Olympiad problems

2012 ELMO Shortlist, 1

Tags: trigonometry , inequalities

Let $x_1,x_2,x_3,y_1,y_2,y_3$ be nonzero real numbers satisfying $x_1+x_2+x_3=0, y_1+y_2+y_3=0$. Prove that \[\frac{x_1x_2+y_1y_2}{\sqrt{(x_1^2+y_1^2)(x_2^2+y_2^2)}}+\frac{x_2x_3+y_2y_3}{\sqrt{(x_2^2+y_2^2)(x_3^2+y_3^2)}}+\frac{x_3x_1+y_3y_1}{\sqrt{(x_3^2+y_3^2)(x_1^2+y_1^2)}} \ge -\frac32.\] [i]Ray Li, Max Schindler.[/i]

2008 Junior Balkan Team Selection Tests - Romania, 2

Tags: geometry , geometric transformation , reflection , inequalities , induction , rectangle , combinatorics

Let $ m,n$ be two natural nonzero numbers and sets $ A \equal{} \{ 1,2,...,n\}, B \equal{} \{1,2,...,m\}$. We say that subset $ S$ of Cartesian product $ A \times B$ has property $ (j)$ if $ (a \minus{} x)(b \minus{} y)\le 0$ for each pairs $ (a,b),(x,y) \in S$. Prove that every set $ S$ with propery $ (j)$ has at most $ m \plus{} n \minus{} 1$ elements. [color=#FF0000]The statement was edited, in order to reflect the actual problem asked. The sign of the inequality was inadvertently reversed into $ (a \minus{} x)(b \minus{} y)\ge 0$, and that accounts for the following two posts.[/color]

Kyiv City MO Juniors 2003+ geometry, 2021.9.51

Tags: geometry , circles , perpendicular

Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line passing through point $B$ intersects $\omega_1$ for the second time at point $C$ and $\omega_2$ at point $D$. The line $AC$ intersects circle $\omega_2$ for the second time at point $F$, and the line $AD$ intersects the circle $\omega_1$ for the second time at point $E$ . Let point $O$ be the center of the circle circumscribed around $\vartriangle AEF$. Prove that $OB \perp CD$.

PEN H Problems, 31

Tags: diophantine equation , ring theory

Determine all integer solutions of the system \[2uv-xy=16,\] \[xv-yu=12.\]

2001 Federal Math Competition of S&M, Problem 4

Tags: inequalities , algebra

Let $S$ be the set of all $n$-tuples of real numbers, with the property that among the numbers $x_1,\frac{x_1+x_2}2,\ldots,\frac{x_1+x_2+\ldots+x_n}n$ the least is equal to $0$, and the greatest is equal to $1$. Determine $$\max_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j)\qquad\text{and}\min_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j).$$

2014-2015 SDML (High School), 8

Tags: geometry , ellipse , conic

What is the maximum area of a triangle that can be inscribed in an ellipse with semi-axes $a$ and $b$? $\text{(A) }ab\frac{3\sqrt{3}}{4}\qquad\text{(B) }ab\qquad\text{(C) }ab\sqrt{2}\qquad\text{(D) }\left(a+b\right)\frac{3\sqrt{3}}{4}\qquad\text{(E) }\left(a+b\right)\sqrt{2}$

2008 ITAMO, 2

Tags: geometry

Let $ ABC$ be a triangle, all of whose angles are greater than $ 45^{\circ}$ and smaller than $ 90^{\circ}$. (a) Prove that one can fit three squares inside $ ABC$ in such a way that: (i) the three squares are equal (ii) the three squares have common vertex $ K$ inside the triangle (iii) any two squares have no common point but $ K$ (iv) each square has two opposite vertices onthe boundary of $ ABC$, while all the other points of the square are inside $ ABC$. (b) Let $ P$ be the center of the square which has $ AB$ as a side and is outside $ ABC$. Let $ r_{C}$ be the line symmetric to $ CK$ with respect to the bisector of $ \angle BCA$. Prove that $ P$ lies on $ r_{C}$.

Kvant 2021, M2660

Tags: combinatorial geometry , combinatorics

4 tokens are placed in the plane. If the tokens are now at the vertices of a convex quadrilateral $P$, then the following move could be performed: choose one of the tokens and shift it in the direction perpendicular to the diagonal of $P$ not containing this token; while shifting tokens it is prohibited to get three collinear tokens. Suppose that initially tokens were at the vertices of a rectangle $\Pi$, and after a number of moves tokens were at the vertices of one another rectangle $\Pi'$ such that $\Pi'$ is similar to $\Pi$ but not equal to $\Pi $. Prove that $\Pi$ is a square.

2013 AMC 10, 24

Tags: 3d geometry , geometry

A positive integer $n$ is [i]nice[/i] if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$) such that the sum of the four divisors is equal to $n$. How many numers in the set $\{2010, 2011, 2012,\ldots,2019\}$ are nice? ${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $

2010 Spain Mathematical Olympiad, 3

Tags: inequalities , perimeter , geometry

Let $ABCD$ be a convex quadrilateral. $AC$ and $BD$ meet at $P$, with $\angle APD=60^{\circ}$. Let $E,F,G$, and $H$ be the midpoints of $AB,BC,CD$ and $DA$ respectively. Find the greatest positive real number $k$ for which \[EG+3HF\ge kd+(1-k)s \] where $s$ is the semi-perimeter of the quadrilateral $ABCD$ and $d$ is the sum of the lengths of its diagonals. When does the equality hold?

1993 Taiwan National Olympiad, 6

Tags: trigonometry , induction , irrational number , algebra

Let $m$ be equal to $1$ or $2$ and $n<10799$ be a positive integer. Determine all such $n$ for which $\sum_{k=1}^{n}\frac{1}{\sin{k}\sin{(k+1)}}=m\frac{\sin{n}}{\sin^{2}{1}}$.

