Found problems: 85335
1984 AIME Problems, 7
The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3 & \text{if } n\ge 1000 \\ f(f(n+5)) & \text{if } n<1000\end{cases} \] Find $f(84)$.
1998 Gauss, 20
Each of the 12 edges of a cube is coloured either red or green. Every face of the cube has at least one
red edge. What is the smallest number of red edges?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$
MOAA Team Rounds, 2022.2
While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$.
2007 Chile National Olympiad, 3
Two players, Aurelio and Bernardo, play the following game. Aurelio begins by writing the number $1$. Next it is Bernardo's turn, who writes number $2$. From then on, each player chooses whether to add $1$ to the number just written by the previous player, or whether multiply that number by $2$. Then write the result and it's the other player's turn. The first player to write a number greater than $ 2007$ loses the game. Determine if one of the players can ensure victory no matter what the other does.
2018 Estonia Team Selection Test, 4
Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$
1993 All-Russian Olympiad, 1
For integers $x$, $y$, and $z$, we have $(x-y)(y-z)(z-x)=x+y+z$. Prove that $27|x+y+z$.
2002 JBMO ShortLists, 11
Let $ ABC$ be an isosceles triangle with $ AB\equal{}AC$ and $ \angle A\equal{}20^\circ$. On the side $ AC$ consider point $ D$ such that $ AD\equal{}BC$. Find $ \angle BDC$.
III Soros Olympiad 1996 - 97 (Russia), 9.1
Without using a calculator, find out which number is greater:
$$|\sqrt[3]{5}-\sqrt3|-\sqrt3| \,\,\,\, \text{or} \,\,\,\, 0.01$$
1999 Harvard-MIT Mathematics Tournament, 6
You want to sort the numbers 5 4 3 2 1 using block moves. In other words, you can take any set of numbers that appear consecutively and put them back in at any spot as a block. For example, [i]6 5 3[/i] 4 2 1 -> 4 2 [i]6 5 3[/i] 1 is a valid block move for 6 numbers. What is the minimum number of block moves necessary to get 1 2 3 4 5?
1968 All Soviet Union Mathematical Olympiad, 106
Medians divide the triangle onto $6$ smaller ones. $4$ of the circles inscribed in those small ones are equal. Prove that the triangle is equilateral.
2020 Balkan MO Shortlist, A2
Given are positive reals $a, b, c$, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$. Prove that
$\frac{\sqrt{a+\frac{b}{c}}+\sqrt{b+\frac{c}{a}}+\sqrt{c+\frac{a}{b}}}{3}\leq \frac{a+b+c-1}{\sqrt{2}}$.
[i]Albania[/i]
2021 Nigerian Senior MO Round 2, 2
$N$ boxes are arranged in a circle and are numbered $1,2,3,.....N$ In a clockwise direction. A ball is assigned a number from${1,2,3,....N}$ and is placed in one of the boxes.A round consist of the following; if the current number on the ball is $n$, the ball is moved $n$ boxes in the clockwise direction and the number on the ball is changed to $n+1$ if $n<N$ and to $1$ if $n=N$. Is it possible to choose $N$, the initial number on the ball, and the first position of the ball in such a way that the ball gets back to the same box with the same number on it for the first time after exactly $2020$ rounds
2024 Yasinsky Geometry Olympiad, 3
Let \( H \) be the orthocenter of an acute triangle \( ABC \), and let \( AT \) be the diameter of the circumcircle of this triangle. Points \( X \) and \( Y \) are chosen on sides \( AC \) and \( AB \), respectively, such that \( TX = TY \) and \( \angle XTY + \angle XAY = 90^\circ \). Prove that \( \angle XHY = 90^\circ \).
[i]
Proposed by Matthew Kurskyi[/i]
1989 IMO Shortlist, 19
A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.
2023 AIME, 6
Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside this region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
size(2.5cm);
draw((0,0)--(0,2)--(1,2)--(1,1)--(2,1)--(2,0)--cycle);
draw((0,1)--(1,1)--(1,0), dotted);
[/asy]
2024 Kyiv City MO Round 2, Problem 1
For some positive integer $n$, Katya wrote on the board next to each other numbers $2^n$ and $14^n$ (in this order), thus forming a new number $A$. Can the number $A - 1$ be prime?
[i]Proposed by Oleksii Masalitin[/i]
2015 Baltic Way, 9
Let $n>2$ be an integer. A deck contains $\frac{n(n-1)}{2}$ cards,numbered \[1,2,3,\cdots , \frac{n(n-1)}{2}\] Two cards form a [i]magic pair[/i] if their numbers are consecutive , or if their numbers are $1$ and $\frac{n(n+1)}{2}$. For which $n$ is it possible to distribute the cards into $n$ stacks in such a manner that, among the cards in any two stacks , there is exactly one [i]magic pair[/i]?
2008 Alexandru Myller, 3
For a convex pentagon, prove that $ \frac{\text{area} (ABC)}{\text{area} (ABCD)} +\frac{\text{area} (CDE)}{\text{area} (BCDE)} <1. $
[i]Dan Ismailescu[/i]
2019 Canadian Mathematical Olympiad Qualification, 4
Let $n$ be a positive integer. For a positive integer $m$, we partition the set $\{1, 2, 3,...,m\}$ into $n$ subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of $n$, fi nd the largest positive integer $m$ for which such a partition exists.
2016 Iran Team Selection Test, 3
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
[i]Proposed by Finland[/i]
2004 239 Open Mathematical Olympiad, 3
Prove that for any integer $a$ there exist infinitely many positive integers $n$ such that $a^{2^n}+2^n$ is not a prime.
[b]proposed by S. Berlov[/b]
1997 Mexico National Olympiad, 4
What is the minimum number of planes determined by $6$ points in space which are not all coplanar, and among which no three are collinear?
Novosibirsk Oral Geo Oly VIII, 2016.3
A square is drawn on a sheet of grid paper on the sides of the cells $ABCD$ with side $8$. Point $E$ is the midpoint of side $BC$, $Q$ is such a point on the diagonal $AC$ such that $AQ: QC = 3: 1$. Find the angle between straight lines $AE$ and $DQ$.
2013 NIMO Problems, 6
Let $n$ and $k$ be integers satisfying $\binom{2k}{2} + n = 60$. It is known that $n$ days before Evan's 16th birthday, something happened. Compute $60-n$.
[i]Proposed by Evan Chen[/i]
2010 Costa Rica - Final Round, 5
Let $C_1$ be a circle with center $O$ and let $B$ and $C$ be points in $C_1$ such that $BOC$ is an equilateral triangle. Let $D$ be the midpoint of the minor arc $BC$ of $C_1$. Let $C_2$ be the circle with center $C$ that passes through $B$ and $O$. Let $E$ be the second intersection of $C_1$ and $C_2$. The parallel to $DE$ through $B$ intersects $C_1$ for second time in $A$. Let $C_3$ be the circumcircle of triangle $AOC$. The second intersection of $C_2$ and $C_3$ is $F$. Show that $BE$ and $BF$ trisect the angle $\angle ABC$.