Olympiad problems

2003 Greece Junior Math Olympiad, 4

Tags: number theory

Find all positive integers which can be written in the form $(mn+1)/(m+n)$, where $m,n$ are positive integers.

2013 Thailand Mathematical Olympiad, 6

Tags: functional equation , functional , algebra

Determine all functions $f$ : $\mathbb R\to\mathbb R$ satisfying $(x^2+y^2)f(xy)=f(x)f(y)f(x^2+y^2)$ $\forall x,y\in\mathbb R$

2021 Taiwan Mathematics Olympiad, 1.

Tags: combinatorics

Find the largest $K$ satisfying the following: Given any closed intervals $A_1,\ldots, A_N$ of length $1$ where $N$ is an arbitrary positive integer. If their union is $[0,2021]$, then we can always find $K$ intervals from $A_1,\ldots, A_N$ such that the intersection of any two of them is empty.

1999 AIME Problems, 6

Tags: analytic geometry , geometry , floor function , inversion

A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\sqrt{x},\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find the greatest integer that does not exceed $k.$

2003 Tournament Of Towns, 1

Tags: combinatorics

$2003$ dollars are placed into $N$ purses, and the purses are placed into $M$ pockets. It is known that $N$ is greater than the number of dollars in any pocket. Is it true that there is a purse with less than $M$ dollars in it?

2011 Croatia Team Selection Test, 3

Tags: circumcircle , ratio , geometric transformation , homothety , geometry

Triangle $ABC$ is given with its centroid $G$ and cicumcentre $O$ is such that $GO$ is perpendicular to $AG$. Let $A'$ be the second intersection of $AG$ with circumcircle of triangle $ABC$. Let $D$ be the intersection of lines $CA'$ and $AB$ and $E$ the intersection of lines $BA'$ and $AC$. Prove that the circumcentre of triangle $ADE$ is on the circumcircle of triangle $ABC$.

2010 Argentina National Olympiad, 3

Tags: algebra , inequalities , number theory

The positive integers $a,b,c$ are less than $99$ and satisfy $a^2+b^2=c^2+99^2$. . Find the minimum and maximum value of $a+b+c$.

2007 Bosnia and Herzegovina Junior BMO TST, 2

Tags: number theory , divisible , divide

Find all pairs of relatively prime numbers ($x, y$) such that $x^2(x + y)$ is divisible by $y^2(y - x)^2$. .

LMT Team Rounds 2021+, 4

Tags: number theory

Fred chooses a positive two-digit number with distinct nonzero digits. Laura takes Fred’s number and swaps its digits. She notices that the sum of her number and Fred’s number is a perfect square and the positive difference between them is a perfect cube. Find the greater of the two numbers.

Estonia Open Senior - geometry, 2011.2.1

Tags: analytic geometry , square

A square $ABCD$ lies in the coordinate plane with its vertices $A$ and $C$ lying on different coordinate axes. Prove that one of the vertices $B$ or $D$ lies on the line $y = x$ and the other one on $y = -x$.

1981 Kurschak Competition, 2

Tags: combinatorics , coloring

Let $n > 2$ be an even number. The squares of an $n\times n$ chessboard are coloured with $\frac12 n^2$ colours in such a way that every colour is used for colouring exactly two of the squares. Prove that one can place $n$ rooks on squares of $n$ different colours such that no two of the rooks can take each other.

2019 CMIMC, 2

Tags: number theory

For all positive integers $n$, let $f(n)$ return the smallest positive integer $k$ for which $\tfrac{n}{k}$ is not an integer. For example, $f(6) = 4$ because $1$, $2$, and $3$ all divide $6$ but $4$ does not. Determine the largest possible value of $f(n)$ as $n$ ranges over the set $\{1,2,\ldots, 3000\}$.

2018 Korea Junior Math Olympiad, 2

Tags: number theory

Find all positive integer $N$ which has not less than $4$ positive divisors, such that the sum of squares of the $4$ smallest positive divisors of $N$ is equal to $N$.

