This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 396

2024 AMC 10, 12
Tags: pigeonhole principle
A group of $100$ students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students $A$ and $B$, student $A$ speaks some language that student $B$ does not speak, and student $B$ speaks some language that student $A$ does not speak. What is the least possible total number of languages spoken by all the students? $ \textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }12 \qquad \textbf{(D) }51 \qquad \textbf{(E) }100 \qquad $

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?

1978 IMO Shortlist, 1
Tags: pigeonhole principle , partition , set systems , extremal combinatorics , combinatorics
The set $M = \{1, 2, . . . , 2n\}$ is partitioned into $k$ nonintersecting subsets $M_1,M_2, \dots, M_k,$ where $n \ge k^3 + k.$ Prove that there exist even numbers $2j_1, 2j_2, \dots, 2j_{k+1}$ in $M$ that are in one and the same subset $M_i$ $(1 \le i \le k)$ such that the numbers $2j_1 - 1, 2j_2 - 1, \dots, 2j_{k+1} - 1$ are also in one and the same subset $M_j (1 \le j \le k).$

2002 All-Russian Olympiad, 2
Tags: induction , pigeonhole principle , combinatorics
We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$. Initially, the pack is situated on the red cell and arranged in an arbitrary order. In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell. Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?

2003 Manhattan Mathematical Olympiad, 4
Tags: pigeonhole principle
Prove that from any set of one hundred different whole numbers one can choose either one number which is divisible by $100$, or several numbers whose sum is divisible by $100$.

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!$.

2007 Federal Competition For Advanced Students, Part 2, 1
Tags: polynomial , pigeonhole principle , absolute value , algebra
Let $ M$ be the set of all polynomials $ P(x)$ with pairwise distinct integer roots, integer coefficients and all absolut values of the coefficients less than $ 2007$. Which is the highest degree among all the polynomials of the set $ M$?

1985 IMO Longlists, 49
Tags: pigeonhole principle , combinatorics , mean
Given a set $M$ of $1985$ positive integers, none of which has a prime divisor larger than $26$, prove that the set has four distinct elements whose geometric mean is an integer.

2012 China Girls Math Olympiad, 4
Tags: geometry , trapezoid , pigeonhole principle , inequalities , combinatorics
There is a stone at each vertex of a given regular $13$-gon, and the color of each stone is black or white. Prove that we may exchange the position of two stones such that the coloring of these stones are symmetric with respect to some symmetric axis of the $13$-gon.

2004 Romania Team Selection Test, 9
Tags: pigeonhole principle , linear algebra , matrix , inequalities , combinatorics
Let $n\geq 2$ be a positive integer, and $X$ a set with $n$ elements. Let $A_{1},A_{2},\ldots,A_{101}$ be subsets of $X$ such that the union of any $50$ of them has more than $\frac{50}{51}n$ elements. Prove that among these $101$ subsets there exist $3$ subsets such that any two of them have a common element.

2024 Auckland Mathematical Olympiad, 8
Tags: pigeonhole principle
There are $25$ points on the plane, and among any three of them there are two at a distance less than $1$. Prove that there is a circle of radius $1$ containing at least $13$ of these points.

2006 Mathematics for Its Sake, 1
Tags: geometry , area , pigeonhole principle , combinatorics
Let be the points $ K,L,M $ on the sides $ BC,CA,AB, $ respectively, of a triangle $ ABC. $ Show that at least one of the areas of the triangles $ MAL,KBM,LCK $ doesn't surpass a fourth of the area of $ ABC. $

1978 IMO Longlists, 1
Tags: pigeonhole principle , combinatorics , partition , set systems , extremal combinatorics
The set $M = \{1, 2, . . . , 2n\}$ is partitioned into $k$ nonintersecting subsets $M_1,M_2, \dots, M_k,$ where $n \ge k^3 + k.$ Prove that there exist even numbers $2j_1, 2j_2, \dots, 2j_{k+1}$ in $M$ that are in one and the same subset $M_i$ $(1 \le i \le k)$ such that the numbers $2j_1 - 1, 2j_2 - 1, \dots, 2j_{k+1} - 1$ are also in one and the same subset $M_j (1 \le j \le k).$

