Found problems: 85335
2015 Dutch IMO TST, 5
Let $N$ be the set of positive integers.
Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers
2024 Ukraine National Mathematical Olympiad, Problem 1
Solomiya wrote the numbers $1, 2, \ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from the board and write the sum $a+b$ instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this?
[i]Proposed by Oleksiy Masalitin[/i]
2024 India National Olympiad, 3
Let $p$ be an odd prime and $a,b,c$ be integers so that the integers $$a^{2023}+b^{2023},\quad b^{2024}+c^{2024},\quad a^{2025}+c^{2025}$$ are divisible by $p$.
Prove that $p$ divides each of $a,b,c$.
$\quad$
Proposed by Navilarekallu Tejaswi
2012 Romania Team Selection Test, 2
Let $ABCD$ be a convex circumscribed quadrilateral such that $\angle ABC+\angle ADC<180^{\circ}$ and $\angle ABD+\angle ACB=\angle ACD+\angle ADB$. Prove that one of the diagonals of quadrilateral $ABCD$ passes through the other diagonals midpoint.
2024 LMT Fall, 8
The LHS Math Team is doing Karaoke. William sings every song, David sings every other song, Peter sings every third song, and Muztaba sings every fourth song. If they sing $600$ songs, find the average number of people singing each song.
Geometry Mathley 2011-12, 13.3
Let $ABCD$ be a quadrilateral inscribed in circle $(O)$. Let $M,N$ be the midpoints of $AD,BC$. A line through the intersection $P$ of the two diagonals $AC,BD$ meets $AD,BC$ at $S, T$ respectively. Let $BS$ meet $AT$ at $Q$. Prove that three lines $AD,BC,PQ$ are concurrent if and only if $M, S, T,N$ are on the same circle.
Đỗ Thanh Sơn
2016 Singapore Junior Math Olympiad, 4
A group of tourists get on $10$ buses in the outgoing trip. The same group of tourists get on $8$ buses in the return trip. Assuming each bus carries at least $1$ tourist, prove that there are at least $3$ tourists such that each of them has taken a bus in the return trip that has more people than the bus he has taken in the outgoing trip.
1998 German National Olympiad, 3
For each nonnegative integer $k$ find all nonnegative integers $x,y,z$ such that $x^2 +y^2 +z^2 = 8^k$
1985 IMO Longlists, 96
Determine all functions $f : \mathbb R \to \mathbb R$ satisfying the following two conditions:
(a) $f(x + y) + f(x - y) = 2f(x)f(y)$ for all $x, y \in \mathbb R$, and
(b) $\lim_{x\to \infty} f(x) = 0$.
2002 Tournament Of Towns, 6
There's a large pile of cards. On each card a number from $1,2,\ldots n$ is written. It is known that sum of all numbers on all of the cards is equal to $k\cdot n!$ for some $k$. Prove that it is possible to arrange cards into $k$ stacks so that sum of numbers written on the cards in each stack is equal to $n!$.
2003 USA Team Selection Test, 4
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that \[ f(m+n)f(m-n) = f(m^2) \] for $m,n \in \mathbb{N}$.
2013 Turkey MO (2nd round), 3
Let $G$ be a simple, undirected, connected graph with $100$ vertices and $2013$ edges. It is given that there exist two vertices $A$ and $B$ such that it is not possible to reach $A$ from $B$ using one or two edges. We color all edges using $n$ colors, such that for all pairs of vertices, there exists a way connecting them with a single color. Find the maximum value of $n$.
2021 Malaysia IMONST 1, 9
Find the sum of (decimal) digits of the number $(10^{2021} + 2021)^2$?
1994 Canada National Olympiad, 1
Evaluate $\sum_{n=1}^{1994}{\left((-1)^{n}\cdot\left(\frac{n^2 + n + 1}{n!}\right)\right)}$ .
