Olympiad problems

1965 All Russian Mathematical Olympiad, 070

Prove that the sum of the lengths of the polyhedron edges exceeds its tripled diameter (distance between two farest vertices).

1949 Moscow Mathematical Olympiad, 158

Tags: diophantine , number theory

a) Prove that $x^2 + y^2 + z^2 = 2xyz$ for integer $x, y, z$ only if $x = y = z = 0$. b) Find integers $x, y, z, u$ such that $x^2 + y^2 + z^2 + u^2 = 2xyzu$.

Ukraine Correspondence MO - geometry, 2014.10

Tags: circumcircle , incenter , geometry

In the triangle $ABC$, it is known that $AC <AB$. Let $\ell$ be tangent to the circumcircle of triangle $ABC$ drawn at point $A$. A circle with center $A$ and radius $AC$ intersects segment $AB$ at point $D$, and line $\ell$ at points $E$ and $F$. Prove that one of the lines $DE$ and $DF$ passes through the center inscribed circle of triangle $ABC$.

2015 Silk Road, 4

Tags: geometry

Let O be a circumcenter of an acute-angled triangle ABC. Consider two circles ω and Ω inscribed in the angle BAC in such way that ω is tangent from the outside to the arc BOC of a circle circumscribed about the triangle BOC; and the circle Ω is tangent internally to a circumcircle of triangle ABC. Prove that the radius of Ω is twice the radius ω.

Fractal Edition 1, P4

Tags: geometry

In triangle $ ABC $, $ D $, $ E $, and $ F $ are the feet of the perpendiculars from the vertices $ A $, $ B $, and $ C $, respectively. The parallel to $ EF $ through $ D $ intersects $ AB $ at $ P_B $ and $ AC $ at $ P_C $. Let $ X $ be the intersection of $ EF $ and $ BC $. Prove that the circumcircle of triangle $ P_B P_C X $ passes through the midpoint of side $ BC $.

2011 Indonesia TST, 3

Tags: coloring , combinatorics

Given a board consists of $n \times n$ unit squares ($n \ge 3$). Each unit square is colored black and white, resembling a chessboard. In each step, TOMI can choose any $2 \times 2$ square and change the color of every unit square chosen with the other color (white becomes black and black becomes white). Find every $n$ such that after a finite number of moves, every unit square on the board has a same color.

2014 Brazil National Olympiad, 4

Tags: induction , polynomial , algebra

The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as \[P_0(x)=x,\quad P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1),\quad n\ge 1.\] Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$.

2013 ELMO Shortlist, 4

Tags: number theory

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

1993 Korea - Final Round, 3

Tags: modular arithmetic , number theory

Find the smallest $x \in\mathbb{N}$ for which $\frac{7x^{25}-10}{83}$ is an integer.

2015 Balkan MO Shortlist, A2

Tags: circumradius , semiperimeter , geometric inequality , inequalities , inradius , geometry

Let $a,b,c$ be sidelengths of a triangle and $r,R,s$ be the inradius, the circumradius and the semiperimeter respectively of the same triangle. Prove that: $$\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c} \leq \frac{r}{16Rs}+\frac{s}{16Rr} + \frac{11}{8s}$$ (Albania)

2014 Harvard-MIT Mathematics Tournament, 5

Tags: counting , distinguishability

Eli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the $4$ people. The following constraints are imposed: $\bullet$ Every person must get a positive integer number of shares, and all $16$ shares must be given out. $\bullet$ No one person can have more shares than the other three people combined. Assuming that shares are indistinguishable, but people are distinguishable, in how many ways can the shares be given out?

1987 Flanders Math Olympiad, 2

Tags: geometric inequality , geometry

Two parallel lines $a$ and $b$ meet two other lines $c$ and $d$. Let $A$ and $A'$ be the points of intersection of $a$ with $c$ and $d$, respectively. Let $B$ and $B'$ be the points of intersection of $b$ with $c$ and $d$, respectively. If $X$ is the midpoint of the line segment $A A'$ and $Y$ is the midpoint of the segment $BB'$, prove that $$|XY| \le \frac{|AB|+|A'B'|}{2}.$$

1995 AMC 8, 5

Tags:

Find the smallest whole number that is larger than the sum \[2\dfrac{1}{2}+3\dfrac{1}{3}+4\dfrac{1}{4}+5\dfrac{1}{5}.\] $\text{(A)}\ 14 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 16 \qquad \text{(D)}\ 17 \qquad \text{(E)}\ 18$

1966 Dutch Mathematical Olympiad, 3

Tags: perfect square , number theory

How many natural numbers are there whose square is a thirty-digit number which has the following curious property: If that thirty-digit number is divided from left to right into three groups of ten digits, then the numbers given by the middle group and the right group formed numbers are both four times the number formed by the left group?