2021 Thailand Mathematical Olympiad, 4
Tags: combinatorics , pigeonhole principle
Kan Krao Park is a circular park that has $21$ entrances and a straight line walkway joining each pair of two entrances. No three walkways meet at a single point. Some walkways are paved with bricks, while others are paved with asphalt. At each intersection of two walkways, excluding the entrances, is planted lotus if the two walkways are paved with the same material, and is planted waterlily if the two walkways are paved with different materials. Each walkway is decorated with lights if and only if the same type of plant is placed at least $45$ different points along that walkway. Prove that there are at least $11$ walkways decorated with lights and paved with the same material.

2007 APMO, 1
Tags: geometry , analytic geometry , vector , pigeonhole principle , modular arithmetic , combinatorics
Let $S$ be a set of $9$ distinct integers all of whose prime factors are at most $3.$ Prove that $S$ contains $3$ distinct integers such that their product is a perfect cube.

PEN A Problems, 63
Tags: induction , pigeonhole principle , divisibility theory
There is a large pile of cards. On each card one of the numbers $1$, $2$, $\cdots$, $n$ is written. It is known that the sum of all numbers of all the cards is equal to $k \cdot n!$ for some integer $k$. Prove that it is possible to arrange cards into $k$ stacks so that the sum of numbers written on the cards in each stack is equal to $n!$.

2003 Balkan MO, 1
Tags: pigeonhole principle , algebra , polynomial , induction , number theory , greatest common divisor
Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?

2014 Contests, 2
Tags: pigeonhole principle , number theory
Prove that among any $16$ perfect cubes we can always find two cubes whose difference is divisible by $91$.

2000 Kurschak Competition, 3
Tags: pigeonhole principle , modular arithmetic , number theory
Let $k\ge 0$ be an integer and suppose the integers $a_1,a_2,\dots,a_n$ give at least $2k$ different residues upon division by $(n+k)$. Show that there are some $a_i$ whose sum is divisible by $n+k$.

1997 All-Russian Olympiad, 4
Tags: geometry , geometric transformation , pigeonhole principle , combinatorics
The numbers from $1$ to $100$ are arranged in a $10\times 10$ table so that any two adjacent numbers have sum no larger than $S$. Find the least value of $S$ for which this is possible. [i]D. Hramtsov[/i]

2014 APMO, 4
Tags: pigeonhole principle , floor function , number theory , algebra , combinatorics
Let $n$ and $b$ be positive integers. We say $n$ is $b$-discerning if there exists a set consisting of $n$ different positive integers less than $b$ that has no two different subsets $U$ and $V$ such that the sum of all elements in $U$ equals the sum of all elements in $V$. (a) Prove that $8$ is $100$-discerning. (b) Prove that $9$ is not $100$-discerning. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

2019 Bangladesh Mathematical Olympiad, 8
Tags: pigeonhole principle , combinatorics
The set of natural numbers $\mathbb{N}$ are partitioned into a finite number of subsets.Prove that there exists a subset of $S$ so that for any natural numbers $n$,there are infinitely many multiples of $n$ in $S$.

2013 Moldova Team Selection Test, 2
Tags: graph theory , pigeonhole principle , combinatorics
Let $a_n=1+n!(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!})$ for any $n\in \mathbb{Z}^{+}$. Consider $a_n$ points in the plane,no $3$ of them collinear.The segments between any $2$ of them are colored in one of $n$ colors. Prove that among them there exist $3$ points forming a monochromatic triangle.

2001 AIME Problems, 14
Tags: combinatorics , pigeonhole principle , recursion
A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible?

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]