Olympiad problems

2015 Canada National Olympiad, 2

Let $ABC$ be an acute-angled triangle with altitudes $AD,BE,$ and $CF$. Let $H$ be the orthocentre, that is, the point where the altitudes meet. Prove that \[\frac{AB\cdot AC+BC\cdot CA+CA\cdot CB}{AH\cdot AD+BH\cdot BE+CH\cdot CF}\leq 2.\]

2011 IMO, 1

Tags: number theory , set

Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i +a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$. [i]Proposed by Fernando Campos, Mexico[/i]

2012 Baltic Way, 9

Tags: combinatorics

Zeroes are written in all cells of a $5 \times 5$ board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it. Is it possible to obtain the number 2012 in all cells simultaneously?

2016 Azerbaijan BMO TST, 2

Tags: geometry

İn triangle $ABC$ the bisector of $\angle BAC$ intersects the side $BC$ at the point $D$.The circle $\omega $ passes through $A$ and tangent to the side $BC$ at $D$.$AC$ and $\omega $ intersects at $M$ second time , $BM$ and $\omega $ intersects at $P$ second time. Prove that point $P$ lies on median of triangle $ABD$.

2006 Switzerland - Final Round, 6

Tags: combinatorics

At least three players have participated in a tennis tournament. Evey two players have played each other exactly once, and each player has at least one match won. Show that there are three players $A,B,C$ such that $A$ won against $B$, $B$ won against $C$ and $C$ won against $A$.

2004 Polish MO Finals, 5

Tags: combinatorial geometry , geometry

Find the greatest possible number of lines in space that all pass through a single point and the angle between any two of them is the same.

2010 All-Russian Olympiad, 3

Tags: polynomial , function , algebra

Polynomial $P(x)$ with degree $n \geq 3$ has $n$ real roots $x_1 < x_2 < x_3 <...< x_n$, such that $x_2-x_1<x_3-x_2<....<x_n-x_{n-1}$. Prove that the maximum of the function $y=|P(x)|$ where $x$ is on the interval $[ x_1, x_n ]$, is in the interval $[x_n-1, x_n]$.

1975 Chisinau City MO, 110

Tags: octagon , symmetry , geometry

Prove that any centrally symmetric convex octagon has a diagonal passing through the center of symmetry that is not parallel to any of its sides.

2021-2022 OMMC, 6

Tags:

Calvin makes a number. He starts with $1$, and on each move, he multiplies his current number by $3$, then adds $5$. After $10$ moves, find the sum of the digits (in base $10$) when Calvin's resulting number is expressed in base $9$. [i]Proposed by Calvin Wang [/i]

2016 Tournament Of Towns, 2

Tags: geometry , locus , circles

On plane there is fixed ray $s$ with vertex $A$ and a point $P$ not on the line which contains $s$. We choose a random point $K$ which lies on ray. Let $N$ be a point on a ray outside $AK$ such that $NK=1$. Let $M$ be a point such that $NM=1,M \in PK$ and $M!=K.$ Prove that all lines $NM$, provided by some point $K$, touch some fixed circle.

2000 India Regional Mathematical Olympiad, 6

Tags:

(i) Consider two positive integers $a$ and $b$ which are such that $a^a b^b$ is divisible by $2000$. What is the least possible value of $ab$? (ii) Consider two positive integers $a$ and $b$ which are such that $a^b b^a$ is divisible by $2000$. What is the least possible value of $ab$?

2015 JBMO Shortlist, C3

Tags: positive integer , combinatorics , array , table

Positive integers are put into the following table. \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & & \\ \hline 2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & & \\ \hline 4 & 8 & 13 & 19 & 26 & 34 & 43 & 53 & & \\ \hline 7 & 12 & 18 & 25 & 33 & 42 & & & & \\ \hline 11 & 17 & 24 & 32 & 41 & & & & & \\ \hline 16 & 23 & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline \end{tabular} Find the number of the line and column where the number $2015$ stays.

2004 Alexandru Myller, 2

Tags: geometry , 3d geometry , tetrahedron

The medians from $ A $ to the faces $ ABC,ABD,ACD $ of a tetahedron $ ABCD $ are pairwise perpendicular. Show that the edges from $ A $ have equal lengths. [i]Dinu Șerbănescu[/i]

