Olympiad problems

2015 India National Olympiad, 4

There are four basketball players $A,B,C,D$. Initially the ball is with $A$. The ball is always passed from one person to a different person. In how many ways can the ball come back to $A$ after $\textbf{seven}$ moves? (for example $A\rightarrow C\rightarrow B\rightarrow D\rightarrow A\rightarrow B\rightarrow C\rightarrow A$, or $A\rightarrow D\rightarrow A\rightarrow D\rightarrow C\rightarrow A\rightarrow B\rightarrow A)$.

2021 IMO Shortlist, C2

Tags: pigeonhole principle , c2

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

2011 Croatia Team Selection Test, 1

Tags: modular arithmetic , pigeonhole principle , induction , algebra

We define a sequence $a_n$ so that $a_0=1$ and \[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + d & \textrm{ otherwise. } \end{cases} \] for all postive integers $n$. Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$.

2010 Contests, 1

Tags: pigeonhole principle , ceiling function , combinatorics

The integer number $n > 1$ is given and a set $S \subset \{0, 1, 2, \ldots, n-1\}$ with $|S| > \frac{3}{4} n$. Prove that there exist integer numbers $a, b, c$ such that the remainders after the division by $n$ of the numbers: \[a, b, c, a+b, b+c, c+a, a+b+c\] belong to $S$.

2010 Romanian Masters In Mathematics, 6

Tags: algebra , polynomial , induction , pigeonhole principle , number theory , relatively prime

Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$. Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set. [i]Dan Schwarz, Romania[/i]

1998 IMO Shortlist, 3

Tags: modular arithmetic , pigeonhole principle , number theory , divisibility , combinatorics

Determine the smallest integer $n\geq 4$ for which one can choose four different numbers $a,b,c$ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$.

2004 Baltic Way, 9

Tags: pigeonhole principle , number theory

A set $S$ of $n-1$ natural numbers is given ($n\ge 3$). There exist at least at least two elements in this set whose difference is not divisible by $n$. Prove that it is possible to choose a non-empty subset of $S$ so that the sum of its elements is divisible by $n$.

2010 Czech-Polish-Slovak Match, 1

Tags: pigeonhole principle , system of equations , algebra

Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations \[ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c. \]

2005 Hong kong National Olympiad, 1

Tags: pigeonhole principle , induction , combinatorics

On a planet there are $3\times2005!$ aliens and $2005$ languages. Each pair of aliens communicates with each other in exactly one language. Show that there are $3$ aliens who communicate with each other in one common language.

2003 Poland - Second Round, 4

Tags: number theory , combinatorics , combinatorial number theory , pigeonhole principle , prime number

Prove that for any prime number $p > 3$ exist integers $x, y, k$ that meet conditions: $0 < 2k < p$ and $kp + 3 = x^2 + y^2$.

2014 Contests, 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]

2014 USAJMO, 4

Tags: function , pigeonhole principle , inequalities

Let $b\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$. Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer.

2002 Manhattan Mathematical Olympiad, 2

Tags: pigeonhole principle , number theory , relatively prime

Let us consider the sequence $1,2, 3, \ldots , 2002$. Somebody choses $1002$ numbers from the sequence. Prove that there are two of the chosen numbers which are relatively prime (i.e. do not have any common divisors except $1$).

2022 Taiwan TST Round 1, C

Tags: pigeonhole principle , c2

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

2003 USA Team Selection Test, 1

Tags: pigeonhole principle , floor function , algebra

For a pair of integers $a$ and $b$, with $0 < a < b < 1000$, set $S\subseteq \{ 1, 2, \dots , 2003\}$ is called a [i]skipping set[/i] for $(a, b)$ if for any pair of elements $s_1, s_2 \in S$, $|s_1 - s_2|\not\in \{ a, b\}$. Let $f(a, b)$ be the maximum size of a skipping set for $(a, b)$. Determine the maximum and minimum values of $f$.

1993 IberoAmerican, 2

Tags: pigeonhole principle , geometry , circumcircle , rhombus , perpendicular bisector , combinatorics

Let $P$ and $Q$ be two distinct points in the plane. Let us denote by $m(PQ)$ the segment bisector of $PQ$. Let $S$ be a finite subset of the plane, with more than one element, that satisfies the following properties: (i) If $P$ and $Q$ are in $S$, then $m(PQ)$ intersects $S$. (ii) If $P_1Q_1, P_2Q_2, P_3Q_3$ are three diferent segments such that its endpoints are points of $S$, then, there is non point in $S$ such that it intersects the three lines $m(P_1Q_1)$, $m(P_2Q_2)$, and $m(P_3Q_3)$. Find the number of points that $S$ may contain.

2014 India IMO Training Camp, 1

Tags: binomial theorem , pigeonhole principle , algebra

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

2012 Canadian Mathematical Olympiad Qualification Repechage, 1

Tags: pigeonhole principle , combinatorics

The front row of a movie theatre contains $45$ seats. [list] [*] (a) If $42$ people are sitting in the front row, prove that there are $10$ consecutive seats that are all occupied. [*] (b) Show that this conclusion doesn’t necessarily hold if only $41$ people are sitting in the front row.[/list]

2001 USA Team Selection Test, 3

Tags: induction , pigeonhole principle , combinatorics , probabilistic method

For a set $S$, let $|S|$ denote the number of elements in $S$. Let $A$ be a set of positive integers with $|A| = 2001$. Prove that there exists a set $B$ such that (i) $B \subseteq A$; (ii) $|B| \ge 668$; (iii) for any $u, v \in B$ (not necessarily distinct), $u+v \not\in B$.

2014 India IMO Training Camp, 1

Tags: algebra , binomial theorem , pigeonhole principle

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

2012 Turkey Junior National Olympiad, 4

Tags: inequalities , induction , pigeonhole principle , geometry , geometric transformation , combinatorics

We want to place $2012$ pockets, including variously colored balls, into $k$ boxes such that [b]i)[/b] For any box, all pockets in this box must include a ball with the same color or [b]ii)[/b] For any box, all pockets in this box must include a ball having a color which is not included in any other pocket in this box Find the smallest value of $k$ for which we can always do this placement whatever the number of balls in the pockets and whatever the colors of balls.

2005 USAMO, 6

Tags: induction , modular arithmetic , ceiling function , floor function , pigeonhole principle , logarithm , inequalities

For $m$ a positive integer, let $s(m)$ be the sum of the digits of $m$. For $n\ge 2$, let $f(n)$ be the minimal $k$ for which there exists a set $S$ of $n$ positive integers such that $s\left(\sum_{x\in X} x\right)=k$ for any nonempty subset $X\subset S$. Prove that there are constants $0<C_1<C_2$ with \[C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.\]

2009 Iran Team Selection Test, 6

Tags: function , geometry , pigeonhole principle , combinatorics

We have a closed path on a vertices of a $ n$×$ n$ square which pass from each vertice exactly once . prove that we have two adjacent vertices such that if we cut the path from these points then length of each pieces is not less than quarter of total path .

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.

2008 Brazil National Olympiad, 1

Tags: pigeonhole principle , number theory

A positive integer is [i]dapper[/i] if at least one of its multiples begins with $ 2008$. For example, $ 7$ is dapper because $ 200858$ is a multiple of $ 7$ and begins with $ 2008$. Observe that $ 200858 \equal{} 28694\times 7$. Prove that every positive integer is dapper.