Olympiad problems

2016 IMO Shortlist, G6

Tags: geometry

Let $ABCD$ be a convex quadrilateral with $\angle ABC = \angle ADC < 90^{\circ}$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $E$ and $F$ respectively, and meet each other at point $P$. Let $M$ be the midpoint of $AC$ and let $\omega$ be the circumcircle of triangle $BPD$. Segments $BM$ and $DM$ intersect $\omega$ again at $X$ and $Y$ respectively. Denote by $Q$ the intersection point of lines $XE$ and $YF$. Prove that $PQ \perp AC$.

1952 Moscow Mathematical Olympiad, 219

Tags: inequalities , algebra

Prove that $(1 - x)^n + (1 + x)^n < 2^n$ for an integer $n \ge 2$ and $|x| < 1$.

1991 Arnold's Trivium, 69

Tags: function

Prove that the solid angle based on a given closed contour is a function of the vertex of the angle that is harmonic outside the contour.

1999 Czech and Slovak Match, 5

Tags: limit , function , algebra

Find all functions $f: (1,\infty)\text{to R}$ satisfying $f(x)-f(y)=(y-x)f(xy)$ for all $x,y>1$. [hide="hint"]you may try to find $f(x^5)$ by two ways and then continue the solution. I have also solved by using this method.By finding $f(x^5)$ in two ways I found that $f(x)=xf(x^2)$ for all $x>1$.[/hide]

2019 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: geometry

Let $ABC$ be a triangle with $AB<AC$ and circuncircle $\omega$. Let $M$ and $N$ be the midpoints of $AC$ and $AB$ respectively and $G$ is the centroid of $ABC$. Let $P$ be the foot of perpendicular of $A$ to the line $BC$, and the point $Q$ is the intersection of $GP$ and $\omega$($Q,P,G$ are collinears in this order). The line $QM$ cuts $\omega$ in $M_1$ and the line $QN$ cuts $\omega$ in $N_1$. If $K$ is the intersection of $BM_1$ and $CN_1$ prove that $P$, $G$ and $K$ are collinears.

2022 Ecuador NMO (OMEC), 5

Tags: geometry , similar triangles

Let $ABC$ be a 90-degree triangle with hypotenuse $BC$. Let $D$ and $E$ distinct points on segment $BC$ and $P, Q$ be the foot of the perpendicular from $D$ to $AB$ and $E$ to $AC$, respectively. $DP$ and $EQ$ intersect at $R$. Lines $CR$ and $AB$ intersect at $M$ and lines $BR$ and $AC$ intersect at $N$. Prove that $MN \parallel BC$ if and only if $BD=CE$.

2019 Czech-Austrian-Polish-Slovak Match, 6

Tags: geometry

Let $ABC$ be an acute triangle with $AB<AC$ and $\angle BAC=60^{\circ}$. Denote its altitudes by $AD,BE,CF$ and its orthocenter by $H$. Let $K,L,M$ be the midpoints of sides $BC,CA,AB$, respectively. Prove that the midpoints of segments $AH, DK, EL, FM$ lie on a single circle.

2008 ISI B.Math Entrance Exam, 9

Tags: algebra

For $n\geq 3$ , determine all real solutions of the system of n equations : $x_1+x_2+...+x_{n-1}=\frac{1}{x_n}$ ....................... $x_1+x_2+...+x_{i-1}+x_{i+1}+...+x_n=\frac{1}{x_i}$ ....................... $x_2+...+x_{n-1}+x_n=\frac{1}{x_1}$

2012 India IMO Training Camp, 3

Tags: discrete continous , combinatorics

Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half. [i]Proposed by Gerhard Wöginger, Austria[/i]

2022 Belarusian National Olympiad, 9.1

Tags: geometry , similar triangles

Given an isosceles triangle $ABC$ with base $BC$. On the sides $BC$, $AC$ and $AB$ points $X,Y$ and $Z$ are chosen respectively such that triangles $ABC$ and $YXZ$ are similar. Point $W$ is symmetric to point $X$ with respect to the midpoint of $BC$. Prove that points $X,Y,Z$ and $W$ lie on a circle.

MathLinks Contest 6th, 7.2

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral. Let $M, N$ be the midpoints of the diagonals $AC$ and $BD$ and let $P$ be the midpoint of $MN$. Let $A',B',C',D'$ be the intersections of the rays $AP$, $BP$, $CP$ and $DP$ respectively with the circumcircle of the quadrilateral $ABCD$. Find, with proof, the value of the sum \[ \sigma = \frac{ AP}{PA'} + \frac{BP}{PB'} + \frac{CP}{PC'} + \frac{DP}{PD'} . \]

Russian TST 2016, P2

Tags: geometry

$ABCDEF$ is a cyclic hexagon with $AB=BC=CD=DE$. $K$ is a point on segment $AE$ satisfying $\angle BKC=\angle KFE, \angle CKD = \angle KFA$. Prove that $KC=KF$.

2022 Princeton University Math Competition, A4 / B6

Tags: combinatorics

Let $C_n$ denote the $n$-dimensional unit cube, consisting of the $2^n$ points $$\{(x_1, x_2, \ldots, x_n) \mid x_i \in \{0, 1\} \text{ for all } 1 \le i \le n\}.$$ A tetrahedron is [i]equilateral[/i] if all six side lengths are equal. Find the smallest positive integer $n$ for which there are four distinct points in $C_n$ that form a non-equilateral tetrahedron with integer side lengths.

