Found problems: 476
2016 Bosnia and Herzegovina Team Selection Test, 6
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
2017 Bosnia and Herzegovina Team Selection Test, Problem 2
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]
2008 Estonia Team Selection Test, 3
Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that
\[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}.
\]
[i]Author: Juhan Aru, Estonia[/i]
2001 IMO Shortlist, 7
A pile of $n$ pebbles is placed in a vertical column. This configuration is modified according to the following rules. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as a column with 0 pebbles.) At each stage, choose a pebble from among those that can be moved (if there are any) and place it at the top of the column to its right. If no pebbles can be moved, the configuration is called a [i]final configuration[/i]. For each $n$, show that, no matter what choices are made at each stage, the final configuration obtained is unique. Describe that configuration in terms of $n$.
[url=http://www.mathlinks.ro/Forum/viewtopic.php?p=119189]IMO ShortList 2001, combinatorics problem 7, alternative[/url]
1984 IMO Longlists, 20
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.
2000 IMO, 1
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.
2013 Brazil Team Selection Test, 2
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.
2020 USA TSTST, 4
Find all pairs of positive integers $(a,b)$ satisfying the following conditions:
[list]
[*] $a$ divides $b^4+1$,
[*] $b$ divides $a^4+1$,
[*] $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$.
[/list]
[i]Yang Liu[/i]
2018 USA TSTST, 1
As usual, let ${\mathbb Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : {\mathbb Z}[x] \to {\mathbb Z}$ such that for any polynomials $p,q \in {\mathbb Z}[x]$, [list] [*]$\theta(p+1) = \theta(p)+1$, and [*]if $\theta(p) \neq 0$ then $\theta(p)$ divides $\theta(p \cdot q)$. [/list]
[i]Evan Chen and Yang Liu[/i]
2016 USAJMO, 6
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
$$(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$$
2020 IMO Shortlist, G5
Let $ABCD$ be a cyclic quadrilateral. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ such that $KLMN$ is a rhombus with $KL \parallel AC$ and $LM \parallel BD$. Let $\omega_A, \omega_B, \omega_C, \omega_D$ be the incircles of $\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN$.
Prove that the common internal tangents to $\omega_A$, and $\omega_C$ and the common internal tangents to $\omega_B$ and $\omega_D$ are concurrent.
2016 Greece Team Selection Test, 3
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
2017 Estonia Team Selection Test, 8
Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$
[i]Proposed by Tigran Margaryan, Armenia[/i]
2022 USAMO, 6
There are $2022$ users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.)
Starting now, Mathbook will only allow a new friendship to be formed between two users if they have [i]at least two[/i] friends in common. What is the minimum number of friendships that must already exist so that every user could eventually become friends with every other user?
2017 EGMO, 2
Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties:
$(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$
$(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$
[i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]
2004 IMO Shortlist, 5
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]
2014 USAJMO, 3
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
2017 CCA Math Bonanza, I12
Let $a_1,a_2,\ldots,a_{2017}$ be the $2017$ distinct complex numbers which satisfy $a_i^{2017}=a_i+1$ for $i=1,2,\ldots,2017$. Compute $$\displaystyle\sum_{i=1}^{2017}\frac{a_i}{a_i^2+1}.$$
[i]2017 CCA Math Bonanza Individual Round #12[/i]
2014 India IMO Training Camp, 2
Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.
1997 USAMO, 5
Prove that, for all positive real numbers $ a$, $ b$, $ c$, the inequality
\[ \frac {1}{a^3 \plus{} b^3 \plus{} abc} \plus{} \frac {1}{b^3 \plus{} c^3 \plus{} abc} \plus{} \frac {1}{c^3 \plus{} a^3 \plus{} abc} \leq \frac {1}{abc}
\]
holds.
2015 USAMO, 6
Consider $0<\lambda<1$, and let $A$ be a multiset of positive integers. Let $A_n=\{a\in A: a\leq n\}$. Assume that for every $n\in\mathbb{N}$, the set $A_n$ contains at most $n\lambda$ numbers. Show that there are infinitely many $n\in\mathbb{N}$ for which the sum of the elements in $A_n$ is at most $\frac{n(n+1)}{2}\lambda$. (A multiset is a set-like collection of elements in which order is ignored, but repetition of elements is allowed and multiplicity of elements is significant. For example, multisets $\{1, 2, 3\}$ and $\{2, 1, 3\}$ are equivalent, but $\{1, 1, 2, 3\}$ and $\{1, 2, 3\}$ differ.)
2007 China Girls Math Olympiad, 5
Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.
2021 China National Olympiad, 4
In acute triangle $ABC (AB>AC)$, $M$ is the midpoint of minor arc $BC$, $O$ is the circumcenter of $(ABC)$ and $AK$ is its diameter. The line parallel to $AM$ through $O$ meets segment $AB$ at $D$, and $CA$ extended at $E$. Lines $BM$ and $CK$ meet at $P$, lines $BK$ and $CM$ meet at $Q$. Prove that $\angle OPB+\angle OEB =\angle OQC+\angle ODC$.
2003 IMO Shortlist, 6
Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.
2016 IMO Shortlist, A5
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.
(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.