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

A lattice cube is a cube in which all vertices have integer coordinates.

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

Given a circle C and an integer n 1, we perform the following operations.

Let displaystyle S(n)=sumlimits{k=0}^{n}binom{n}{k}k^n.

A set of disks numbered 1 through 100 are placed in a line in random order.

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

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

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

For a positive integer k, define d(k) as the sum of the digits of k in its usual decimal representation.

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

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

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.

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

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

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

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

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 a1, a2, dots, an be an integer sequence of length n such that: - a1 = 6 - for all 1 le i lt n: phi(ai) lt phi(a{i +

Let tk be the tribonacci numbers defined as: quad t0 = t1 = 0; quad t2 = 1; quad tk = t{k-1} + t{k-2} + t{k-3} quad text

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

The Fibonacci sequence is defined by the recurrence relation: Fn = F{n - 1} + F{n - 2}, where F1 = 1 and F2 = 1.

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.

Define F(m,n) as the number of n-tuples of positive integers for which the product of the elements doesn't exceed m.

Let's call an integer sided triangle with exactly one angle of 60 degrees a 60-degree triangle.

Consider the number 142857.

Let ABCD be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows: A(a, 0), B(0, b),

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

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

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

Consider the fraction, dfrac n d, where n and d are positive integers.

The blancmange curve is the set of points (x, y) such that 0 le x le 1 and y = sum limits{n = 0}^{infty} {dfrac{s(2^n x)

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

Using a combination of grey square tiles and oblong tiles chosen from: red tiles (measuring two units), green tiles (mea

Triangle numbers Tk are integers of the form frac{k(k+1)} 2.

An n times n grid of squares contains n^2 ants, one ant per square.

Alice and Bob have enjoyed playing Nim every day.

Let d(n) be the number of divisors of n.

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

Let's call S the (infinite) string that is made by concatenating the consecutive positive integers (starting from 1) wri

Consider the region constrained by 1 le x and 0 le y le 1/x.

It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different orde

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

An integer partition of a number n is a way of writing n as a sum of positive integers.

The numbers 545, 5995 and 15151 are the three smallest palindromes divisible by 109.

Let f be a function from a finite set S to itself.

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

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

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

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.

A rectilinear grid is an orthogonal grid where the spacing between the gridlines does not have to be equidistant.

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

It is possible to write five as a sum in exactly six different ways: How many different ways can one hundred be written

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

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

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.

k defects are randomly distributed amongst n integrated-circuit chips produced by a factory (any number of defects may b

"What? Where? When?" is a TV game show in which a team of experts attempt to answer questions.

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

This problem involves an iterative procedure that begins with a circle of nge 3 integers.

For a number written in Roman numerals to be considered valid there are basic rules which must be followed.

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

Let f(n) be the largest positive integer x less than 10^9 such that the last 9 digits of n^x form the number x (includin

For any prime p the number N(p, q) is defined by N(p, q) = sum{n = 0}^q Tn cdot p^n with Tn generated by the following r

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

We wish to tile a rectangle whose length is twice its width.

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

Consider the number 45656.

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

There is a grid of length and width 50515093 points.

A geoboard (of order N) is a square board with equally-spaced pins protruding from the surface, representing an integer

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

Let W(n,k) be the number of ways in which n can be written as the product of k distinct positive integers.

For positive integers n and m, we define two polynomials Fn(x) = x^n and Gm(x) = (x-1)^m.

The smallest positive integer n for which the numbers n^2 + 1, n^2 + 3, n^2 + 7, n^2 + 9, n^2 + 13, and n^2 + 27 are con

A game is played with many identical, round coins on a flat table.

For positive real numbers a,b, an atimes b torus is a rectangle of width a and height b, with left and right sides ident

The first number n for which phi(n)=13! is 6227180929.

Inside a rope of length n, n - 1 points are placed with distance 1 from each other and from the endpoints.

A robot moves in a series of one-fifth circular arcs (72^circ), with a free choice of a clockwise or an anticlockwise ar

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

12n musicians participate at a music festival.

Let S = 2, 3, 5, dots, 4999 be the set of prime numbers less than 5000.

Fermat's Last Theorem states that no three positive integers a, b, c satisfy the equation for any integer value of n gre

The inversion count of a sequence of digits is the smallest number of adjacent pairs that must be swapped to sort the se

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

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

Suppliers 'A' and 'B' provided the following numbers of products for the luxury hamper market: | Product | 'A' | 'B' | |

Consider the fraction, dfrac n d, where n and d are positive integers.

Consider the natural numbers having at least 5 prime factors, which don't have to be distinct.

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

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

The complexity of an ntimes n binary matrix is the number of distinct rows and columns.

Let us call an integer sided triangle with sides a le b le c barely acute if the sides satisfy a^2 + b^2 = c^2 + 1.

Let's call a lattice point (x, y) inadmissible if x, y and x+y are all positive perfect squares.

Consider equations of the form: a^2 + b^2 = N, 0 le a le b, a, b and N integer.

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

We define two sequences S = S(1), S(2), ..., S(n) and S2 = S2(1), S2(2), ..., S2(n): S(k) = (pk)^k bmod 10007 where pk i

For a positive integer n, let sigma2(n) be the sum of the squares of its divisors.