Olympiad problems

2001 Austrian-Polish Competition, 2

Let $n$ be a positive integer greater than $2$. Solve in nonnegative real numbers the following system of equations \[x_{k}+x_{k+1}=x_{k+2}^{2}\quad , \quad k=1,2,\cdots,n\] where $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$.

2000 Korea - Final Round, 1

Tags: pigeonhole principle , quadratic , number theory

Prove that for any prime $p$, there exist integers $x,y,z,$ and $w$ such that $x^2+y^2+z^2-wp=0$ and $0<w<p$

2008 Bulgaria National Olympiad, 3

Tags: inequalities , trigonometry , modular arithmetic , induction , ceiling function , pigeonhole principle , algebra

Let $n\in\mathbb{N}$ and $0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi$ and $b_1,b_2,\ldots ,b_n$ are real numbers for which the following inequality is satisfied : \[\left|\sum_{i\equal{}1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\] for all $ k\in\mathbb{N}$. Prove that $ b_1\equal{}b_2\equal{}\ldots \equal{}b_n\equal{}0$.

2010 USAJMO, 2

Tags: pigeonhole principle , arithmetic sequence , induction

Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties: (a). $x_1 < x_2 < \cdots < x_{n-1}$ ; (b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$; (c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.

PEN P Problems, 6

Tags: function , inequalities , pigeonhole principle , floor function , additive number theory

Show that every integer greater than $1$ can be written as a sum of two square-free integers.

2012 Online Math Open Problems, 20

Tags: algorithm , inequalities , induction , pigeonhole principle , strong induction

The numbers $1, 2, \ldots, 2012$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number $N$ remains. Find the remainder when the maximum possible value of $N$ is divided by 1000. [i]Victor Wang.[/i]

1976 AMC 12/AHSME, 12

Tags: pigeonhole principle , probability , expected value

A supermarket has $128$ crates of apples. Each crate contains at least $120$ apples and at most $144$ apples. What is the largest integer $n$ such that there must be at least $n$ crates containing the same number of apples? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }24\qquad \textbf{(E) }25$

2005 All-Russian Olympiad, 2

Tags: analytic geometry , pigeonhole principle , geometry

Do there exist 12 rectangular parallelepipeds $P_1,\,P_2,\ldots,P_{12}$ with edges parallel to coordinate axes $OX,\,OY,\,OZ$ such that $P_i$ and $P_j$ have a common point iff $i\ne j\pm 1$ modulo 12?

2008 IMO Shortlist, 3

Tags: geometry , combinatorics , extremal combinatorics , point set , bezout s identity , pigeonhole principle

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

PEN O Problems, 9

Tags: pigeonhole principle , induction , absolute value

Let $n$ be an integer, and let $X$ be a set of $n+2$ integers each of absolute value at most $n$. Show that there exist three distinct numbers $a, b, c \in X$ such that $c=a+b$.

2009 Indonesia TST, 1

Tags: pigeonhole principle , search , number theory

a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime? b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?

2008 Mongolia Team Selection Test, 1

Tags: counting , derangement , pigeonhole principle , combinatorics

How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3?

2011 Romania Team Selection Test, 2

Tags: floor function , pigeonhole principle , arithmetic sequence , ramsey theory , number theory

Prove that the set $S=\{\lfloor n\pi\rfloor \mid n=0,1,2,3,\ldots\}$ contains arithmetic progressions of any finite length, but no infinite arithmetic progressions. [i]Vasile Pop[/i]

2009 Germany Team Selection Test, 1

Tags: geometry , combinatorics , extremal combinatorics , point set , bezout s identity , pigeonhole principle

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

2006 USAMO, 1

Tags: inequalities , floor function , modular arithmetic , pigeonhole principle , number theory

Let $p$ be a prime number and let $s$ be an integer with $0 < s < p.$ Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and \[ \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p} \] if and only if $s$ is not a divisor of $p-1$. Note: For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of x.

2018 AMC 10, 17

Tags: pigeonhole principle

Let $S$ be a set of 6 integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$? $\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7$

2014 Contests, 2

Tags: pigeonhole principle , global , combinatorics

Let $n \ge 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is [i]peaceful[/i] if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k \times k$ square which does not contain a rook on any of its $k^2$ unit squares.

2008 Argentina National Olympiad, 1

Tags: pigeonhole principle , modular arithmetic , number theory

$ 101$ positive integers are written on a line. Prove that we can write signs $ \plus{}$, signs $ \times$ and parenthesis between them, without changing the order of the numbers, in such a way that the resulting expression makes sense and the result is divisible by $ 16!$.

1988 USAMO, 3

Tags: pigeonhole principle , ceiling function , function , boolean lattice method

A function $f(S)$ assigns to each nine-element subset of $S$ of the set $\{1,2,\ldots, 20\}$ a whole number from $1$ to $20$. Prove that regardless of how the function $f$ is chosen, there will be a ten-element subset $T\subset\{1,2,\ldots, 20\}$ such that $f(T - \{k\})\neq k$ for all $k\in T$.

1993 India National Olympiad, 7

Tags: pigeonhole principle , combinatorics

Let $A = \{ 1,2, 3 , \ldots, 100 \}$ and $B$ be a subset of $A$ having $53$ elements. Show that $B$ has 2 distinct elements $x$ and $y$ whose sum is divisible by $11$.

2004 China Girls Math Olympiad, 8

Tags: pigeonhole principle , geometry , rectangle , combinatorics

When the unit squares at the four corners are removed from a three by three squares, the resulting shape is called a cross. What is the maximum number of non-overlapping crosses placed within the boundary of a $ 10\times 11$ chessboard? (Each cross covers exactly five unit squares on the board.)