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A firecracker explodes at a height of pu{100 m} above level ground.

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

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

Let f(N) be the number of points with integer coordinates that are on a circle passing through (0,0), (N,0),(0,N), and (

A window into a matrix is a contiguous sub matrix.

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

Barbara is a mathematician and a basketball player.

Let S(n, k, b) represent the number of valid solutions to x1 + x2 + cdots + xk le n, where 0 le xm le b^m for all 1 le m

We say that a d-digit positive number (no leading zeros) is a one-child number if exactly one of its sub-strings is divi

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

For each integer p gt 1 coprime to 10 there is a positive divisibility multiplier m lt p which preserves divisibility by

The following undirected network consists of seven vertices and twelve edges with a total weight of 243.

For a positive integer n gt 1, let p(n) be the smallest prime dividing n, and let alpha(n) be its p-adic order, i.e.

You are given a pizza (perfect circle) that has been cut into m cdot n equal pieces and you want to have exactly one top

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

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

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

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

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

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

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

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

Let fn(k) = e^{k/n} - 1, for all non-negative integers k.

Alice and Bob play the game Nim Square.

The binomial coefficients displaystyle binom n k can be arranged in triangular form, Pascal's triangle, like this: | | |

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

On the parabola y = x^2/k, three points A(a, a^2/k), B(b, b^2/k) and C(c, c^2/k) are chosen.

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

For fixed integers a, b, c, define the crazy function F(n) as follows: F(n) = n - c for all n gt b F(n) = F(a + F(a + F(

A set of lattice points S is called a titanic set if there exists a line passing through exactly two points in S.

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

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

Bob is very familiar with the famous mathematical puzzle/game, "Tower of Hanoi," which consists of three upright rods an

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 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

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

Let us consider mixtures of three substances: A, B and C.

Given the set 1,2,dots,n, we define f(n, k) as the number of its k-element subsets with an odd sum of elements.

The nth harmonic number Hn is defined as the sum of the multiplicative inverses of the first n positive integers, and ca

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

A number consisting entirely of ones is called a repunit.

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

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

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

A number consisting entirely of ones is called a repunit.

A company specialises in producing large rectangular metal sheets, starting from unit square metal plates.

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

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

For any set A of numbers, let operatorname{sum}(A) be the sum of the elements of A.

For any N, let f(N) be the last twelve hexadecimal digits before the trailing zeroes in N!.

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

Phil the confectioner is making a new batch of chocolate covered candy.

Albert chooses a positive integer k, then two real numbers a, b are randomly chosen in the interval [0,1] with uniform d

Circles A and B are tangent to each other and to line L at three distinct points.

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

Using only a six-sided fair dice and a five-sided fair dice, we would like to emulate an n-sided fair dice.

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

Let F(r, c, n) be the number of ways to colour a rectangular grid with r rows and c columns using at most n colours such

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

For every n ge 1 the prime-counting function pi(n) is equal to the number of primes not exceeding n.

Consider the number 15.

We use xoplus y to be the bitwise XOR of x and y.

Using all of the digits 1 through 9 and concatenating them freely to form decimal integers, different sets can be formed

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

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

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

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

Consider a single game of Ramvok: Let t represent the maximum number of turns the game lasts.

Let Sk be the set containing 2 and 5 and the first k primes that end in 7.

By replacing the 1st digit of the 2-digit number 3, it turns out that six of the nine possible values: 13, 23, 43, 53, 7

Let n be a positive integer.

Mamma Triangolo baked a triangular pizza.

A certain type of tile comes in three different sizes - 1 times 1, 1 times 2, and 1 times 3 - and in four different colo

Given a non-square integer d, any real x can be approximated arbitrarily close by quadratic integers a+bsqrt{d}, where a

A 3-smooth number is an integer which has no prime factor larger than 3.

Let Sm = (x1, x2, dots , xm) be the m-tuple of positive real numbers with x1 + x2 + cdots + xm = m for which Pm = x1 cdo

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

The Carmichael function lambda(n) is defined as the smallest positive integer m such that a^m = 1 modulo n for all integ

The positive integral solutions of the equation x^y=y^x are (2,4), (4,2) and (k,k) for all k 0.

When wrapping several cubes in paper, it is more efficient to wrap them all together than to wrap each one individually.

The kernel of a polygon is defined by the set of points from which the entire polygon's boundary is visible.

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

The binomial coefficient displaystyle{binom{10^{18}}{10^9}} is a number with more than 9 billion (9times 10^9) digits.

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

Consider an infinite row of boxes.

A unit fraction contains 1 in the numerator.

Define f(0)=1 and f(n) to be the number of ways to write n as a sum of powers of 2 where no power occurs more than twice

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,

In this problem we consider triangles drawn on a hexagonal lattice, where each lattice point in the plane has six neighb

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

ABCD is a convex, integer sided quadrilateral with 1 le AB lt BC lt CD lt AD.

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

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

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

Both 169 and 961 are the square of a prime.

In Plato's heaven, there exist an infinite number of bowls in a straight line.

Let a0, a1, dots be an integer sequence defined by: - a0 = 1; - for n ge 1, an is the sum of the digits of all preceding

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

The game of Mahjong is played with tiles belonging to s suits.

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