Olympiad problems

Russian TST 2017, P2

Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

2010 USA Team Selection Test, 8

Tags: induction , algebra , polynomial , combinatorics , Hi

Let $m,n$ be positive integers with $m \geq n$, and let $S$ be the set of all $n$-term sequences of positive integers $(a_1, a_2, \ldots a_n)$ such that $a_1 + a_2 + \cdots + a_n = m$. Show that \[\sum_S 1^{a_1} 2^{a_2} \cdots n^{a_n} = {n \choose n} n^m - {n \choose n-1} (n-1)^m + \cdots + (-1)^{n-2} {n \choose 2} 2^m + (-1)^{n-1} {n \choose 1}.\]

2020 Taiwan TST Round 2, 3

Tags: combinatorics , Taiwan , Hi

There are $N$ acute triangles on the plane. Their vertices are all integer points, their areas are all equal to $2^{2020}$, but no two of them are congruent. Find the maximum possible value of $N$. Note: $(x,y)$ is an integer point if and only if $x$ and $y$ are both integers. [i]Proposed by CSJL[/i]

2017 USAMO, 1

Tags: number theory , USAJMO , USAMO , 2017 USAMO , Hi

Prove that there are infinitely many distinct pairs $(a, b)$ of relatively prime integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.

2018 China Team Selection Test, 2

Tags: number theory , arithmetic series , number of divisors , China TST , Hi

A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.

2017 Bosnia Herzegovina Team Selection Test, 2

Tags: number theory , IMO Shortlist , function , Divisibility , Hi

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2016 China Team Selection Test, 4

Tags: number theory , floor function , Hi

Set positive integer $m=2^k\cdot t$, where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$. Prove that for any positive integer $n$ and for any positive odd number $a\le n$, $\prod_{m=1}^n f(m)$ is a multiple of $a$.

2013 ELMO Shortlist, 8

Tags: combinatorics , Chip firing , Hi

There are 20 people at a party. Each person holds some number of coins. Every minute, each person who has at least 19 coins simultaneously gives one coin to every other person at the party. (So, it is possible that $A$ gives $B$ a coin and $B$ gives $A$ a coin at the same time.) Suppose that this process continues indefinitely. That is, for any positive integer $n$, there exists a person who will give away coins during the $n$th minute. What is the smallest number of coins that could be at the party? [i]Proposed by Ray Li[/i]

2023 USAJMO, 1

Tags: USAJMO , Hi

Find all triples of positive integers $(x,y,z)$ that satisfy the equation $$2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023.$$

2000 USA Team Selection Test, 2

Tags: geometry , cyclic quadrilateral , geometric transformation , geometry solved , USA TST , steiner line , Hi

Let $ ABCD$ be a cyclic quadrilateral and let $ E$ and $ F$ be the feet of perpendiculars from the intersection of diagonals $ AC$ and $ BD$ to $ AB$ and $ CD$, respectively. Prove that $ EF$ is perpendicular to the line through the midpoints of $ AD$ and $ BC$.

2000 Belarus Team Selection Test, 8.2

Tags: modular arithmetic , number theory , Integer sequence , Divisibility , Sequence , IMO Shortlist , Hi

Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.

2022 Germany Team Selection Test, 1

Tags: algebra , real analysis , inequalities , Hi

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

2018 All-Russian Olympiad, 3

Tags: combinatorics , Hi

A positive integer $k$ is given. Initially, $N$ cells are marked on an infinite checkered plane. We say that the cross of a cell $A$ is the set of all cells lying in the same row or in the same column as $A$. By a turn, it is allowed to mark an unmarked cell $A$ if the cross of $A$ contains at least $k$ marked cells. It appears that every cell can be marked in a sequence of such turns. Determine the smallest possible value of $N$.

2022 Germany Team Selection Test, 1

Tags: algebra , real analysis , inequalities , Hi

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

2012 ELMO Shortlist, 1

Tags: geometry , circumcircle , power of a point , radical axis , cyclic quadrilateral , Elmo , Hi

In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$. [i]Ray Li.[/i]

2011 USAMO, 6

Tags: USA(J)MO , USAMO , linear algebra , matrix , inequalities , function , Hi

Let $A$ be a set with $|A|=225$, meaning that $A$ has 225 elements. Suppose further that there are eleven subsets $A_1, \ldots, A_{11}$ of $A$ such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$ for $1\leq i<j\leq11$. Prove that $|A_1\cup A_2\cup\ldots\cup A_{11}|\geq 165$, and give an example for which equality holds.

2022 Taiwan TST Round 1, 2

Tags: number theory , IMO Shortlist , Divisibility , cubefree , Hi

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

2020 Thailand TST, 1

Tags: combinatorics , binomial coefficients , IMO Shortlist , IMO Shortlist 2019 , trivial , Hi

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

2017 ELMO Shortlist, 1

Tags: number theory , greatest common divisor , ELMO 2017 , Elmo , Hi

Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$ [i]Proposed by Daniel Liu[/i]

2022 Thailand TSTST, 2

Tags: number theory , IMO Shortlist , Divisibility , cubefree , Hi

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

2005 IMO Shortlist, 6

Tags: modular arithmetic , number theory , Divisibility , IMO Shortlist , Chinese Remainder Theorem , Hi

Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]

2022 Greece Team Selection Test, 1

Tags: number theory , IMO Shortlist , Divisibility , cubefree , Hi

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

2005 IMO Shortlist, 3

Tags: algebra , polynomial , number theory , composite number , prime numbers , IMO Shortlist , Hi

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

2023 USAMO, 3

Tags: AMC , USA(J)MO , USAMO , Hi

Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a [i]maximal grid-aligned configuration[/i] on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find all possible values of $k(C)$ as a function of $n$. [i]Proposed by Holden Mui[/i]

2012 USA TSTST, 5

Tags: pigeonhole principle , modular arithmetic , number theory , relatively prime , CIGARETTES , Hi

A rational number $x$ is given. Prove that there exists a sequence $x_0, x_1, x_2, \ldots$ of rational numbers with the following properties: (a) $x_0=x$; (b) for every $n\ge1$, either $x_n = 2x_{n-1}$ or $x_n = 2x_{n-1} + \textstyle\frac{1}{n}$; (c) $x_n$ is an integer for some $n$.