Olympiad problems

2015 Geolympiad Summer, 3.

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Let $ABC$ be an acute scalene triangle with incenter $I$, circumcircle $w_1$, and denote the circumcircle of $BIC$ as $w_2$. Suppose point $P$ lies on $w_2$ and is inside $w_1$. Let $X,Y$ lie on $BC$ with $XP \perp BP, YP \perp PC$. Circles $O_1, O_2$ are drawn tangent to $w_1$ at points on the same side of $BC$ as $A$ and tangent to $BC$ at $X,Y$ respectively. Let the centers of those two circles be $Z_1, Z_2$. Let $D$ be the point on $w_2$ opposite to $P$ and let $E$ be the foot of the altitude from $P$ to $BC$. Show that $DE \perp Z_1Z_2$

2007 Mathematics for Its Sake, 3

Let be three positive real numbers $ a,b,c, $ a natural number $ n, $ and the functions $ f:\mathbb{R}\longrightarrow\mathbb{R} ,g:(0,\infty )\longrightarrow\mathbb{R} $ defined as: $$ f(x)=\frac{2(n+1)x^n(x^{n+1}-a) +nx^{n+1} +2a^2x+a}{x^{2n+2}-2ax^{n+1} +a^2x^2+a^2} , $$ $$ g(x)=\frac{a+bx^n}{x+cx^{2n+1}} $$ Calculate the antiderivatives of $ f $ and $ g. $ [i]Nicolae Sanda[/i]

2005 Bosnia and Herzegovina Team Selection Test, 6

Tags: integer , number theory , perfect cube

Let $a$, $b$ and $c$ are integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$. Prove that $abc$ is a perfect cube of an integer.

2024 Harvard-MIT Mathematics Tournament, 6

Tags: geometry

In triangle $ABC$, circle $\omega$ with center $O$ passes through $B$ and $C$ and it intersects segments $\overline{AB}$ and $\overline{AC}$ again at $B^{\prime}$ and $C^{\prime}$, respectively. Suppose the circles with diameters $\overline{BB^{\prime}}$ and $\overline{CC^{\prime}}$ are externally tangent to each other at $T$ with $AB=18$, $AC=36$, and $AT=12$. Find $AO$.

2009 Romania Team Selection Test, 2

Tags: combinatorics , pigeonhole principle

A square of side $N=n^2+1$, $n\in \mathbb{N}^*$, is partitioned in unit squares (of side $1$), along $N$ rows and $N$ columns. The $N^2$ unit squares are colored using $N$ colors, $N$ squares with each color. Prove that for any coloring there exists a row or a column containing unit squares of at least $n+1$ colors.

2013 Middle European Mathematical Olympiad, 8

Tags: number theory

The expression \[ \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box \] is written on the blackboard. Two players, $ A $ and $ B $, play a game, taking turns. Player $ A $ takes the first turn. In each turn, the player on turn replaces a symbol $ \Box $ by a positive integer. After all the symbols $\Box$ are replace, player $A$ replaces each of the signs $\pm$ by either + or -, independently of each other. Player $ A $ wins if the value of the expression on the blackboard is not divisible by any of the numbers $ 11, 12, \cdots, 18 $. Otherwise, player $ B$ wins. Determine which player has a winning strategy.

MathLinks Contest 7th, 6.3

Tags: geometry , circumcircle , geometric transformation , homothety , linear algebra , matrix , incenter

Let $ \Omega$ be the circumcircle of triangle $ ABC$. Let $ D$ be the point at which the incircle of $ ABC$ touches its side $ BC$. Let $ M$ be the point on $ \Omega$ such that the line $ AM$ is parallel to $ BC$. Also, let $ P$ be the point at which the circle tangent to the segments $ AB$ and $ AC$ and to the circle $ \Omega$ touches $ \Omega$. Prove that the points $ P$, $ D$, $ M$ are collinear.

2016 PUMaC Combinatorics B, 4

Tags: probability

$32$ teams, ranked $1$ through $32$, enter a basketball tournament that works as follows: the teams are randomly paired and in each pair, the team that loses is out of the competition. The remaining $16$ teams are randomly paired, and so on, until there is a winner. A higher ranked team always wins against a lower-ranked team. If the probability that the team ranked $3$ (the third-best team) is one of the last four teams remaining can be written in simplest form as $\dfrac{m}{n}$, compute $m+n$.

2023 Azerbaijan IMO TST, 6

Tags: combinatorics

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

2025 All-Russian Olympiad Regional Round, 9.5

Tags: number theory

Find all pairs of integer numbers $m$ and $n>2$ such that $((n-1)!-n)(n-2)!=m(m-2)$. [i]A. Kuznetsov[/i]

2014 SEEMOUS, Problem 1

Tags: linear algebra , matrix , function , determinant

Let $n$ be a nonzero natural number and $f:\mathbb R\to\mathbb R\setminus\{0\}$ be a function such that $f(2014)=1-f(2013)$. Let $x_1,x_2,x_3,\ldots,x_n$ be real numbers not equal to each other. If $$\begin{vmatrix}1+f(x_1)&f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&1+f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&f(x_2)&1+f(x_3)&\cdots&f(x_n)\\\vdots&\vdots&\vdots&\ddots&\vdots\\f(x_1)&f(x_2)&f(x_3)&\cdots&1+f(x_n)\end{vmatrix}=0,$$prove that $f$ is not continuous.

