brain

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

A common security method used for online banking is to ask the user for three random characters from a passcode.

For integers a and b, we define D(a, b) as the domain enclosed by the parabola y = x^2 and the line y = acdot x + b: D(a

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

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

Consider the divisors of 30: 1,2,3,5,6,10,15,30.

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

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

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

Consider the infinite polynomial series AG(x) = x G1 + x^2 G2 + x^3 G3 + cdots, where Gk is the kth term of the second o

Define d(n,k) to be the number of ways to write n as a product of k ordered integers Further define D(N,K) to be the sum

Consider the number 142857.

The Fibonacci numbers fn, n ge 0 are defined recursively as fn = f{n-1} + f{n-2} with base cases f0 = 0 and f1 = 1.

A Hilbert number is any positive integer of the form 4k+1 for integer kgeq 0.

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

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

Three distinct points are plotted at random on a Cartesian plane, for which -1000 le x, y le 1000, such that a triangle

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

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

The minimum number of cubes to cover every visible face on a cuboid measuring 3 times 2 times 1 is twenty-two.

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 =

A positive integer, n, is divided by d and the quotient and remainder are q and r respectively.

Alice and Bob are playing a modified game of Nim called Scatterstone Nim, with Alice going first, alternating turns with

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

Let f(n) be the largest prime factor of n.

Given the positive integers, x, y, and z, are consecutive terms of an arithmetic progression, the least value of the pos

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.

Find the number of non-empty subsets of 1^1, 2^2, 3^3,dots, 250250^{250250}, the sum of whose elements is divisible by 2

Let g(a, n, b, m) be the smallest non-negative solution x to the system: x = a bmod n x = b bmod m if such a solution ex

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

Let T(r) be the number of integer quadruplets x, y, z, t such that x^2 + y^2 + z^2 + t^2 le r^2.

Consider the number 50.

A positive integer will be called reachable if it can result from an arithmetic expression obeying the following rules:

Two friends A and B are great fans of Chess.

Let p = p1 p2 p3 cdots be an infinite sequence of random digits, selected from 0,1,2,3,4,5,6,7,8,9 with equal probabilit

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

There are N seats in a row.

For a non-negative integer k, the triple (p,q,r) of positive integers is called a k-shifted Pythagorean triple if (p, q,

Consider a directed graph made from an orthogonal lattice of Htimes W nodes.

Let D(m,n)=displaystylesum{dmid m}sum{k=1}^nsigma0(kd) where d runs through all divisors of m and sigma0(n) is the numbe

The radical of n, operatorname{rad}(n), is the product of distinct prime factors of n.

5-smooth numbers are numbers whose largest prime factor doesn't exceed 5.

A unitary divisor d of a number n is a divisor of n that has the property gcd(d, n/d) = 1.

Consider the number 45656.

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

Let S(n) be the sum of all positive integers m not exceeding n having the following property: a^{m + 4} equiv a pmod m f

For every real number a gt 1 is given the sequence ga by: g{a}(x)=1 for x lt a g{a}(x)=g{a}(x-1)+ga(x-a) for x ge a G(n)

A graph is made up of vertices and coloured edges.

Let p(t) denote the (t+1)th prime number.

Let r be the real root of the equation x^3 = x^2 + 1.

It is a common recreational problem to make a target number using a selection of other numbers.

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

Su Doku (Japanese meaning number place) is the name given to a popular puzzle concept.

Consider a rectangle made up of W times H square cells each with area 1.

Consider the triangle with sides sqrt 5, sqrt {65} and sqrt {68}.

A number consisting entirely of ones is called a repunit.

Let f(n) represent the number of ways one can fill a 3 times 3 times n tower with blocks of 2 times 1 times 1.

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

Let W(p,q,r) be the number of words that can be formed using the letter A p times, the letter B q times and the letter C

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

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.

Let F(n) be the number of connected graphs with blue edges (directed) and red edges (undirected) containing: - two verti

Two players are playing a game, alternating turns.

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

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

An integer of the form p^q q^p with prime numbers p neq q is called a hybrid-integer.

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^

Recall that a graph is a collection of vertices and edges connecting the vertices, and that two vertices connected by an

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

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

Consider the isosceles triangle with base length, b = 16, and legs, L = 17.

A rectangular sheet of grid paper with integer dimensions w times h is given.

For any integer n0 and prime number p, define nup(n) as the greatest integer r such that p^r divides n.

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

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

n families, each with four members, a father, a mother, a son and a daughter, were invited to a restaurant.

A modified Collatz sequence of integers is obtained from a starting value a1 in the following way: a{n+1} = frac {an} 3

A square is drawn around a circle as shown in the diagram below on the left.

The sum of the kth powers of the first n positive integers can be expressed as a polynomial of degree k+1 with rational

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

Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.

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

It turns out that pu{12 cm} is the smallest length of wire that can be bent to form an integer sided right angle triangl

Barbara is a mathematician and a basketball player.

Two positive integers x and y (x y) can generate a sequence in the following manner: - ax = y is the first term, - a{z+1

Let S(n) = sum a + b + c over all triples (a, b, c) such that: - a, b and c are prime numbers.

For 0 le x lt 1, define di(x) to be the ith digit after the binary point of the binary representation of x.

Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.

Let F5(n) be the number of strings s such that: - s consists only of '0's and '1's, - s has length at most n, and - s co

Let N and K be two positive integers.

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

"And he came towards a valley, through which ran a river; and the borders of the valley were wooded, and on each side of

Let (p1 p2 ldots pk) denote the permutation of the set {1, ..., k} that maps pimapsto i.

In a 3 times 2 cross-hatched grid, a total of 37 different rectangles could be situated within that grid as indicated in

Let phi be Euler's totient function, i.e.

Consider the set S(r) of points (x,y) with integer coordinates satisfying |x| + |y| le r.

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

Sam and Tom are trying a game of (partially) covering a given line segment of length L by taking turns in placing unit s

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

A number is p-smooth if it has no prime factors larger than p.