Olympiad problems

2010 Contests, 2

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Prove that for every positive integer $n$, there exist integers $a$ and $b$ such that $4a^2 + 9b^2 - 1$ is divisible by $n$.

2019 India IMO Training Camp, P1

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

2023 Costa Rica - Final Round, 3.2

Tags: ordered pairs , number theory , least common multiple

Find all ordered pairs of positive integers $(r, s)$ for which there are exactly $35$ ordered pairs of positive integers $(a, b)$ such that the least common multiple of $a$ and $b$ is $2^r \cdot 3^s$.

1988 AMC 8, 25

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A [b]palindrome[/b] is a whole number that reads the same forwards and backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are: $ \boxed{1:01} $, $ \boxed{12:21} $. How many times during a 12-hour period will be palindromes? $ \text{(A)}\ 57\qquad\text{(B)}\ 60\qquad\text{(C)}\ 63\qquad\text{(D)}\ 90\qquad\text{(E)}\ 93 $

2017 AMC 10, 6

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What is the largest number of solid $2\text{-in}\times 2\text{-in}\times 1\text{-in}$ blocks that can fit in a $3\text{-in}\times 2\text{-in}\times 3\text{-in}$ box? $\textbf{(A) } 3\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 6\qquad \textbf{(E) } 7$

1995 All-Russian Olympiad, 2

Tags: geometry

A chord $CD$ of a circle with center $O$ is perpendicular to a diameter $AB$. A chord $AE$ bisects the radius $OC$. Show that the line $DE$ bisects the chord $BC$ [i]V. Gordon[/i]

2004 Dutch Mathematical Olympiad, 4

Tags: geometry , isosceles , tangent circle

Two circles $C_1$ and $C_2$ touch each other externally in a point $P$. At point $C_1$ there is a point $Q$ such that the tangent line in $Q$ at $C_1$ intersects the circle $C_2$ at points $A$ and $B$. The line $QP$ still intersects $C_2$ at point $C$. Prove that triangle $ABC$ is isosceles.

1992 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: geometry , circumradius , circumcircle

Let $D$ be the center of the circumcircle of the acute triangle $ABC$. If the circumcircle of triangle $ADB$ intersects $AC$ (or its extension) at $M$ and also $BC$ (or its extension) at $N$, show that the radii of the circumcircles of $\triangle ADB$ and $\triangle MNC$ are equal.

2021 Indonesia TST, G

Tags: geometry , geometric transformation

Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.

2023 India National Olympiad, 5

Tags: graph theory , combinatorics , inequalities

Euler marks $n$ different points in the Euclidean plane. For each pair of marked points, Gauss writes down the number $\lfloor \log_2 d \rfloor$ where $d$ is the distance between the two points. Prove that Gauss writes down less than $2n$ distinct values. [i]Note:[/i] For any $d>0$, $\lfloor \log_2 d\rfloor$ is the unique integer $k$ such that $2^k\le d<2^{k+1}$. [i]Proposed by Pranjal Srivastava[/i]

2000 All-Russian Olympiad Regional Round, 11.3

Tags: algebra , integer , number theory

Sequence of real numbers $a_1, a_2, . . . , a_{2000}$ is such that for any natural number $n$, $1\le n \le 2000$, the equality $$a^3_1+ a^3_2+... + a^3_n = (a_1 + a_2 +...+ a_n)^2.$$ Prove that all terms of this sequence are integers.

2010 IFYM, Sozopol, 4

Tags: inequalities

For $x,y,z > 0$ and $xyz=1$, prove that \[\frac{x^{9}+y^{9}}{x^{6}+x^{3}y^{3}+y^{6}}+\frac{x^{9}+z^{9}}{x^{6}+x^{3}z^{3}+z^{6}}+\frac{y^{9}+z^{9}}{y^{6}+y^{3}z^{3}+z^{6}}\geq 2\]

2006 All-Russian Olympiad Regional Round, 10.6

Tags: geometry , similar triangles

Through the point of intersection of the altitudes of an acute triangle $ABC$ three circles pass through, each of which touches one of the sides triangle at the foot of the altitude . Prove that the second intersection points of the circles are the vertices of a triangle similar to the original one.

2009 Argentina Iberoamerican TST, 2

Tags: combinatorics , analytic geometry

There are $ m \plus{} 1$ horizontal lines and $ m$ vertical lines on the plane so that $ m(m \plus{} 1)$ intersections are made. A mark is placed at one of the $ m$ points of the lowest horizontal line. 2 players play the game of the following rules on this lines and points. 1. Each player moves a mark from a point to a point along the lines in turns. 2. The segment is erased after a mark moved along it. 3. When a player cannot make a move, then he loses. Prove that the lead always wins the game. PS I haven't found a student who solved it. There can be no one.