2018 China Second Round Olympiad, 1

Tags: algebra

Let $a,b \in \mathbb R,f(x)=ax+b+\frac{9}{x}.$ Prove that there exists $x_0 \in \left[1,9 \right],$ such that $|f(x_0)| \ge 2.$

2004 Romania National Olympiad, 4

Tags: function , real analysis

(a) Build a function $f : \mathbb R \to \mathbb R_+$ with the property $\left( \mathcal P \right)$, i.e. all $x \in \mathbb Q$ are local, strict minimum points. (b) Build a function $f : \mathbb Q \to \mathbb R_+$ such that every point is a local, strict minimum point and such that $f$ is unbounded on $I \cap \mathbb Q$, where $I$ is a non-degenerate interval. (c) Let $f: \mathbb R \to \mathbb R_+$ be a function unbounded on every $I \cap \mathbb Q$, where $I$ is a non-degenerate interval. Prove that $f$ doesn't have the property $\left( \mathcal P \right)$.

2015 Iran MO (3rd round), 2

Tags: number theory

$M_0 \subset \mathbb{N}$ is a non-empty set with a finite number of elements. Ali produces sets $ M_1,M_2,...,M_n $ in the following order: In step $n$, Ali chooses an element of $M_{n-1} $ like $b_n$ and defines $M_n$ as $$M_n = \left \{ b_nm+1 \vert m\in M_{n-1} \right \}$$ Prove that at some step Ali reaches a set which no element of it divides another element of it.

2022 IMAR Test, 3

Tags: geometry , parallelogram

Given is a parallelogram $XYZT$, and the variable points $A, B, C, D$ lie on the sides $XY, XT, TZ, ZY$ respectively, so that $ABCD$ is cyclic with circumcenter $O$, $AC \parallel XT$, and $BD \parallel XY$. Let $P$ be the intersection point of the lines $AD$ and $BC$, and let $Q$ be the intersection of the lines $AB$ and $CD$. Prove that the circle $(POQ)$ passes through a fixed point.

2024 Assara - South Russian Girl's MO, 7

Tags: combinatorics

There is a chip in one of the squares on the checkered board. In one move, she can move either $1$ square to the right, or diagonally $1$ to the left and $1$ up, or $1$ to the left and $3$ down (see Fig.). The chip made $n$ moves and returned to the starting square. Prove that a) $n$ is divisible by $2$, b) $n$ is divisible by $8$. [i]K.A.Sukhov[/i]

2014 Bosnia Herzegovina Team Selection Test, 1

Tags: trigonometry , geometry

Let $k$ be the circle and $A$ and $B$ points on circle which are not diametrically opposite. On minor arc $AB$ lies point arbitrary point $C$. Let $D$, $E$ and $F$ be foots of perpendiculars from $C$ on chord $AB$ and tangents of circle $k$ in points $A$ and $B$. Prove that $CD= \sqrt {CE \cdot CF}$

2021 MIG, 2

Tags:

Solve for $x$ if $20x + 21 = 121$. $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }7$

2017 VJIMC, 1

Tags:

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function satisfying \[f(x+2y)=2f(x)f(y)\] for every $x,y \in \mathbb{R}$. Prove that $f$ is constant.

2005 AMC 12/AHSME, 24

Tags: analytic geometry , parabola , graphing lines , slope , trigonometry , number theory , conic

All three vertices of an equilateral triangle are on the parabola $ y \equal{} x^2$, and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum of $ m/n$, where $ m$ and $ n$ are relatively prime positive integers. What is the value of $ m \plus{} n$? $ \textbf{(A)}\ 14\qquad \textbf{(B)}\ 15\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 17\qquad \textbf{(E)}\ 18$

2016 Harvard-MIT Mathematics Tournament, 2

Tags:

For positive integers $n$, let $c_n$ be the smallest positive integer for which $n^{c_n}-1$ is divisible by $210$, if such a positive integer exists, and $c_n = 0$ otherwise. What is $c_1 + c_2 + \dots + c_{210}$?

2023 Peru MO (ONEM), 4

Tags: geometry , incenter

Let $ABC$ be an acute scalene triangle and $K$ be a point inside it that belongs to the bisector of the angle $\angle ABC$. Let$ P$ be the point where the line $AK$ intersects the line perpendicular to $AB$ that passes through $B$, and let $Q$ be the point where the line $CK$ intersects the line perpendicular to $CB$ that passes through $B$. Let $L$ be the foot of the perpendicular drawn from $K$ on the line $AC$. Prove that if $P Q$ is perpendicular to $BL$, then $K$ is the incenter of $ABC$.

2009 Korea Junior Math Olympiad, 7

Tags: combinatorics , circular permutations

There are $3$ students from Korea, China, and Japan, so total of $9$ students are present. How many ways are there to make them sit down in a circular table, with equally spaced and equal chairs, such that the students from the same country do not sit next to each other? If array $A$ can become array $B$ by rotation, these two arrays are considered equal.

2014 BAMO, 5

Tags: game , tournament , combinatorics

A chess tournament took place between $2n+1$ players. Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly $k$ games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than $n-\sqrt{2k}$ and no more than $n+\sqrt{2k}$ games.

2025 Bulgarian Spring Mathematical Competition, 11.4

Tags: algebra , polynomial

We call two non-constant polynomials [i]friendly[/i] if each of them has only real roots, and every root of one polynomial is also a root of the other. For two friendly polynomials $ P(x), Q(x) $ and a constant $ C \in \mathbb{R}, C \neq 0 $, it is given that $ P(x) + C $ and $ Q(x) + C $ are also friendly polynomials. Prove that $ P(x) \equiv Q(x) $.