2016 Spain Mathematical Olympiad, 4
Let $m$ be a positive integer and $a$ and $b$ be distinct positive integers strictly greater than $m^2$ and strictly less than $m^2+m$. Find all integers $d$ such that $m^2 < d < m^2+m$ and $d$ divides $ab$.
2020 Brazil Undergrad MO, Problem 6
Let $f(x) = 2x^2 + x - 1, f^{0}(x) = x$, and $f^{n+1}(x) = f(f^{n}(x))$ for all real $x>0$ and $n \ge 0$ integer (that is, $f^{n}$ is $f$ iterated $n$ times).
a) Find the number of distinct real roots of the equation $f^{3}(x) = x$
b) Find, for each $n \ge 0$ integer, the number of distinct real solutions of the equation $f^{n}(x) = 0$
2023-24 IOQM India, 24
A trapezium in the plane is a quadrilateral in which a pair of opposite sides are parallel. A trapezium is said to be non-degenerate if it has positive area. Find the number of mutually non-congruent, non-degenerate trapeziums whose sides are four distinct integers from the set $\{5,6,7,8,9,10\}$
2013 ELMO Shortlist, 4
Positive reals $a$, $b$, and $c$ obey $\frac{a^2+b^2+c^2}{ab+bc+ca} = \frac{ab+bc+ca+1}{2}$. Prove that \[ \sqrt{a^2+b^2+c^2} \le 1 + \frac{\lvert a-b \rvert + \lvert b-c \rvert + \lvert c-a \rvert}{2}. \][i]Proposed by Evan Chen[/i]
2009 AMC 10, 16
Points $ A$ and $ C$ lie on a circle centered at $ O$, each of $ \overline{BA}$ and $ \overline{BC}$ are tangent to the circle, and $ \triangle ABC$ is equilateral. The circle intersects $ \overline{BO}$ at $ D$. What is $ \frac {BD}{BO}$?
$ \textbf{(A)}\ \frac {\sqrt2}{3} \qquad \textbf{(B)}\ \frac {1}{2} \qquad \textbf{(C)}\ \frac {\sqrt3}{3} \qquad \textbf{(D)}\ \frac {\sqrt2}{2} \qquad \textbf{(E)}\ \frac {\sqrt3}{2}$
2018 Saudi Arabia GMO TST, 4
In a graph with $8$ vertices that contains no cycle of length $4$, at most how many edges can there be?
2019 Caucasus Mathematical Olympiad, 7
On sides $BC$, $CA$, $AB$ of a triangle $ABC$ points $K$, $L$, $M$ are chosen, respectively, and a point $P$ is inside $ABC$ is chosen so that $PL\parallel BC$, $PM\parallel CA$, $PK\parallel AB$. Determine if it is possible that each of three trapezoids $AMPL$, $BKPM$, $CLPK$ has an inscribed circle.
2011 Gheorghe Vranceanu, 2
Let $ a\ge 3 $ and a polynom $ P. $ Show that:
$$ \max_{1\le k\le \text{grad} P} \left| a^{k-1}-P(k-1) \right| \ge 1 $$
1990 Federal Competition For Advanced Students, P2, 1
Determine the number of integers $ n$ with $ 1 \le n \le N\equal{}1990^{1990}$ such that $ n^2\minus{}1$ and $ N$ are coprime.
2014 Contests, 4
Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$
2022 Harvard-MIT Mathematics Tournament, 9
Consider permutations $(a_0, a_1, . . . , a_{2022})$ of $(0, 1, . . . , 2022)$ such that
$\bullet$ $a_{2022} = 625$,
$\bullet$ for each $0 \le i \le 2022$, $a_i \ge \frac{625i}{2022}$ ,
$\bullet$ for each $0 \le i \le 2022$, $\{a_i, . . . , a_{2022}\}$ is a set of consecutive integers (in some order).
The number of such permutations can be written as $\frac{a!}{b!c!}$ for positive integers $a, b, c$, where $b > c$ and $a$ is minimal. Compute $100a + 10b + c$.