Olympiad problems

2017 Moldova Team Selection Test, 8

Tags: combinatorics

At a summer school there are $7$ courses. Each participant was a student in at least one course, and each course was taken by exactly $40$ students. It is known that for each $2$ courses there were at most $9$ students who took them both. Prove that at least $120$ students participated at this summer school.

1998 Italy TST, 4

Tags: algebra , polynomial

Find all polynomials $P(x) = x^n +a_1x^{n-1} +...+a_n$ whose zeros (with their multiplicities) are exactly $a_1,a_2,...,a_n$.

2014 USAJMO, 6

Tags: geometry , incenter , circumcircle , vector , trigonometry , ratio

Let $ABC$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M,N,P$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ and let $E,F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$, respectively. Let $U,V$ be the intersections of line $EF$ with line $MN$ and line $MP$, respectively, and let $X$ be the midpoint of arc $BAC$ of $\Gamma$. (a) Prove that $I$ lies on ray $CV$. (b) Prove that line $XI$ bisects $\overline{UV}$.

2010 Dutch BxMO TST, 2

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Find all functions $f : R \to R$ satisfying $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.

2020 Online Math Open Problems, 25

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Let $n$ be a positive integer with exactly twelve positive divisors $1=d_1 < \cdots < d_{12}=n$. We say $n$ is [i]trite[/i] if \[ 5 + d_6(d_6+d_4) = d_7d_4. \] Compute the sum of the two smallest trite positive integers. [i]Proposed by Brandon Wang[/i]

2009 China Team Selection Test, 3

Tags: induction , inequalities

Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y.$ Prove that $ 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|$

2018-2019 Fall SDPC, 7

Tags: geometry

The incircle of $\triangle{ABC}$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Point $P$ is chosen on $EF$ such that $AP$ is parallel to $BC$, and $AD$ intersects the incircle of $\triangle{ABC}$ again at $G$. Show that $\angle AGP = 90^{\circ}$.

2022 Belarusian National Olympiad, 9.1

Tags: geometry , similar triangles

Given an isosceles triangle $ABC$ with base $BC$. On the sides $BC$, $AC$ and $AB$ points $X,Y$ and $Z$ are chosen respectively such that triangles $ABC$ and $YXZ$ are similar. Point $W$ is symmetric to point $X$ with respect to the midpoint of $BC$. Prove that points $X,Y,Z$ and $W$ lie on a circle.

2012 IMO Shortlist, C2

Tags: double counting , combinatorics , extremal combinatorics

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

2013 NIMO Problems, 7

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Tyler has two calculators, both of which initially display zero. The first calculators has only two buttons, $[+1]$ and $[\times 2]$. The second has only the buttons $[+1]$ and $[\times 4]$. Both calculators update their displays immediately after each keystroke. A positive integer $n$ is called [i]ambivalent[/i] if the minimum number of keystrokes needed to display $n$ on the first calculator equals the minimum number of keystrokes needed to display $n$ on the second calculator. Find the sum of all ambivalent integers between $256$ and $1024$ inclusive. [i]Proposed by Joshua Xiong[/i]