Olympiad problems

1994 Iran MO (2nd round), 3

Find all functions $ f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}$ such that for all $ x,y \in \mathbb{Z}\setminus\{0\}$: \[ f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}\]

2016 Sharygin Geometry Olympiad, P9

Tags: geometry , incircle , right triangle , concurrency

Let $ABC$ be a right-angled triangle and $CH$ be the altitude from its right angle $C$. Points $O_1$ and $O_2$ are the incenters of triangles $ACH$ and $BCH$ respectively, $P_1$ and $P_2$ are the touching points of their incircles with $AC$ and $BC$. Prove that lines $O_1P_1$ and $O_2P_2$ meet on $AB$.

1966 IMO Longlists, 35

Tags: algebra , polynomial , irrational number , root

Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.

1997 May Olympiad, 3

Tags: combinatorics

On an $8 \times 8$ board, $10$ checkers have been placed, each occupying a square. On each square without a token, a number between $0$ and $8$ is written, which is equal to the number of tokens placed on its neighboring squares. Neighboring cells are those that have a side or a vertex in common. Give a distribution of the tiles that makes the sum of the numbers written on the board the greatest possible.

2012-2013 SDML (High School), 5

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Jimmy invites Kima, Lester, Marlo, Namond, and Omar to dinner. There are nine chairs at Jimmy's round dinner table. Jimmy sits in the chair nearest the kitchen. How many different ways can Jimmy's five dinner guests arrange themselves in the remaining $8$ chairs at the table if Kima and Marlo refuse to be seated in adjacent chairs?

2011 JBMO Shortlist, 1

Tags: number theory , algebra , integer equation , diophantine equation , modular arithmetic

Solve in positive integers the equation $1005^x + 2011^y = 1006^z$.

2015 Caucasus Mathematical Olympiad, 2

Tags: geometry , equal angles , equal segments , convex quadrilateral

In the convex quadrilateral $ABCD$, point $K$ is the midpoint of $AB$, point $L$ is the midpoint of $BC$, point $M$ is the midpoint of CD, and point $N$ is the midpoint of $DA$. Let $S$ be a point lying inside the quadrilateral $ABCD$ such that $KS = LS$ and $NS = MS$ .Prove that $\angle KSN = \angle MSL$.

2019 Centers of Excellency of Suceava, 1

Tags: inequalities

For $ a,b,c,d $ positive, prove: $$ \frac{2a}{a^2+bc} +\frac{2b}{b^2+cd} +\frac{2c}{c^2+da} +\frac{2d}{d^2+ab}\le \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d} $$ [i]Dan Popescu[/i]

2014 Putnam, 6

Tags: function , slope , integration , real analysis

Let $f:[0,1]\to\mathbb{R}$ be a function for which there exists a constant $K>0$ such that $|f(x)-f(y)|\le K|x-y|$ for all $x,y\in [0,1].$ Suppose also that for each rational number $r\in [0,1],$ there exist integers $a$ and $b$ such that $f(r)=a+br.$ Prove that there exist finitely many intervals $I_1,\dots,I_n$ such that $f$ is a linear function on each $I_i$ and $[0,1]=\bigcup_{i=1}^nI_i.$

1974 IMO Longlists, 47

Tags: ratio , geometry

Given two points $A,B$ outside of a given plane $P,$ find the positions of points $M$ in the plane $P$ for which the ratio $\frac{MA}{MB}$ takes a minimum or maximum.

1999 Greece JBMO TST, 4

Tags: perfect power , number theory , perfect cube

Examine whether exists $n \in N^*$, such that: (a) $3n$ is perfect cube, $4n$ is perfect fourth power and $5n$ perfect fifth power (b) $3n$ is perfect cube, $4n$ is perfect fourth power, $5n$ perfect fifth power and $6n$ perfect sixth power

1984 IMO Longlists, 10

Tags: 3d geometry , tetrahedron , ratio , vector , inequalities , geometry

Assume that the bisecting plane of the dihedral angle at edge $AB$ of the tetrahedron $ABCD$ meets the edge $CD$ at point $E$. Denote by $S_1, S_2, S_3$, respectively the areas of the triangles $ABC, ABE$, and $ABD$. Prove that no tetrahedron exists for which $S_1, S_2, S_3$ (in this order) form an arithmetic or geometric progression.

2019 Kyiv Mathematical Festival, 4

Tags: combinatorics

99 dwarfs stand in a circle, some of them wear hats. There are no adjacent dwarfs in hats and no dwarfs in hats with exactly 48 dwarfs standing between them. What is the maximal possible number of dwarfs in hats?

