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For a set of positive integers a, a+1, a+2, dots , b, let C(a,b) be the number of non-empty subsets in which the product

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

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

We are trying to find a hidden number selected from the set of integers 1, 2, dots, n by asking questions.

We call the convex area enclosed by two circles a lenticular hole if: - The centres of both circles are on lattice point

Let a, b and c be the sides of an integer sided triangle with one angle of 120 degrees, a le b le c and b-a le 100.

Every divisor d of a number n has a complementary divisor n/d.

A sliding block puzzle is a puzzle where pieces are confined to a grid and by sliding the pieces a final configuration i

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

Let ai be the sequence defined by ai=153^i bmod 10000019 for i ge 1.

The above is an example of a cryptic kakuro (also known as cross sums, or even sums cross) puzzle, with its final soluti

2^7=128 is the first power of two whose leading digits are "12".

Let us define a balanced sculpture of order n as follows: - A polyominoAn arrangement of identical squares connected thr

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

A triomino is a shape consisting of three squares joined via the edges.

If a triple of positive integers (a, b, c) satisfies a^2+b^2=c^2, it is called a Pythagorean triple.

Let ABCD be a convex quadrilateral, with diagonals AC and BD.

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

At Euler University, each of the n students (numbered from 1 to n) occupies a bed in the dormitory and uses a desk in th

Rand48 is a pseudorandom number generator used by some programming languages.

Consider the real number sqrt 2 + sqrt 3.

The first 15 Fibonacci numbers are: 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610.

Given the values of integers 1 < a1 < a2 < dots < an, consider the linear combination q1 a1+q2 a2 + dots + qn an=b, usin

For a positive integer d, let f(d) be the number created by sorting the digits of d in ascending order, removing any zer

Let f(n) be the number of 6-tuples (x1,x2,x3,x4,x5,x6) such that: - All xi are integers with 0 leq xi < n - gcd(x1^2+x2^

How many 18-digit numbers n (without leading zeros) are there such that no digit occurs more than three times in n?

Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are

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

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

The function f is defined for all positive integers as follows: - f(1)=1 - f(3)=3 - f(2n)=f(n) - f(4n + 1)=2f(2n + 1) -

A group of p people decide to sit down at a round table and play a lottery-ticket trading game.

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

Consider the infinite integer sequence S starting with: S = 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 8, 4, 9, 1, 10, 11, 5

A definition for an ellipse is: Given a circle c with centre M and radius r and a point G such that d(G,M) lt r, the loc

Consider the following binary quadratic form: A positive integer q has a primitive representation if there exist positiv

Denote the average of k numbers x1, ..., xk by bar{x} = frac{1}{k} sumi xi.

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

The number sequence game starts with a sequence S of N numbers written on a line.

Consider a function f(k) defined for all positive integers k0.

There are 1111 ways in which five 6-sided dice (sides numbered 1 to 6) can be rolled so that the top three sum to 15.

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

A partition of n is a set of positive integers for which the sum equals n.

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

A root or zero of a polynomial P(x) is a solution to the equation P(x) = 0.

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

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

Let F(N) be the maximum number of lattice points in an axis-aligned Ntimes N square that the graph of a single strictly

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

Peter moves in a hallway with N + 1 doors consecutively numbered from 0 through N.

For a prime p let S(p) = (sum (p-k)!) bmod (p) for 1 le k le 5.

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

Consider all the triangles having: - All their vertices on lattice pointsInteger coordinates.

Coprime Nim is just like ordinary normal play Nim, but the players may only remove a number of stones from a pile that i

Consider the triangle with sides 6, 8, and 10.

Let S be the set consisting of the four letters texttt{A'},texttt{E'},texttt{F'},texttt{R'}.

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

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

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

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

Let R(a, b, c) be the maximum area covered by three non-overlapping circles inside a triangle with edge lengths a, b and

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.

A number consisting entirely of ones is called a repunit.

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

Let sigma(n) be the sum of all the divisors of the positive integer n, for example: sigma(10) = 1+2+5+10 = 18.

A row measuring seven units in length has red blocks with a minimum length of three units placed on it, such that any tw

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

Given a set, L, of unique lines, let M(L) be the number of lines in the set and let S(L) be the sum over every line of t

We define a block to be a rectangle with a height of 1 and an integer-valued length.

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

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

When we calculate 8^n modulo 11 for n=0 to 9 we get: 1, 8, 9, 6, 4, 10, 3, 2, 5, 7.

By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top

Let n be an integer and S(n) be the set of factors of n.

A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle.

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

For a positive integer n, define f(n) to be the number of non-empty substrings of n that are divisible by 3.

Any positive real number x can be decomposed into integer and fractional parts lfloor x rfloor + x, where lfloor x rfloo

A composite is a number containing at least two prime factors.

A sequence is defined as: - gk = 1, for 0 le k le 1999 - gk = g{k-2000} + g{k - 1999}, for k ge 2000.

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 -

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

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

Adam plays the following game with his birthday cake.

A positive integer n is powerful if p^2 is a divisor of n for every prime factor p in n.

Christopher Robin and Pooh Bear love the game of Poohsticks so much that they invented a new version which allows them t

A dynamical polynomial is a monicleading coefficient is 1 polynomial f(x) with integer coefficients such that f(x) divid

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

We call a natural number a duodigit if its decimal representation uses no more than two different digits.

Given n and k two positive integers we begin with an urn that contains kn white balls.

The Pythagorean tree is a fractal generated by the following procedure: Start with a unit square.

In the game of darts a player throws three darts at a target board which is split into twenty equal sized sections numbe

We want to tile a board of length n and height 1 completely, with either 1 times 2 blocks or 1 times 1 blocks with a sin

Consider the following variant of "The Chase" game.

Consider the number 15.

Define the sequence a1, a2, a3, dots as: - a1 = 1 - a{n+1} = 6an^2 + 10an + 3 for n ge 1.

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

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

Let f(n) be the number of ways an ntimes n square grid can be coloured, each cell either black or white, such that each

Consider a three dimensional grid of cubes.