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.

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Found problems: 64

2022-IMOC, N4
Tags: number theory , IMOC
Find all pair of positive integers $(m,n)$ such that $$mn(m^2+6mn+n^2)$$is a perfect square. [i]Proposed by Li4 and Untro368[/i]

2022-IMOC, C1
Tags: combinatorics , IMOC
Given a positive integer $k$, a pigeon and a seagull play a game on an $n\times n$ board. The pigeon goes first, and they take turns doing the operations. The pigeon will choose $m$ grids and lay an egg in each grid he chooses. The seagull will choose a $k\times k$ grids and eat all the eggs inside them. If at any point every grid in the $n\times n $ board has an egg in it, then the pigeon wins. Else, the seagull wins. For every integer $n\geq k$, find all $m$ such that the pigeon wins. [i]Proposed by amano_hina[/i]

2019-IMOC, N2
Tags: number theory , IMOC
Find all pairs of positive integers $(m, n)$ such that $$m^n * n^m = m^m + n^n$$

2021-IMOC, C11
Tags: combinatorics , grid , IMOC
In an $m \times n$ grid, each square is either filled or not filled. For each square, its [i]value[/i] is defined as $0$ if it is filled and is defined as the number of neighbouring filled cells if it is not filled. Here, two squares are neighbouring if they share a common vertex or side. Let $f(m,n)$ be the largest total value of squares in the grid. Determine the minimal real constant $C$ such that $$\frac{f(m,n)}{mn} \le C$$holds for any positive integers $m,n$ [i]CSJL[/i]

2019-IMOC, C4
Tags: combinatorics , IMOC
Determine the largest $k$ such that for all competitive graph with $2019$ points, if the difference between in-degree and out-degree of any point is less than or equal to $k$, then this graph is strongly connected.

2020-IMOC, A5
Tags: inequalities , IMOC
Let $0<c<1$ be a given real number. Determine the least constant $K$ such that the following holds: For all positive real $M$ that is greater than $1$, there exists a strictly increasing sequence $x_0, x_1, \ldots, x_n$ (of arbitrary length) such that $x_0=1, x_n\geq M$ and \[\sum_{i=0}^{n-1}\frac{\left(x_{i+1}-x_i\right)^c}{x_i^{c+1}}\leq K.\] (From 2020 IMOCSL A5. I think this problem is particularly beautiful so I want to make a separate thread for it :D )

2021-IMOC, A2
Tags: algebra , IMOC
For any positive integers $n$, find all $n$-tuples of complex numbers $(a_1,..., a_n)$ satisfying $$(x+a_1)(x+a_2)\cdots (x+a_n)=x^n+\binom{n}{1}a_1 x^{n-1}+\binom{n}{2}a_2^2 x^{n-2}+\cdots +\binom{n}{n-1} a_{n-1}^{n-1}+\binom{n}{n}a_n^n.$$ Proposed by USJL.

2022-IMOC, N3
Tags: number theory , IMOC
Find all positive integer $n$ satifying $$2n+3|n!-1$$ [i]Proposed by ltf0501[/i]

2021-IMOC qualification, G3
Tags: IMOC , geometry
Given a $\triangle ABC$, $\angle A=45^\circ$, $O$ is the circumcenter and $H$ is the orthocenter of $\triangle ABC$. $M$ is the midpoint of $\overline{BC}$, and $N$ is the midpoint of $\overline{OH}$. Prove that $\angle BAM=\angle CAN$.

2020-IMOC, N5
Tags: functional equation , number theory , IMOC , Taiwan
$\textbf{N5.}$ Find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all $a,b,c \in \mathbb{N}$ $f(a)+f(b)+f(c)-ab-bc-ca \mid af(a)+bf(b)+cf(c)-3abc$

2024-IMOC, G6
Tags: geometry , IMOC
$ABCD$ is a cyclic quadrilateral and $AC$ intersects $BD$ at $E$. $M, N$ are the midpoints of $AB, CD$, respectively. $\odot(AMN)$ meets $\odot(ABCD)$ again at $P$. $\odot(CMN)$ meets $\odot(ABCD)$ again at $Q$. $\odot(PEQ)$ meets $BD$ again at $T$. Prove that $M,N,T$ are colinear. [i]Proposed by chengbilly[/i]

2022-IMOC, N1
Tags: number theory , IMOC
Find all positive integer $n$ such that for all $i=1,2,\cdots,n$, $\frac{n!}{i!(n-i+1)!}$ is an integer. [i]Proposed by ckliao914[/i]

2022-IMOC, C5
Tags: combinatorics , IMOC
Define a "ternary sequence" is a sequence that every number is $0,1$ or $2$. ternary sequence $(x_1,x_2,x_3,\cdots,x_n)$, define its difference to be $$(|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|)$$ A difference will make the length of the sequence decrease by $1$, so we define the "feature value" of a ternary sequence with length $n$ is the number left after $n-1$ differences. How many ternary sequences has length $2023$ and feature value $0$? [i]Proposed by CSJL[/i]

2021-IMOC qualification, C1
Tags: IMOC , combinatorics
There are $3n$ $A$s and $2n$ $B$s in a string, where $n$ is a positive integer, prove that you can find a substring in this string that contains $3$ $A$s and $2$ $B$s.

