of prime numbers less than 5000.

Let S(n) be the number of pairs (a,b) of distinct divisors of n such that a divides b.

A positive integer, n, is a near power sum if there exists a positive integer, k, such that the sum of the kth powers of

The smallest number m such that 10 divides m! is m=5.

Let varphi(n) be Euler's totient function.

We stack n plates into k non-empty piles where each pile is a different size.

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

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

Let d(i,b) be the digit sum of the number i in base b.

A number chain is created by continuously adding the square of the digits in a number to form a new number until it has

A prime number p is called a Panaitopol prime if p = dfrac{x^4 - y^4}{x^3 + y^3} for some positive integers x and y.

It is possible to write ten as the sum of primes in exactly five different ways: What is the first value which can be wr

A hexagonal tile with number 1 is surrounded by a ring of six hexagonal tiles, starting at "12 o'clock" and numbering th

Define two functions on lattice points: r(x,y) = (x+1,2y) s(x,y) = (2x,y+1) A path to equality of length n for a pair (a

Card Stacking is a game on a computer starting with an array of N cards labelled 1,2,ldots,N.

Let n be a natural number and p1^{alpha1}cdot p2^{alpha2}cdots pk^{alphak} its prime factorisation.

Let Si be an integer sequence produced with the following pseudo-random number generator: - S0 = 290797 - S{i+1} = Si ^2

Let E(x0, y0) be the number of steps it takes to determine the greatest common divisor of x0 and y0 with Euclid's algori

Let S(n) be the sum of all contiguous integer-substrings that can be formed from the integer n.

For a positive integer n we define q(n) to be the number of solutions to: where 0 leq ai, bi lt n.

Oregon licence plates consist of three letters followed by a three digit number (each digit can be from [0..9]).

A horizontal row comprising of 2n + 1 squares has n red counters placed at one end and n blue counters at the other end,

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

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