2024 District Olympiad, P2

Tags: integral , derivative , real analysis

Let $f:[0,1]\to(0,\infty)$ be a continous function on $[0,1]$ and let $A=\int_0^1 f(t)\mathrm{d}t.$[list=a] [*]Consider the function $F:[0,1]\to[0,A]$ defined by \[F(x)=\int_0^xf(t)\mathrm{d}t.\]Prove that $F(x)$ has an inverse function, which is differentiable. [*]Prove that there exists a unique function $g:[0,1]\to[0,1]$ for which\[\int_0^xf(t)\mathrm{d}t=\int_{g(x)}^1f(t)\mathrm{d}t\]is satisfied for every $x\in [0,1].$ [*]Prove that there exists $c\in[0,1]$ for which\[\lim_{x\to c}\frac{g(x)-c}{x-c}=-1,\]whre $g$ is the function uniquely determined at b. [/list]

2024 Malaysia IMONST 2, 5

Tags: geometry

A duck drew a square $ABCD$, then he reflected $C$ across $B$ to obtain a point $E$. He also drew the center of the square to be $F$. Then, he drew a point $G$ on ray $EF$ beyond $F$ such that $\angle AGC = 135^{\circ}$. Help the Duck prove that $\angle CGD = 135^{\circ}$ as well.

2008 Iran MO (2nd Round), 1

Tags: combinatorics

In how many ways, can we draw $n-3$ diagonals of a $n$-gon with equal sides and equal angles such that: $i)$ none of them intersect each other in the polygonal. $ii)$ each of the produced triangles has at least one common side with the polygonal.

1996 IMO Shortlist, 1

Tags: inequalities , three variable inequality , algebra

Suppose that $a, b, c > 0$ such that $abc = 1$. Prove that \[ \frac{ab}{ab + a^5 + b^5} + \frac{bc}{bc + b^5 + c^5} + \frac{ca}{ca + c^5 + a^5} \leq 1. \]

2024 Brazil EGMO TST, 4

Tags: geometry , bicentric quadrilateral , tangent , projective geometry

Let $ABCD$ be a cyclic quadrilateral with all distinct sides that has an inscribed circle. The incircle of $ABCD$ has center $I$ and is tangent to $AB$, $BC$, $CD$, and $DA$ at points $W$, $X$, $Y$, and $Z$, respectively. Let $K$ be the intersection of the lines $WX$ and $YZ$. Prove that $KI$ is tangent to the circumcircle of triangle $AIC$.

OMMC POTM, 2022 7

Tags: diophantine equation , number theory

Find all ordered triples of positive integers $(a,b,c)$ where $$\left(a+\frac{1}{a}\right)\left(b+\frac{1}{b}\right)=c+\frac{1}{c}.$$ [i]Proposed by vsamc[/i]

2018 Azerbaijan JBMO TST, 2

Tags: geometry

Let $ABC$ be an acute triangle and let $M$ be the midpoint of side $BC$. Let $D,E$ be the excircles of triangles $AMB,AMC$ respectively, towards $M$. Circumcirscribed circle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. Circumcirscribed circles of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF=CG$. by Petru Braica, Romania

2022 AMC 10, 18

Tags: transformation

Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself? $\textbf{(A) } 359 \qquad \textbf{(B) } 360\qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721$

2021 China Girls Math Olympiad, 4

Tags: number theory , infinite sequence , integer sequence

Call a sequence of positive integers $(a_n)_{n \ge 1}$ a "CGMO sequence" if $(a_n)_{n \ge 1}$ strictly increases, and for all integers $n \ge 2022$, $a_n$ is the smallest integer such that there exists a non-empty subset of $\{a_{1}, a_{2}, \cdots, a_{n-1} \}$ $A_n$ where $a_n \cdot \prod\limits_{a \in A_n} a$ is a perfect square. Proof: there exists $c_1, c_2 \in \mathbb{R}^{+}$ s.t. for any "CGMO sequence" $(a_n)_{n \ge 1}$ , there is a positive integer $N$ that satisfies any $n \ge N$, $c_1 \cdot n^2 \le a_n \le c_2 \cdot n^2$.

Cono Sur Shortlist - geometry, 2020.G2

Tags: geometry , incircle , circumcircle

Let $ABC$ be a triangle whose inscribed circle is $\omega$. Let $r_1$ be the line parallel to $BC$ and tangent to $\omega$, with $r_1 \ne BC$ and let $r_2$ be the line parallel to $AB$ and tangent to $\omega$ with $r_2 \ne AB$. Suppose that the intersection point of $r_1$ and $r_2$ lies on the circumscribed circle of triangle $ABC$. Prove that the sidelengths of triangle $ABC$ form an arithmetic progression.

2021 IMO Shortlist, C2

Tags: pigeonhole principle , c2

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

1972 Swedish Mathematical Competition, 2

Tags: combinatorial geometry , combinatorics

A rectangular grid of streets has $m$ north-south streets and $n$ east-west streets. For which $m, n > 1$ is it possible to start at an intersection and drive through each of the other intersections just once before returning to the start?