2015 Indonesia MO Shortlist, C3

Tags: coloring , color problem , combinatorics

We have $2015$ marbles in a box, where each marble has one color from red, green or blue. At each step, we are allowed to take $2$ different colored marbles, then replace it with $2$ marbles with the third color. For example, we take one blue marble and one green marble, and we fill with $2$ red marbles. Prove that we can always do a series of steps so that all marbles in the box have the same color.

2004 Romania National Olympiad, 4

Tags: geometry

Let $\displaystyle \left( P_n \right)_{n \geq 1}$ be an infinite family of planes and $\displaystyle \left( X_n \right)_{n \geq 1}$ be a family of non-void, finite sets of points such that $\displaystyle X_n \subset P_n$ and the projection of the set $\displaystyle X_{n+1}$ on the plane $\displaystyle P_n$ is included in the set $X_n$, for all $n$. Prove that there is a sequence of points $\displaystyle \left( p_n \right)_{n \geq 1}$ such that $\displaystyle p_n \in P_n$ and $p_n$ is the projection of $p_{n+1}$ on the plane $P_n$, for all $n$. Does the conclusion of the problem remain true if the sets $X_n$ are infinite? [i]Claudiu Raicu[/i]

1997 Czech and Slovak Match, 5

Tags: integer , sum of powers , number theory

The sum of several integers (not necessarily distinct) equals $1492$. Decide whether the sum of their seventh powers can equal (a) $1996$; (b) $1998$.

2009 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: circumcircle , tangent , isosceles , geometry

Let $C$ be a circle with center $O$ and radius $R$. From point $A$ of circle $C$ we construct a tangent $t$ on circle $C$. We construct line $d$ through point $O$ whch intersects tangent $t$ in point $M$ and circle $C$ in points $B$ and $D$ ($B$ lies between points $O$ and $M$). If $AM=R\sqrt{3}$, prove: $a)$ Triangle $AMD$ is isosceles $b)$ Circumcenter of $AMD$ lies on circle $C$

1990 IMO Longlists, 71

Tags: geometry

Given a point $P = (p_1, p_2, \ldots, p_n)$ in $n$-dimensional space . Find point $X = (x_1, x_2, \ldots, x_n)$, such that $x_1 \leq x_2 \leq\cdots \leq x_n$ and $\sqrt{(x_1-p_1)^2 + (x_2-p_2)^2+\cdots+(x_n-p_n)^2}$ is minimal.

2011 Morocco National Olympiad, 1

Tags: inequalities , trigonometry

Let $x$, $y$, and $z$ be three real positive numbers such that $x^{2}+y^{2}+z^{2}+2xyz=1$. Prove that $2(x+y+z)\leq 3$.

2018 District Olympiad, 1

Tags: matrices , linear algebra

Show that if $n\ge 2$ is an integer, then there exist invertible matrices $A_1, A_2, \ldots, A_n \in \mathcal{M}_2(\mathbb{R})$ with non-zero entries such that: \[A_1^{-1} + A_2^{-1} + \ldots + A_n^{-1} = (A_1 + A_2 + \ldots + A_n)^{-1}.\] [hide=Edit.] The $77777^{\text{th}}$ topic in College Math :coolspeak: [/hide]

1985 Dutch Mathematical Olympiad, 3

Tags: combinatorics

In a factory, square tables of $ 40 \times 40$ are tiled with four tiles of size $ 20 \times 20$. All tiles are the same and decorated in the same way with an asymmetric pattern such as the letter $ J$. How many different types of tables can be produced in this way?

2020 Thailand TST, 5

Tags: algebra

Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\] Define the set $A$ by \[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\] Prove that, if $A$ is not empty, then \[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]

2009 Today's Calculation Of Integral, 497

Tags: calculus , calculus computations , geometry , integration , trigonometry

Consider a parameterized curve $ C: x \equal{} e^{ \minus{} t}\cos t,\ y \equal{} e^{ \minus{} t}\sin t\ \left(0\leq t\leq \frac {\pi}{2}\right).$ (1) Find the length $ L$ of $ C$. (2) Find the area $ S$ of the region bounded by $ C$, the $ x$ axis and $ y$ axis. You may not use the formula $ \boxed{\int_a^b \frac {1}{2}r(\theta)^2d\theta }$ here.

2024 Canadian Open Math Challenge, C2

Tags:

a) How many ways are there to pair up the elements of $\{1,2,\dots,14\}$ into seven pairs so that each pair has sum at least $15$? b) How many ways are there to pair up the elements of $\{1,2,\dots,14\}$ into seven pairs so that each pair has sum at least $13$? c) How many ways are there to pair up the elements of $\{1,2,\dots,2024\}$ into $1012$ pairs so that each pair has sum at least $2001$?

1966 German National Olympiad, 3

Tags: counting , trigonometry , geometry , area of a triangle , combinatorics

Consider all segments dividing the area of a triangle $ABC$ in two equal parts. Find the length of the shortest segment among them, if the side lengths $a,$ $b,$ $c$ of triangle $ABC$ are given. How many of these shortest segments exist ?