2022-IMOC, C2
Tags: combinatorics , IMOC
There are $2022$ stones on a table. At the start of the game, Teacher Tseng will choose a positive integer $m$ and let Ming and LTF play a game. LTF is the first to move, and he can remove at most $m$ stones on his round. Then the two people take turns removing stone, each round they must remove at least one stone, and they cannot remove more than twice the amount of stones the last person removed. The player unable to move loses. Find the smallest positive integer $m$ such that LTF has a winning strategy. [i]Proposed by ltf0501[/i]

2021-IMOC qualification, C3
Tags: IMOC , combinatorics
There are n cards on a table numbered from $1$ to $n$, where $n$ is an even number. Two people take turns taking away the cards. The first player will always take the card with the largest number on it, but the second player will take a random card. Prove: the probability that the first player takes the card with the number $i$ is $ \frac{i-1}{n-1} $

2021-IMOC qualification, N1
Tags: IMOC , number theory
Prove: if $2^{2^n-1}-1$ is a prime, then $n$ is a prime.

2020-IMOC, A2
Tags: algebra , functional equation , function , IMOC
Find all function $f:\mathbb{R}^+$ $\rightarrow \mathbb{R}^+$ such that: $f(f(x) + y)f(x) = f(xy + 1) \forall x, y \in \mathbb{R}^+$

2020-IMOC, N6
Tags: number theory , greatest common divisor , IMOC
$\textbf{N6.}$ Let $a,b$ be positive integers. If $a,b$ satisfy that \begin{align*} \frac{a+1}{b} + \frac{b+1}{a} \end{align*} is also a positive integer, show that \begin{align*} \frac{a+b}{gcd(a,b)^2} \end{align*} is a Fibonacci number. [i]Proposed by usjl[/i]

2024-IMOC, G1
Tags: geometry , IMOC
Given quadrilateral $ABCD$. $AC$ and $BD$ meets at $E$, and $M, N$ are the midpoints of $AC, BD$, respectively. Let the circumcircles of $ABE$ and $CDE$ meets again at $X\neq E$. Prove that $E, M, N, X$ are concyclic. [i]Proposed by chengbilly[/i]

2020-IMOC, A6
Tags: algebra , polynomial , IMOC
$\definecolor{A}{RGB}{255,0,0}\color{A}\fbox{A6.}$ Let $ P (x)$ be a polynomial with real coefficients such that $\deg P \ge 3$ is an odd integer. Let $f : \mathbb{R}\rightarrow\mathbb{Z}$ be a function such that $$\definecolor{A}{RGB}{0,0,200}\color{A}\forall_{x\in\mathbb{R}}\ f(P(x)) = P(f(x)).$$ $\definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(a)}$ Prove that the range of $f$ is finite. $\definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(b)}$ Show that for any positive integer $n$, there exist $P$, $f$ that satisfies the above condition and also that the range of $f$ has cardinality $n$. [i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b]. [color=#3D9186]#1735[/color]

2021-IMOC qualification, C2
Tags: IMOC , combinatorics
Find the largest positive integer $n$ such that no two adjacent digits are the same, and for any two distinct digits $0 \leq a,b \leq 9 $, you can't get the string $abab$ just by removing digits from $n$.

2019-IMOC, N4
Tags: number theory , IMOC
Given a sequence of prime numbers $p_1, p_2,\cdots$ , with the following property: $p_{n+2}$ is the largest prime divisor of $p_n+p_{n+1}+2018$ Show that the set $\{p_i\}_{i\in \mathbb{N}}$ is finite.

2019-IMOC, A5
Tags: functional equation , Positive integer FE , iterated FE , algebra , IMOC
Find all functions $f : \mathbb N \mapsto \mathbb N$ such that the following identity $$f^{x+1}(y)+f^{y+1}(x)=2f(x+y)$$ holds for all $x,y \in \mathbb N$

2021-IMOC, C9
Tags: combinatorics , graph theory , IMOC
In a simple graph, there exist two vertices $A,B$ such that there are exactly $100$ shortest paths from $A$ to $B$. Find the minimum number of edges in the graph. [i]CSJL[/i]