Olympiad problems

1918 Eotvos Mathematical Competition, 2

Find three distinct natural numbers such that the sum of their reciprocals is an integer.

2009 Indonesia TST, 1

Tags: inequalities

Let $ x_1,x_2,\dots,x_n$ be positive real numbers. Let $ m\equal{}\min\{x_1,x_2,\dots,x_n\}$, $ M\equal{}\max\{x_1,x_2,\dots,x_n\}$, $ A\equal{}\frac{1}{n}(x_1\plus{}x_2\plus{}\dots\plus{}x_n)$, and $ G\equal{}\sqrt[n]{x_1x_2 \dots x_n}$. Prove that \[ A\minus{}G \ge \frac{1}{n}(\sqrt{M}\minus{}\sqrt{m})^2.\]

2022 239 Open Mathematical Olympiad, 1

Tags: cell , combinatorics , board , diagonal

A piece is placed in the lower left-corner cell of the $15 \times 15$ board. It can move to the cells that are adjacent to the sides or the corners of its current cell. It must also alternate between horizontal and diagonal moves $($the first move must be diagonal$).$ What is the maximum number of moves it can make without stepping on the same cell twice$?$

1935 Moscow Mathematical Olympiad, 008

Tags: arithmetic progression , inradius , altitude , geometry , arithmetic sequence

Prove that if the lengths of the sides of a triangle form an arithmetic progression, then the radius of the inscribed circle is one third of one of the heights of the triangle.

2021 Iran MO (3rd Round), 2

Tags: geometry , geometric transformation , reflection , circumcircle , tangent circle

Given an acute triangle $ABC$, let $AD$ be an altitude and $H$ the orthocenter. Let $E$ denote the reflection of $H$ with respect to $A$. Point $X$ is chosen on the circumcircle of triangle $BDE$ such that $AC\| DX$ and point $Y$ is chosen on the circumcircle of triangle $CDE$ such that $DY\| AB$. Prove that the circumcircle of triangle $AXY$ is tangent to that of $ABC$.

2014 IPhOO, 3

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Which of the following derived units is equivalent to units of velocity? $ \textbf {(A) } \dfrac {\text {W}}{\text {N}} \qquad \textbf {(B) } \dfrac {\text {N}}{\text {W}} \qquad \textbf {(C) } \dfrac {\text {W}}{\text {N}^2} \qquad \textbf {(D) } \dfrac {\text {W}^2}{\text {N}} \qquad \textbf {(E) } \dfrac {\text {N}^2}{\text {W}^2} $ [i]Problem proposed by Ahaan Rungta[/i]

VMEO IV 2015, 12.3

Tags: geometry , circumcircle , equal segments , symmedian

Triangle $ABC$ is inscribed in circle $(O)$. $ P$ is a point on arc $BC$ that does not contain $ A$ such that $AP$ is the symmedian of triangle $ABC$. $E ,F$ are symmetric of $P$ wrt $CA, AB$ respectively . $K$ is symmetric of $A$ wrt $EF$. $L$ is the projection of $K$ on the line passing through $A$ and parallel to $BC$. Prove that $PA=PL$.

2009 All-Russian Olympiad Regional Round, 11.1

Tags: algebra , polynomial

Square trinomial $f(x)$ is such that the polynomial (f(x))^5 - f(x) has exactly three real roots. Find the ordinate of the vertex of the graph of this trinomial.

1981 Tournament Of Towns, (008) 2

Tags: symmetry , geometry , geometric transformation

$M$ is a finite set of points in a plane. Point $O$ in the plane is called an “almost centre of symmetry” of set $M$ if it is possible to remove from $M$ one point in such a way that among the remaining members $O$ is the centre of symmetry in the usual sense. How many such “almost centres of symmetry” may a finite point set in a plane have? Indicate all such points. (V Prasolov, Moscow)

2000 Belarus Team Selection Test, 2.2

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Real numbers $a$, $b$, $c$ satisfy the equation $$2a^3-b^3+2c^3-6a^2b+3ab^2-3ac^2-3bc^2+6abc=0$$. If $a<b$, find which of the numbers $b$, $c$ is larger.

2008 National Chemistry Olympiad, 9

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How many moles of oxygen gas are produced by the decomposition of $245$ g of potassium chlorate? \[\ce{2KClO3(s)} \rightarrow \ce{2KCl(s)} + \ce{3O2(g)}\] Given: Molar Mass/ $\text{g} \cdot \text{mol}^{-1}$ $\ce{KClO3}$: $122.6$ $ \textbf{(A)}\hspace{.05in}1.50 \qquad\textbf{(B)}\hspace{.05in}2.00 \qquad\textbf{(C)}\hspace{.05in}2.50 \qquad\textbf{(D)}\hspace{.05in}3.00 \qquad $

2025 Taiwan TST Round 1, N

Tags: permutation , number theory

Find all positive integers $n$ such that there exist two permutations $a_0,a_1,\ldots,a_{n-1}$ and $b_0,b_1,\ldots,b_{n-1}$ of the set $\lbrace0,1,\ldots,n-1\rbrace$, satisfying the condition $$ia_i\equiv b_i\pmod{n}$$ for all $0\le i\le n-1$. [i]Proposed by Fysty[/i]

