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Let T(n, m) be the number of m-tuples of positive integers such that the sum of any two neighbouring elements of the tup

We define the Matrix Sum of a matrix as the maximum possible sum of matrix elements such that none of the selected eleme

A dominating number is a positive integer that has more than half of its digits equal.

Let C(n) be the number of squarefree integers of the form x^2 + 1 such that 1 le x le n.

In the classic "Crossing Ladders" problem, we are given the lengths x and y of two ladders resting on the opposite walls

The number 209 can be expressed as a^2 + 3ab + b^2 in two distinct ways: qquad 209 = 8^2 + 3cdot 8cdot 5 + 5^2 qquad 209

The largest integer le 100 that is only divisible by both the primes 2 and 3 is 96, as 96=32times 3=2^5 times 3.

Amidakuji (Japanese: 阿弥陀籤) is a method for producing a random permutation of a set of objects.

An even positive integer N will be called admissible, if it is a power of 2 or its distinct prime factors are consecutiv

Let A be an affine plane over a radically integral local field F with residual characteristic p.

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

The regular star polygon p/q, for coprime integers p,q with p gt 2q gt 0, is a polygon formed from p edges of equal leng

It is well known that if the square root of a natural number is not an integer, then it is irrational.

How many 20 digit numbers n (without any leading zero) exist such that no three consecutive digits of n have a sum great

Having three black objects B and one white object W they can be grouped in 7 ways like this: | | | | | | | | |--------|-

Consider writing a natural number as product of powers of natural numbers with given exponents, additionally requiring d

Three points, P1, P2 and P3, are randomly selected within a unit square.

Consider a wtimes h grid.

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

In a room N chairs are placed around a round table.

A random generator produces a sequence of symbols drawn from the set {I, V, X, L, C, D, M, }.

Both 169 and 961 are the square of a prime.

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

Given any positive integer n, we can construct a new integer by inserting plus signs between some of the digits of the b

Bob is a manufacturer of nanobots and wants to impress his customers by giving them a ball coloured by his new nanobots

Let pn be the nth prime: 2, 3, 5, 7, 11, dots, and let r be the remainder when (pn - 1)^n + (pn + 1)^n is divided by pn^

As we all know the equation x^2=-1 has no solutions for real x.

Consider the following Diophantine equation: where x, y and z are positive integers.

Leonhard Euler was born on 15 April 1707.

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

We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and hor

!0315clocks.gif Sam and Max are asked to transform two digital clocks into two "digital root" clocks.

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

The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) eac

Define A(n) to be the 10 decimal digits from the nth digit onward.

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

Let pi(x) be the prime counting function, i.e.

On the Euclidean plane, an ant travels from point A(0, 1) to point B(d, 1) for an integer d.

Consider the sequence of real numbers an defined by the starting value a0 and the recurrence displaystyle a{n+1}=an-frac

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

Two cars are on a circular track of total length 2n, facing the same direction, initially distance n apart.

A non-decreasing sequence of integers an can be generated from any positive real value theta by the following procedure:

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.

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

Jeff eats a pie in an unusual way.

Define function P(n, k) = 1 if n can be written as the sum of k prime numbers (with repetitions allowed), and P(n, k) =

Let H(n) denote the number of sets of positive integers such that the least common multiple of the integers in the set e

A snowflake of order n is formed by overlaying an equilateral triangle (rotated by 180 degrees) onto each equilateral tr

Consider the term small sqrt{x+sqrt{y}+sqrt{z}} that is representing a nested square root.

A standard 52-card deck comprises 13 ranks in four suits.

A mountain range consists of a line of mountains with slopes of exactly 45^circ, and heights governed by the prime numbe

A row of n squares contains a frog in the leftmost square.

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

Let ABC be a triangle with all interior angles being less than 120 degrees.

Starting with three numbers a, b, c, at each step do one of the three operations: - change a to 2(b + c) - a; - change b

Consider a unit circlecircle with radius 1 C0 on the plane that does not enclose the origin.

Alice plays the following game, she starts with a list of integers L and on each step she can either: - remove two eleme

Fred the farmer arranges to have a new storage silo installed on his farm and having an obsession for all things square

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

Siegbert and Jo take turns playing a game with a heap of N pebbles: 1.

For an odd prime p, define f(p) = leftlfloorfrac{2^{(2^p)}}{p}rightrfloorbmod{2^p} For example, when p=3, lfloor 2^8/3rf

In the following equation x, y, and n are positive integers.

Construct triangle ABC such that: - Vertices A, B and C are lattice points inside or on the circle of radius r centered

Two players play a game with a deck of cards which contains s suits with each suit containing n cards numbered from 1 to

The following equation represents the continuous topography of a mountainous region, giving the elevationheight above se

A segment is uniquely defined by its two endpoints.

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

Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for examp

Let f(n) be the number of couples (x, y) with x and y positive integers, x le y and the least common multiple of x and y

What is the length of the shortest pipe, of internal radius pu{50 mm}, that can fully contain 21 balls of radii pu{30 mm

The proper divisors of a number are all the divisors excluding the number itself.

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

All positive integers can be partitioned in such a way that each and every term of the partition can be expressed as 2^i

We shall define a pythagorean polygon to be a convex polygon with the following properties: - there are at least three v

Euler's totient function, phi(n) [sometimes called the phi function], is used to determine the number of positive number

Consider n coins arranged in a circle where each coin shows heads or tails.

Let y0, y1, y2, dots be a sequence of random unsigned 32-bit integers (i.e.

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

The McCarthy 91 function is defined as follows: We can generalize this definition by abstracting away the constants into

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

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

A hexagonal orchard of order n is a triangular lattice made up of points within a regular hexagon with side n.

For two positive integers a and b, the Ulam sequence U(a,b) is defined by U(a,b)1 = a, U(a,b)2 = b and for k gt 2, U(a,b

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

We create an array of points Pn in a two dimensional plane using the following random number generator: s0=290797 s{n+1}

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

Consider the alphabet A made out of the letters of the word "text{project}": A=text c,text e,text j,text o,text p,text r

Ea is an ellipse with an equation of the form x^2 + 4y^2 = 4a^2.

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

One variant of N.G. de Bruijn's silver dollar game can be described as follows: On a strip of squares a number of coins

The following diagram shows a billiard table of a special quadrilateral shape.

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

Let b(n) be the largest power of 2 that divides n.

The (a,b,m)-sequence, where 0 leq a,b lt m, is defined as $begin{align} g(0)&=a g(1)&=b g(n)&= big(g(n-1) + g(n-2)big) b

The number 512 is interesting because it is equal to the sum of its digits raised to some power: 5 + 1 + 2 = 8, and 8^3

A unit fraction contains 1 in the numerator.

Nim is a game played with heaps of stones, where two players take it in turn to remove any number of stones from any hea

Tom has built a random generator that is connected to a row of n light bulbs.

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

We define a pseudo-geometric sequence to be a finite sequence a0, a1, dotsc, an of positive integers, satisfying the fol