1995 AMC 8, 12

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A ''lucky'' year is one in which at least one date, when written in the form month/day/year, has the following property: ''The product of the month times the day equals the last two digits of the year''. For example, 1956 is a lucky year because it has the date 7/8/56 and $7\times 8 = 56$. Which of the following is NOT a lucky year? $\text{(A)}\ 1990 \qquad \text{(B)}\ 1991 \qquad \text{(C)}\ 1992 \qquad \text{(D)}\ 1993 \qquad \text{(E)}\ 1994$

1974 Miklós Schweitzer, 7

Tags: algebra , polynomial , trigonometry , integration , real analysis

Given a positive integer $ m$ and $ 0 < \delta <\pi$, construct a trigonometric polynomial $ f(x)\equal{}a_0\plus{} \sum_{n\equal{}1}^m (a_n \cos nx\plus{}b_n \sin nx)$ of degree $ m$ such that $ f(0)\equal{}1, \int_{ \delta \leq |x| \leq \pi} |f(x)|dx \leq c/m,$ and $ \max_{\minus{}\pi \leq x \leq \pi}|f'(x)| \leq c/{\delta}$, for some universal constant $ c$. [i]G. Halasz[/i]

2006 Canada National Olympiad, 4

Tags: search , combinatorics , graph theory , global

Consider a round-robin tournament with $2n+1$ teams, where each team plays each other team exactly one. We say that three teams $X,Y$ and $Z$, form a [i]cycle triplet [/i] if $X$ beats $Y$, $Y$ beats $Z$ and $Z$ beats $X$. There are no ties. a)Determine the minimum number of cycle triplets possible. b)Determine the maximum number of cycle triplets possible.

2022 Rioplatense Mathematical Olympiad, 1

Tags: number theory

In how many ways can the numbers from $2$ to $2022$ be arranged so that the first number is a multiple of $1$, the second number is a multiple of $2$, the third number is a multiple of $3$, and so on untile the last number is a multiple of $2021$?

1967 AMC 12/AHSME, 20

Tags: ratio , geometric series , geometry

A circle is inscribed in a square of side $m$, then a square is inscribed in that circle, then a circle is inscribed in the latter square, and so on. If $S_n$ is the sum of the areas of the first $n$ circles so inscribed, then, as $n$ grows beyond all bounds, $S_n$ approaches: $\textbf{(A)}\ \frac{\pi m^2}{2}\qquad \textbf{(B)}\ \frac{3\pi m^2}{8}\qquad \textbf{(C)}\ \frac{\pi m^2}{3}\qquad \textbf{(D)}\ \frac{\pi m^2}{4}\qquad \textbf{(E)}\ \frac{\pi m^2}{8}$

1998 Romania Team Selection Test, 2

Tags: arithmetic sequence

Let $ n \ge 3$ be a prime number and $ a_{1} < a_{2} < \cdots < a_{n}$ be integers. Prove that $ a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $ \{0, 1, 2, \cdots \}$ into sets $ A_{1},A_{2},\cdots,A_{n}$ such that \[ a_{1} \plus{} A_{1} \equal{} a_{2} \plus{} A_{2} \equal{} \cdots \equal{} a_{n} \plus{} A_{n},\] where $ x \plus{} A$ denotes the set $ \{x \plus{} a \vert a \in A \}$.

2005 Indonesia MO, 5

Tags: function , floor function , number theory

For an arbitrary real number $ x$, $ \lfloor x\rfloor$ denotes the greatest integer not exceeding $ x$. Prove that there is exactly one integer $ m$ which satisfy $ \displaystyle m\minus{}\left\lfloor \frac{m}{2005}\right\rfloor\equal{}2005$.

1977 Germany Team Selection Test, 4

Tags: modular arithmetic , divisibility , digit , decimal representation , digit sum

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

2008 Chile National Olympiad, 2

Tags: right triangle , isosceles , triangle , geometry

Let $ABC$ be right isosceles triangle with right angle in $A$. Given a point $P$ inside the triangle, denote by $a, b$ and $c$ the lengths of $PA, PB$ and $PC$, respectively. Prove that there is a triangle whose sides have a length of $a\sqrt2 , b$ and $c$.

2015 Peru IMO TST, 12

Tags: combinatorics

Find the least positive real number $\alpha$ with the following property: if the weight of a finite number of pumpkins is $1$ ton and the weight of each pumpkin is not greater than $\alpha$ tons then the pumpkins can be distributed in $50$ boxes (some boxes can be empty) so that there is no more than $\alpha$ tons of pumpkins in each box.

2012 Miklós Schweitzer, 4

Tags: geometry

Let $K$ be a convex shape in the $n$ dimensional space, having unit volume. Let $S \subset K$ be a Lebesgue measurable set with measure at least $1-\varepsilon$, where $0<\varepsilon<1/3$. Prove that dilating $K$ from its centroid by the ratio of $2\varepsilon \ln(1/\varepsilon)$, the shape obtained contains the centroid of $S$.

2005 AIME Problems, 6

Tags: floor function , algebra , polynomial , quadratic , function , trigonometry

Let $P$ be the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005$. Find $\lfloor P\rfloor$.