2019 Jozsef Wildt International Math Competition, W. 67

Tags: geometry , toricelli point , geometric inequality , inequalities

Denote $T$ the Toricelli point of the triangle $ABC$. Prove that $$AB^2 \times BC^2 \times CA^2 \geq 3(TA^2\times TB + TB^2 \times TC + TC^2 \times TA)(TA\times TB^2 + TB \times TC^2 + TC \times TA^2)$$

2004 Manhattan Mathematical Olympiad, 4

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An elevator in a 100 floor building has only two buttons. The UP button makes the elevator go $13$ floors up, and the DOWN button makes the elevator go $8$ floors down. Is it possible to go from the $13^{\text{th}}$ floor to $8^{\text{th}}$ floor?

2016 India Regional Mathematical Olympiad, 1

Tags: sum , number theory , algebra

Suppose in a given collection of $2016$ integer, the sum of any $1008$ integers is positive. Show that sum of all $2016$ integers is positive.

2001 Baltic Way, 9

Tags: geometry , rhombus , parallelogram , circumcircle , hyperbola , cyclic quadrilateral , conic

Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.

1999 Korea Junior Math Olympiad, 4

Tags: geometry , geometric transformation , rotation

$C$ is the unit circle in some plane. $R$ is a square with side $a$. $C$ is fixed and $R$ moves(without rotation) on the plane, in such a way that its center stays inside $C$(including boundaries). Find the maximum value of the area drawn by the trace of $R$.

1962 AMC 12/AHSME, 34

Tags: quadratic

For what real values of $ K$ does $ x \equal{} K^2 (x\minus{}1)(x\minus{}2)$ have real roots? $ \textbf{(A)}\ \text{none} \qquad \textbf{(B)}\ \minus{}2<K<1 \qquad \textbf{(C)}\ \minus{}2 \sqrt{2} < K < 2 \sqrt{2} \qquad \textbf{(D)}\ K>1 \text{ or } K<\minus{}2 \qquad \textbf{(E)}\ \text{all}$

2022 Tuymaada Olympiad, 7

Tags: combinatorics , tiels , domino , rectangle

A $1 \times 5n$ rectangle is partitioned into tiles, each of the tile being either a separate $1 \times 1$ square or a broken domino consisting of two such squares separated by four squares (not belonging to the domino). Prove that the number of such partitions is a perfect fifth power. [i](K. Kokhas)[/i]

2010 CIIM, Problem 2

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In one side of a hall there are $2N$ rooms numbered from 1 to $2N$. In each room $i$ between 1 and $N$ there are $p_i$ beds. Is needed to move every one of this beds to the roms from $N+ 1$ to $2N$, in such a way that for every $j$ between $N+1$ and $2N$ the room $j$ will have $p_j$ beds. Supose that each bed can be move once and the price of moving a bed from room $i$ to room $j$ is $(i-j)^2$. Find a way to move every bed such that the total cost is minimize. Note: The numbers $p_i$ are given and satisfy that $p_1 + p_2 + \cdots + p_N = p_{N+1} + p_{N+2} + \cdots+ p_{2N}.$

2003 Baltic Way, 2

Tags: inequalities , quadratic , algebra , number theory

Prove that any real solution of $x^3+px+q=0$, where $p,q$ are real numbers, satisfies the inequality $4qx\le p^2$.

2007 Mexico National Olympiad, 3

Tags: geometry

Let $ABC$ be a triangle with $AB>BC>CA$. Let $D$ be a point on $AB$ such that $CD=BC$, and let $M$ be the midpoint of $AC$. Show that $BD=AC$ and that $\angle BAC=2\angle ABM.$

2020 Indonesia MO, 4

Tags: rectangle , combinatorics , coloring

Problem 4. A chessboard with $2n \times 2n$ tiles is coloured such that every tile is coloured with one out of $n$ colours. Prove that there exists 2 tiles in either the same column or row such that if the colours of both tiles are swapped, then there exists a rectangle where all its four corner tiles have the same colour.

2015 Junior Balkan Team Selection Tests - Romania, 3

Tags: inequalities

Let $x$,$y$,$z>0$ . Show that : $$\frac{x^3}{z^3+x^2y}+\frac{y^3}{x^3+y^2z}+\frac{z^3}{y^3+z^2x} \geq \frac{3}{2}$$

2005 National Olympiad First Round, 15

Tags: function , vieta

For how many positive real numbers $a$ has the equation $a^2x^2 + ax+1-7a^2 = 0$ two distinct integer roots? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ \text{Infinitely many} \qquad\textbf{(E)}\ \text{None of above} $

2009 IMC, 3

Tags: linear algebra , matrix , algebra , polynomial

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ be two $n \times n$ matrices such that \[ A^2B+BA^2=2ABA \] Prove there exists $k\in \mathbb{N}$ such that \[ (AB-BA)^k=\mathbf{0}_n\] Here $\mathbf{0}_n$ is the null matrix of order $n$.