2023 Francophone Mathematical Olympiad, 3

Tags: geometry , circumcircle , tangent circle

Let $\Gamma$ and $\Gamma'$ be two circles with centres $O$ and $O'$, such that $O$ belongs to $\Gamma'$. Let $M$ be a point on $\Gamma'$, outside of $\Gamma$. The tangents to $\Gamma$ that go through $M$ touch $\Gamma$ in two points $A$ and $B$, and cross $\Gamma'$ again in two points $C$ and $D$. Finally, let $E$ be the crossing point of the lines $AB$ and $CD$. Prove that the circumcircles of the triangles $CEO'$ and $DEO'$ are tangent to $\Gamma'$.

1990 Baltic Way, 1

Tags: inequalities , triangle inequality

Numbers $1, 2, \dots , n$ are written around a circle in some order. What is the smallest possible sum of the absolute differences of adjacent numbers?

2011 Estonia Team Selection Test, 1

Tags: geometric transformation , geometry

Two circles lie completely outside each other.Let $A$ be the point of intersection of internal common tangents of the circles and let $K$ be the projection of this point onto one of their external common tangents.The tangents,different from the common tangent,to the circles through point $K$ meet the circles at $M_1$ and $M_2$.Prove that the line $AK$ bisects angle $M_1 KM_2$.

2009 Kazakhstan National Olympiad, 4

Tags: inequalities

Let $a,b,c,d $-reals positive numbers. Prove inequality: $\frac{a^2+b^2+c^2}{ab+bc+cd}+\frac{b^2+c^2+d^2}{bc+cd+ad}+\frac{a^2+c^2+d^2}{ab+ad+cd}+\frac{a^2+b^2+d^2}{ab+ad+bc} \geq 4$

1991 Arnold's Trivium, 9

Tags: algebra , polynomial

Does every positive polynomial in two real variables attain its lower bound in the plane?

2021 Harvard-MIT Mathematics Tournament., 6

Tags: combi

A light pulse starts at a corner of a reflective square. It bounces around inside the square, reflecting off of the square’s perimeter $n$ times before ending in a different corner. The path of the light pulse, when traced, divides the square into exactly $2021$ regions. Compute the smallest possible value of $n$.

2014 Baltic Way, 6

Tags: combinatorics

In how many ways can we paint $16$ seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd?

1991 Cono Sur Olympiad, 1

Tags: combinatorics

A game consists in $9$ coins (blacks or whites) arrenged in the following position (see picture 1). If you choose $1$ coin on the border of the square, this coin and it's neighbours change their color. If you choose the coin at the centre, it doesn't change it's color, but the other $8$ coins do. Here is an example of $9$ white coins, and the changes of their colors, choosing the coin said: (see picture 2). Is it possible, starting with $9$ white coins, to have $9$ black coins?.

1979 IMO Longlists, 50

Tags: additive number theory , additive combinatorics , system of equations , number theory

Let $m$ positive integers $a_1, \dots , a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \dots , b_n$ such that all sums of distinct $b_k$’s are distinct and all $a_i \ (i \leq m)$ occur among them.

2016 Germany Team Selection Test, 1

Tags: concyclic , geometry , circles

The two circles $\Gamma_1$ and $\Gamma_2$ with the midpoints $O_1$ resp. $O_2$ intersect in the two distinct points $A$ and $B$. A line through $A$ meets $\Gamma_1$ in $C \neq A$ and $\Gamma_2$ in $D \neq A$. The lines $CO_1$ and $DO_2$ intersect in $X$. Prove that the four points $O_1,O_2,B$ and $X$ are concyclic.

2019 Putnam, A4

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Let $f$ be a continuous real-valued function on $\mathbb R^3$. Suppose that for every sphere $S$ of radius $1$, the integral of $f(x,y,z)$ over the surface of $S$ equals zero. Must $f(x,y,z)$ be identically zero?

1979 AMC 12/AHSME, 21

Tags: ratio , geometry

The length of the hypotenuse of a right triangle is $h$ , and the radius of the inscribed circle is $r$. The ratio of the area of the circle to the area of the triangle is $\textbf{(A) }\frac{\pi r}{h+2r}\qquad\textbf{(B) }\frac{\pi r}{h+r}\qquad\textbf{(C) }\frac{\pi}{2h+r}\qquad\textbf{(D) }\frac{\pi r^2}{r^2+h^2}\qquad\textbf{(E) }\text{none of these}$

2025 Ukraine National Mathematical Olympiad, 9.3

Tags: number theory

Anton wrote $4$ positive integers on the board. Oleksii calculated their product, while Fedir calculated the sum of their fourth powers. Is it possible that Oleksii's number and Fedir's number have the same number of digits and that these numbers are written as digit-reversals of each other? [i]Proposed by Fedir Yudin and Mykhailo Shtandenko[/i]