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A positive integer with k (decimal) digits is called balanced if its first lceil k/2 rceil digits sum to the same value

2^N binary digits can be placed in a circle so that all the N-digit clockwise subsequences are distinct.

Consider the values of log2(8), log4(64) and log3(27).

Let (a, b, c) represent the three sides of a right angle triangle with integral length sides.

A Harshad or Niven number is a number that is divisible by the sum of its digits.

The most naive way of computing n^{15} requires fourteen multiplications: But using a "binary" method you can compute it

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

The Golomb's self-describing sequence (G(n)) is the only nondecreasing sequence of natural numbers such that n appears e

Numbers of the form n^{15}+1 are composite for every integer n gt 1.

Assuming that two points are chosen randomly (with uniform distribution) within a rectangle, it is possible to determine

Let Sn be the regular n-sided polygon – or shape – whose vertices vk (k = 1, 2, dots, n) have coordinates: Each Sn is to

Find the number of integers 1 lt n lt 10^7, for which n and n + 1 have the same number of positive divisors.

A gozinta chain for n is a sequence 1,a,b,dots,n where each element properly divides the next.

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

A gozinta chain for n is a sequence 1,a,b,dots,n where each element properly divides the next.

Two players, Anton and Bernhard, are playing the following game.

Consider the infinite repeating sequence of digits: 1234321234321234321...

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

Let C(x, y) be a circle passing through the points (x, y), (x, y + 1), (x + 1, y) and (x + 1, y + 1).

It is possible to show that the square root of two can be expressed as an infinite continued fraction.

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

Let p(n) represent the number of different ways in which n coins can be separated into piles.

Given is the arithmetic-geometric sequence u(k) = (900-3k)r^{k - 1}.

Given an odd prime p, put the numbers 1,...,p-1 into frac{p-1}{2} pairs such that each number appears exactly once.

Let A and B be bit strings (sequences of 0's and 1's).

Define F(n) to be the number of integers x≤n that can be written in the form x=a^2b^3, where a and b are integers not ne

A touch-screen device can be unlocked with a "password" consisting of a sequence of two or more distinct spots that the

Consider the Gaussian integer i-1.

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

A composite number can be factored many different ways.

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

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

NOTE: This problem is a significantly more challenging version of Problem 81.

A regular hexagon table of side length N is divided into equilateral triangles of side length 1.

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

Any positive integer can be written as a product of prime powers: p1^{a1} times p2^{a2} times cdots times pk^{ak}, where

The look and say sequence goes 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...

An unbiased single 4-sided die is thrown and its value, T, is noted.

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

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

Larry and Robin play a memory game involving a sequence of random numbers between 1 and 10, inclusive, that are called o

The binomial coefficient displaystyle binom {10} 3 = 120.

Let d(p, n, 0) be the multiplicative inverse of n modulo prime p, defined as n times d(p, n, 0) = 1 bmod p.

We define 123-numbers as follows: - 1 is the smallest 123-number.

Jack has three plates in front of him.

Consider an equilateral triangle in which straight lines are drawn from each vertex to the middle of the opposite side,

Consider the Fibonacci sequence 1,2,3,5,8,13,21,ldots.

Alice enlists the help of some friends to generate a random number, using a single unfair coin.

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

A window into a matrix is a contiguous sub matrix.

You probably know the game Fifteen Puzzle.

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

A list initially contains the numbers 2, 3, dots, n.

Let T(n) be the nth triangle number, so T(n) = dfrac{n(n + 1)}{2}.

Define f(n) as the sum of the factorials of the digits of n.

Taking three different letters from the 26 letters of the alphabet, character strings of length three can be formed.

A triplicate number is a positive integer such that, after repeatedly removing three consecutive identical digits from i

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

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

If we take 47, reverse and add, 47 + 74 = 121, which is palindromic.

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

Given two positive integers a,b, Alex and Bianca play a game in ab rounds.

The RSA encryption is based on the following procedure: Generate two distinct primes p and q.

Every day for the past n days Even Stevens brings home his groceries in a plastic bag.

It is possible to find positive integers A and B such that given any triangular number, Tn, then ATn +B is always a tria

Many numbers can be expressed as the sum of a square and a cube.

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

The hyperexponentiation or tetration of a number a by a positive integer b, denoted by amathbin{uparrow uparrow}b or ^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

70 coloured balls are placed in an urn, 10 for each of the seven rainbow colours.

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

For any positive integer n, the nth weak Goodstein sequence g1, g2, g3, dots is defined as: - g1 = n - for k gt 1, gk is

For an integer n, we define the square prime factors of n to be the primes whose square divides n.

Let f(n) be the largest prime factor of n and displaystyle F(n) = sum{i=2}^n f(i).

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

Consider a circle where 2n distinct points have been marked on its circumference.

A line segment of length 2n-3 is randomly split into n segments of integer length (n ge 3).

A 10-substring of a number is a substring of its digits that sum to 10.

Let Bbb R^2 be the set of pairs of real numbers (x, y).

Let d(k) be the sum of all divisors of k.

Let f5(n) be the largest integer x for which 5^x divides n.

Starting with 1 gram of gold you play a game.

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

There are 5 ways to tile a regular dodecagon of side 1 with regular polygons of side 1.

Here are the records from a busy telephone system with one million users: | RecNr | Caller | Called | |:----------:|:---

The 5-digit number, 16807=7^5, is also a fifth power.

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

A positive integer is called square root smooth if all of its prime factors are strictly less than its square root.

The following is a function defined for all positive rational values of x.

For a positive integer n, d(n) is defined to be the sum of the digits of n.

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

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

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

Susan has a prime frog.

A pythagorean triangle with catheti a and b and hypotenuse c is characterized by the well-known equation a^2+b^2=c^2.

Let n be a positive integer.

Two players play a game with two piles of stones.

For some fixed rho in [0, 1], we begin a sum s at 0 and repeatedly apply a process: With probability rho, we add 1 to s,

The harmonic series 1 + frac 1 2 + frac 1 3 + frac 1 4 + cdots is well known to be divergent.