Olympiad problems

2009 Tournament Of Towns, 1

Tags: induction

One hundred pirates played cards. When the game was over, each pirate calculated the amount he won or lost. The pirates have a gold sand as a currency; each has enough to pay his debt. Gold could only change hands in the following way. Either one pirate pays an equal amount to every other pirate, or one pirate receives the same amount from every other pirate. Prove that after several such steps, it is possible for each winner to receive exactly what he has won and for each loser to pay exactly what he has lost. [i](4 points)[/i]

2022 Saint Petersburg Mathematical Olympiad, 5

Tags: geometry

Altitudes $AA_1, BB_1, CC_1$ of acute triangle $ABC$ intersect at point $H$. On the tangent drawn from point $C$ to the circle $(AB_1C_1)$, the perpendicular $HQ$ is drawn (the point $Q$ lies inside the triangle $ABC$). Prove that the circle passing through the point $B_1$ and touching the line $AB$ at point $A$ is also tangent to line $A_1Q$.

2018 Peru Iberoamerican Team Selection Test, P9

Tags: geometry , circumcircle , collinear , collinearity

Let $\Gamma$ be the circumcircle of a triangle $ABC$ with $AB <BC$, and let $M$ be the midpoint from the side $AC$ . The median of side $AC$ cuts $\Gamma$ at points $X$ and $Y$ ($X$ in the arc $ABC$). The circumcircle of the triangle $BMY$ cuts the line $AB$ at $P$ ($P \ne B$) and the line $BC$ in $Q$ ($Q \ne B$). The circumcircles of the triangles $PBC$ and $QBA$ are cut in $R$ ($R \ne B$). Prove that points $X, B$ and $R$ are collinear.

2019 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: number theory

Prove that there are infinite triples $(a,b,c)$ of positive integers $a,b,c>1$, $gcd(a,b)=gcd(b,c)=gcd(c,a)=1$ such that $a+b+c$ divides $a^b+b^c+c^a$.

1994 APMO, 5

Tags: floor function , number theory , logarithm

You are given three lists $A$, $B$, and $C$. List $A$ contains the numbers of the form $10^k$ in base $10$, with $k$ any integer greater than or equal to $1$. Lists $B$ and $C$ contain the same numbers translated into base $2$ and $5$ respectively: $$\begin{array}{lll} A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}$$ Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists $B$ or $C$ that has exactly $n$ digits.

2013 Switzerland - Final Round, 8

Tags: algebra , inequalities

Let $a, b, c > 0$ be real numbers. Show the following inequality: $$a^2 \cdot \frac{a - b}{a + b}+ b^2\cdot \frac{b - c}{b + c}+ c^2\cdot \frac{c - a}{c + a} \ge 0 .$$ When does equality holds?

2009 Purple Comet Problems, 7

Tags:

How many distinct four letter arrangements can be formed by rearranging the letters found in the word [b]FLUFFY[/b]? For example, FLYF and ULFY are two possible arrangements.

2019 Sharygin Geometry Olympiad, 11

Tags: geometry

Morteza marks six points in the plane. He then calculates and writes down the area of every triangle with vertices in these points ($20$ numbers). Is it possible that all of these numbers are integers, and that they add up to $2019$?

PEN P Problems, 30

Tags: additive number theory

Let $a_{1}, a_{2}, a_{3}, \cdots$ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$, where $i, j, $ and $k$ are not necessarily distinct. Determine $a_{1998}$.

2022 Iran Team Selection Test, 1

Tags: combinatorics

Morteza Has $100$ sets. at each step Mahdi can choose two distinct sets of them and Morteza tells him the intersection and union of those two sets. Find the least steps that Mahdi can find all of the sets. Proposed by Morteza Saghafian

2010 IMO Shortlist, 3

Tags: matrix , combinatorics , chessboard , counting

2500 chess kings have to be placed on a $100 \times 100$ chessboard so that [b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); [b](ii)[/b] each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.) [i]Proposed by Sergei Berlov, Russia[/i]

2009 Cuba MO, 4

Tags: algebra , functional

Determine all the functions $f : R \to R$ such that: $$x + f(xf(y)) = f(y) + yf(x)$$ for all $x, y \in R$.

1966 Polish MO Finals, 5

Tags: geometry , hexagon , concurrency

Each of the diagonals $AD$, $BE$, $CF$ of a convex hexagon $ABCDEF$ bisects the area of the hexagon. Prove that these three diagonals pass through the same point.

1992 Chile National Olympiad, 4

Tags: equilateral , parallel

Given three parallel lines, prove that there are three points, one on each line, which are the vertices of an equilateral triangle.

2008 IberoAmerican Olympiad For University Students, 2

Tags: polynomial , ratio , geometry , algebra

Prove that for each natural number $n$ there is a polynomial $f$ with real coefficients and degree $n$ such that $ p(x)=f(x^2-1)$ is divisible by $f(x)$ over the ring $\mathbb{R}[x]$.

2019 Iran Team Selection Test, 6

Tags: inequalities

$x,y$ and $z$ are real numbers such that $x+y+z=xy+yz+zx$. Prove that $$\frac{x}{\sqrt{x^4+x^2+1}}+\frac{y}{\sqrt{y^4+y^2+1}}+\frac{z}{\sqrt{z^4+z^2+1}}\geq \frac{-1}{\sqrt{3}}.$$ [i]Proposed by Navid Safaei[/i]

2012 IFYM, Sozopol, 5

Tags: number theory , sum of powers , sequence

We are given the following sequence: $a_1=8,a_2=20,a_{n+2}=a_{n+1}^2+12a_n a_{n+1}+11a_n$. Prove that none of the members of the sequence can be presented as a sum of three seventh powers of natural numbers.

2024 Alborz Mathematical Olympiad, P1

Tags: number theory , complete residue system , divisor

Find all positive integers $n$ such that if $S=\{d_1,d_2,\cdots,d_k\}$ is the set of positive integer divisors of $n$, then $S$ is a complete residue system modulo $k$. (In other words, for every pair of distinct indices $i$ and $j$, we have $d_i\not\equiv d_j \pmod{k}$). Proposed by Heidar Shushtari

2004 Putnam, A4

Tags: algebra , polynomial

Show that for any positive integer $n$ there is an integer $N$ such that the product $x_1x_2\cdots x_n$ can be expressed identically in the form \[x_1x_2\cdots x_n=\sum_{i=1}^Nc_i(a_{i1}x_1+a_{i2}x_2+\cdots +a_{in}x_n)^n\] where the $c_i$ are rational numbers and each $a_{ij}$ is one of the numbers, $-1,0,1.$