1958 AMC 12/AHSME, 19

Tags: ratio

The sides of a right triangle are $ a$ and $ b$ and the hypotenuse is $ c$. A perpendicular from the vertex divides $ c$ into segments $ r$ and $ s$, adjacent respectively to $ a$ and $ b$. If $ a : b \equal{} 1 : 3$, then the ratio of $ r$ to $ s$ is: $ \textbf{(A)}\ 1 : 3\qquad \textbf{(B)}\ 1 : 9\qquad \textbf{(C)}\ 1 : 10\qquad \textbf{(D)}\ 3 : 10\qquad \textbf{(E)}\ 1 : \sqrt{10}$

2013 HMIC, 3

Tags: circumcircle , geometry

Triangle $ABC$ is inscribed in a circle $\omega$ such that $\angle A = 60^o$ and $\angle B = 75^o$. Let the bisector of angle $A$ meet $BC$ and $\omega$ at $E$ and $D$, respectively. Let the reflections of $A$ across $D$ and $C$ be $D'$ and $C'$ , respectively. If the tangent to $\omega$ at $A$ meets line $BC$ at $P$, and the circumcircle of $APD'$ meets line $AC$ at $F \ne A$, prove that the circumcircle of $C'FE$ is tangent to $BC$ at $E$.

2022 Brazil National Olympiad, 3

Tags: geometry

Let $ABC$ be a triangle with incenter $I$ and let $\Gamma$ be its circumcircle. Let $M$ be the midpoint of $BC$, $K$ the midpoint of the arc $BC$ which does not contain $A$, $L$ the midpoint of the arc $BC$ which contains $A$ and $J$ the reflection of $I$ by the line $KL$. The line $LJ$ intersects $\Gamma$ again at the point $T\neq L$. The line $TM$ intersects $\Gamma$ again at the point $S\neq T$. Prove that $S, I, M, K$ lie on the same circle.

2005 Bulgaria Team Selection Test, 5

Tags: geometry , circumcircle , reflection , geometric transformation , trigonometry , incenter , trapezoid

Let $ABC$, $AC \not= BC$, be an acute triangle with orthocenter $H$ and incenter $I$. The lines $CH$ and $CI$ meet the circumcircle of $\bigtriangleup ABC$ at points $D$ and $L$, respectively. Prove that $\angle CIH = 90^{\circ}$ if and only if $\angle IDL = 90^{\circ}$

2017 Princeton University Math Competition, 7

Tags: combinatorics

$2017$ voters vote by submitting a ranking of the integers $\{1, 2, ..., 38\}$ from favorite (a vote for that value in $1$st place) to least favorite (a vote for that value in $38$th/last place). Let $a_k$ be the integer that received the most $k$th place votes (the smallest such integer if there is a tie). Find the maximum possible value of $\Sigma_{k=1}^{38} a_k$.

2002 Croatia Team Selection Test, 3

Tags: prime , number theory

Prove that if $n$ is a natural number such that $1 + 2^n + 4^n$ is prime then $n = 3^k$ for some $k \in N_0$.

2015 Tournament of Towns, 4

Tags: combinatorial geometry , combinatorics

A convex$N-$gon with equal sides is located inside a circle. Each side is extended in both directions up to the intersection with the circle so that it contains two new segments outside the polygon. Prove that one can paint some of these new $2N$ segments in red and the rest in blue so that the sum of lengths of all the red segments would be the same as for the blue ones. [i]($8$ points)[/i]

1965 AMC 12/AHSME, 19

Tags: polynomial , vieta , algebra

If $ x^4 \plus{} 4x^3 \plus{} 6px^2 \plus{} 4qx \plus{} r$ is exactly divisible by $ x^3 \plus{} 3x^2 \plus{} 9x \plus{} 3$, the value of $ (p \plus{} q)r$ is: $ \textbf{(A)}\ \minus{} 18 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 45 \qquad$

2013 NZMOC Camp Selection Problems, 1

Tags: algebra , number theory

You have a set of five weights, together with a balance that allows you to compare the weight of two things. The weights are known to be $10$, $20$,$30$,$40$ and $50$ grams, but are otherwise identical except for their labels. The $10$ and $50$ gram weights are clearly labelled, but the labels have been erased on the remaining weights. Using the balance exactly once, is it possible to determine what one of the three unlabelled weights is? If so, explain how, and if not, explain why not.

2018 District Olympiad, 2

Tags: divide , number theory

Find the pairs of integers $(a, b)$ such that $a^2 + 2b^2 + 2a +1$ is a divisor of $2ab$.

2000 Slovenia National Olympiad, Problem 4

Tags: analytic geometry , geometry

All vertices of a convex $n$-gon ($n\ge3$) in the plane have integer coordinates. Show that its area is at least $\frac{n-2}2$.