2011 AIME Problems, 11

Tags: modular arithmetic , invariant , function

Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the remainder when $S$ is divided by $1000$.

VI Soros Olympiad 1999 - 2000 (Russia), 11.6

Tags: algebra , polynomial , integer polynomial

Let $P(x)$ be a polynomial with integer coefficients. It is known that the number $\sqrt2+\sqrt3$ is its root. Prove that the number $\sqrt2-\sqrt3$ is also its root.

2016 JBMO Shortlist, 4

Tags: inequality

If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that $$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$ When does the equality occur? [i]Proposed by Dorlir Ahmeti, Albania[/i]

1998 Belarus Team Selection Test, 1

Tags: geometric inequality , hexagon , geometry

The lengths of the sides of a convex hexagon $ ABCDEF$ satisfy $ AB \equal{} BC$, $ CD \equal{} DE$, $ EF \equal{} FA$. Prove that: \[ \frac {BC}{BE} \plus{} \frac {DE}{DA} \plus{} \frac {FA}{FC} \geq \frac {3}{2}. \]

2019 Durer Math Competition Finals, 6

Tags: number theory , multiple , divide , divisible , digit

Find the smallest multiple of $81$ that only contains the digit $1$. How many $ 1$’s does it contain?

2014 Korea Junior Math Olympiad, 3

Tags: combinatorics

Find the number of $n$-movement on the following graph, starting from $S$. [img]https://cdn.artofproblemsolving.com/attachments/2/0/4a23c83c7f5405575acbe6d09f202d87341337.png[/img]

2009 Sharygin Geometry Olympiad, 8

Tags: geometry , incircle , excircle

A triangle $ABC$ is given, in which the segment $BC$ touches the incircle and the corresponding excircle in points $M$ and $N$. If $\angle BAC = 2 \angle MAN$, show that $BC = 2MN$. (N.Beluhov)

2005 National Olympiad First Round, 3

Tags:

If the difference between the greatest and the smallest root of the equation $x^3 - 6x^2 + 5$ is equal to $F$, which of the following is true? $ \textbf{(A)}\ 0 \leq F < 2 \quad\textbf{(B)}\ 2 \leq F < 4 \quad\textbf{(C)}\ 4 \leq F < 6 \quad\textbf{(D)}\ 6 \leq F < 8 \quad\textbf{(E)}\ 8 \leq F $

2020 USMCA, 11

Tags:

What is the largest real $x$ satisfying $(x+1)(x+2)(x+3)(x+6) = 2x+1$?

2023 Middle European Mathematical Olympiad, 6

Tags: geometry

Let $ABC$ be an acute triangle with $AB < AC$. Let $J$ be the center of the $A$-excircle of $ABC$. Let $D$ be the projection of $J$ on line $BC$. The internal bisectors of angles $BDJ$ and $JDC$ intersectlines $BJ$ and $JC$ at $X$ and $Y$, respectively. Segments $XY$ and $JD$ intersect at $P$. Let $Q$ be the projection of $A$ on line $BC$. Prove that the internal angle bisector of $QAP$ is perpendicular to line $XY$. [i]Proposed by Dominik Burek, Poland[/i]

2001 China Team Selection Test, 1

Tags: perfect square , number theory

For which integer $ h $, are there infinitely many positive integers $ n $ such that $ \lfloor \sqrt{h^2 + 1} \cdot n \rfloor $ is a perfect square? (Here $ \lfloor x \rfloor $ denotes the integer part of the real number $ x $?

1971 Poland - Second Round, 6

Tags: algebra , inequalities

Given an infinite sequence $ \{a_n\} $. Prove that if $$ a_n + a_{n+2} > 2a_{n+1} \ \ for \ \ n = 1, 2 ... $$ then $$ \frac{a_1+a_3+\ldots a_{2n+1}}{n+1} \geq \frac{a_2+a_4+\ldots a_{2n}}{n} $$ for $ n = 1, 2, \ldots $.