Olympiad problems

2010 LMT, 23

Tags:

In how many ways can six marbles be placed in the squares of a $6$-by-$6$ grid such that no two marbles lie in the same row or column?

2003 Iran MO (3rd Round), 9

Tags: number theory

Does there exist an infinite set $ S$ such that for every $ a, b \in S$ we have $ a^2 \plus{} b^2 \minus{} ab \mid (ab)^2$.

Croatia MO (HMO) - geometry, 2022.3

Tags: angle bisector , geometry

Let $ABC$ be an acute-angled triangle in which $|AB| < |AC|$ and let circle $k$ with center $O$ be its circumscribed circle. Let $P$ and $Q$ be points on sides $\overline{BC}$ and $\overline{AB}$ respectively such that $AQPO$ is a parallelogram. Let $K$ and $L$ be the intersections of the perpendicular bisector of $\overline{OP}$ with circle $k$, where $K$ is on the shorter arc $AB$. Let $M$ be the second intersection of the line $KQ$ with the circle $k$. Prove that the point $A$ belongs to the bisector of the angle $\angle QLM$.

2002 Greece Junior Math Olympiad, 1

Tags: geometry , trigonometry

In the exterior of an equilateral triangle $ABC$ of side $\alpha$ we construct an isosceles right-angled triangle $ACD$ with $\angle CAD=90^0.$The lines $DA$ and $CB$ meet at point $E$. (a) Find the angle $\angle DBC.$ (b) Express the area of triangle $CDE$ in terms of $\alpha.$ (c) Find the length of $BD.$

2022 Girls in Math at Yale, 9

Tags: college , quadratic

Suppose that $P(x)$ is a monic quadratic polynomial satisfying $aP(a) = 20P(20) = 22P(22)$ for some integer $a\neq 20, 22$. Find the minimum possible positive value of $P(0)$. [i]Proposed by Andrew Wu[/i] (Note: wording changed from original to specify that $a \neq 20, 22$.)

2017 Kürschák Competition, 1

Tags: probability , plane geometry , ceva theorem , algebra

Let $ABC$ be a triangle. Choose points $A'$, $B'$ and $C'$ independently on side segments $BC$, $CA$ and $AB$ respectively with a uniform distribution. For a point $Z$ in the plane, let $p(Z)$ denote the probability that $Z$ is contained in the triangle enclosed by lines $AA'$, $BB'$ and $CC'$. For which interior point $Z$ in triangle $ABC$ is $p(Z)$ maximised?

2018 JBMO Shortlist, G4

Tags: geometric inequality , inequalities , geometry , incenter , circumradius

Let $ABC$ be a triangle with side-lengths $a, b, c$, inscribed in a circle with radius $R$ and let $I$ be ir's incenter. Let $P_1, P_2$ and $P_3$ be the areas of the triangles $ABI, BCI$ and $CAI$, respectively. Prove that $$\frac{R^4}{P_1^2}+\frac{R^4}{P_2^2}+\frac{R^4}{P_3^2}\ge 16$$

2001 Stanford Mathematics Tournament, 10

Tags: college , function

You know that the binary function $\diamond$ takes in two non-negative integers and has the following properties: \begin{align*}0\diamond a&=1\\ a\diamond a&=0\end{align*} $\text{If } a<b, \text{ then } a\diamond b\&=(b-a)[(a-1)\diamond (b-1)].$ Find a general formula for $x\diamond y$, assuming that $y\gex>0$.

2019 ELMO Shortlist, N1

Tags: number theory

Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]

1966 IMO Longlists, 48

Tags: algebra , equation , number theory , root , parametric equation

For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

2022 Saudi Arabia IMO TST, 1

Tags: game , game strategy , combinatorics

There are a) $2022$, b) $2023$ plates placed around a round table and on each of them there is one coin. Alice and Bob are playing a game that proceeds in rounds indefinitely as follows. In each round, Alice first chooses a plate on which there is at least one coin. Then Bob moves one coin from this plate to one of the two adjacent plates, chosen by him. Determine whether it is possible for Bob to select his moves so that, no matter how Alice selects her moves, there are never more than two coins on any plate.

Ukraine Correspondence MO - geometry, 2005.4

Tags: geometry , angle bisector , angle

The bisectors of the angles $A$ and $B$ of the triangle $ABC$ intersect the sides $BC$ and $AC$ at points $D$ and $E$. It is known that $AE + BD = AB$. Find the angle $\angle C$.

2010 LMT, 7

Tags:

Let $ABCD$ be a square with $AB=6.$ A point $P$ in the interior is $2$ units away from side $BC$ and $3$ units away from side $CD.$ What is the distance from $P$ to $A?$

2023-24 IOQM India, 3

Tags:

Let $\alpha$ and $\beta$ be positive integers such that $$ \frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16} . $$ Find the smallest possible value of $\beta$.

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.