2006 Turkey Team Selection Test, 1

Tags: algebra , geometry , quadratic , pythagorean theorem , circumcircle , trigonometry

Find the maximum value for the area of a heptagon with all vertices on a circle and two diagonals perpendicular.

1998 Harvard-MIT Mathematics Tournament, 10

Tags: probability

In the fourth annual Swirled Series, the Oakland Alphas are playing the San Francisco Gammas. The first game is played in San Francisco and succeeding games alternate in location. San Francisco has a $50\%$ chance of winning their home games, while Oakland has a probability of $60\%$ of winning at home. Normally, the series will stretch on forever until one team gets a three game lead, in which case they are declared the winners. However, after each game in San Francisco there is a $50\%$ chance of an earthquake, which will cause the series to end with the team that has won more games declared the winner. What is the probability that the Gammas will win?

2015 JBMO Shortlist, A2

Tags: algebra , polynomial

If $x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0$ find the value of $x^6-8x^5+13x^4-5x^3+49x^2-137x+2015$ .

2023 Thailand Online MO, 10

Tags: combinatorics , combinatorial game

Let $n$ be an even positive integer. Alice and Bob play the following game. Before the start of the game, Alice chooses a set $S$ containing $m$ integers and announces it to Bob. The players then alternate turns, with Bob going first, choosing $i\in\{1,2,\dots, n\}$ that has not been chosen and setting the value of $v_i$ to either $0$ or $1$. At the end of the game, when all of $v_1,v_2,\dots,v_n$ have been set, the expression $$E=v_1\cdot 2^0 + v_2 \cdot 2^1 + \dots + v_n \cdot 2^{n-1}$$ is calculated. Determine the minimum $m$ such that Alice can always ensure that $E\in S$ regardless of how Bob plays.

2013 Brazil Team Selection Test, 2

Tags: moving points , ptolemy sinus lemma , parallelogram , reflection , geometry

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.

2002 Poland - Second Round, 3

Tags: combinatorics , search , inequalities , geometry

A positive integer $ n$ is given. In an association consisting of $ n$ members work $ 6$ commissions. Each commission contains at least $ \large \frac{n}{4}$ persons. Prove that there exist two commissions containing at least $ \large \frac{n}{30}$ persons in common.

1955 AMC 12/AHSME, 12

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The solution of $ \sqrt{5x\minus{}1}\plus{}\sqrt{x\minus{}1}\equal{}2$ is: $ \textbf{(A)}\ x\equal{}2,x\equal{}1 \qquad \textbf{(B)}\ x\equal{}\frac{2}{3} \qquad \textbf{(C)}\ x\equal{}2 \qquad \textbf{(D)}\ x\equal{}1 \qquad \textbf{(E)}\ x\equal{}0$

KoMaL A Problems 2021/2022, A. 814

Tags: combinatorial geometry , geometry

We are given $666$ points on the plane such that they cannot be covered by $10$ lines. Show that we can choose $66$ out of these points such that they can not be covered by $10$ lines.

2025 Kyiv City MO Round 1, Problem 2

Tags: combinatorics , grid

Is it possible to write positive integers from $1$ to $2025$ in the cells of a $ 45 \times 45 $ grid such that each number is used exactly once, and at the same time, each written number is either greater than all the numbers located in its side-adjacent cells or smaller than all the numbers located in its side-adjacent cells? [i]Proposed by Anton Trygub[/i]

2022 Korea Winter Program Practice Test, 5

Tags: concurrency , cyclic quadrilateral , geometry

Let $ABDC$ be a cyclic quadrilateral inscribed in a circle $\Omega$. $AD$ meets $BC$ at $P$, and $\Omega$ meets lines passing $A$ and parallel to $DB$, $DC$ at $E$, $F$, respectively. $X$ is a point on $\Omega$ such that $PA=PX$. Prove that the lines $BE$, $CF$, and $DX$ are concurrent.

1997 AMC 12/AHSME, 3

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If $ x$,$ y$, and $ z$ are real numbers such that \[(x \minus{} 3)^2 \plus{} (y \minus{} 4)^2 \plus{} (z \minus{} 5)^2 \equal{} 0,\] then $ x \plus{} y \plus{} z \equal{}$ $ \textbf{(A)}\ \minus{}12\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 50$