Olympiad problems

2022 Poland - Second Round, 6

$n$ players took part in badminton tournament, where $n$ is positive and odd integer. Each two players played two matches with each other. There were no draws. Each player has won as many matches as he has lost. Prove that you can cancel half of the matches s.t. each player still has won as many matches as he has lost.

2010 Iran MO (3rd Round), 3

Tags: geometry , geometric transformation , reflection , circumcircle , homothety , ratio , power of a point

in a quadrilateral $ABCD$ digonals are perpendicular to each other. let $S$ be the intersection of digonals. $K$,$L$,$M$ and $N$ are reflections of $S$ to $AB$,$BC$,$CD$ and $DA$. $BN$ cuts the circumcircle of $SKN$ in $E$ and $BM$ cuts the circumcircle of $SLM$ in $F$. prove that $EFLK$ is concyclic.(20 points)

2009 Jozsef Wildt International Math Competition, W. 7

Tags: integration , inequalities , calculus , logarithm

If $0<a<b$ then $$\int \limits_a^b \frac{\left (x^2-\left (\frac{a+b}{2} \right )^2\right )\ln \frac{x}{a} \ln \frac{x}{b}}{(x^2+a^2)(x^2+b^2)} dx > 0$$

2016 Junior Regional Olympiad - FBH, 2

Tags: root , algebra

If $$w=\sqrt{1+\sqrt{-3+2\sqrt{3}}}-\sqrt{1-\sqrt{-3+2\sqrt{3}}}$$ prove that $w=\sqrt{3}-1$

2002 Germany Team Selection Test, 1

Tags: function , quadratic , algebra

Let $P$ denote the set of all ordered pairs $ \left(p,q\right)$ of nonnegative integers. Find all functions $f: P \rightarrow \mathbb{R}$ satisfying \[ f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0, \\ 1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases} \] Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.

2000 Harvard-MIT Mathematics Tournament, 36

Tags: geometry , inradius , trigonometry , trig identities , law of cosines

If, in a triangle of sides $a, b, c$, the incircle has radius $\frac{b+c-a}{2}$, what is the magnitude of $\angle A$?

2017 QEDMO 15th, 6

Tags: number theory , diophantine , diophantine equation

Find all integers $x,y$ satisfy the $x^3 + y^3 = 3xy$.

2017 Iranian Geometry Olympiad, 3

Tags: geometry

In the regular pentagon $ABCDE$, the perpendicular at $C$ to $CD$ meets $AB$ at $F$. Prove that $AE+AF=BE$. [i]Proposed by Alireza Cheraghi[/i]

2008 Germany Team Selection Test, 2

Tags: rectangle , combinatorics , geometry

Tracey baked a square cake whose surface is dissected in a $ 10 \times 10$ grid. In some of the fields she wants to put a strawberry such that for each four fields that compose a rectangle whose edges run in parallel to the edges of the cake boundary there is at least one strawberry. What is the minimum number of required strawberries?

KoMaL A Problems 2018/2019, A. 745

Tags: combinatorics , geometry

A clock hand is attached to every face of a convex polyhedron. Each hand always points towards a neighboring face and every minute, exactly one of the hands turns clockwise to point at the next face. Suppose that the hands on neighboring faces never point towards one another. Show that one of the hands makes only finitely many turns.

2004 Purple Comet Problems, 7

Tags: geometry , rectangle , percent

A rectangle has area $1100$. If the length is increased by ten percent and the width is decreased by ten percent, what is the area of the new rectangle?

2007 Oral Moscow Geometry Olympiad, 6

Tags: geometry , fixed , tangent

A point $P$ is fixed inside the circle. $C$ is an arbitrary point of the circle, $AB$ is a chord passing through point $B$ and perpendicular to the segment $BC$. Points $X$ and $Y$ are projections of point $B$ onto lines $AC$ and $BC$. Prove that all line segments $XY$ are tangent to the same circle. (A. Zaslavsky)

1993 Romania Team Selection Test, 3

Tags: diagonal , symmetry , hexagon , geometry , equal area

Suppose that each of the diagonals $AD,BE,CF$ divides the hexagon $ABCDEF$ into two parts of the same area and perimeter. Does the hexagon necessarily have a center of symmetry?

1986 AMC 12/AHSME, 8

Tags: geometry

The population of the United States in 1980 was $226,504,825$. The area of the country is $3,615,122$ square miles. The are $(5280)^{2}$ square feet in one square mile. Which number below best approximates the average number of square feet per person? $ \textbf{(A)}\ 5,000\qquad\textbf{(B)}\ 10,000\qquad\textbf{(C)}\ 50,000\qquad\textbf{(D)}\ 100,000\qquad\textbf{(E)}\ 500,000 $

2015 Taiwan TST Round 2, 1

Tags: algebra , polynomial

Let $f(x)=\sum_{i=0}^{n}a_ix^i$ and $g(x)=\sum_{i=0}^{n}b_ix^i$, where $a_n$,$b_n$ can be zero. Called $f(x)\ge g(x)$ if exist $r$ such that $\forall i>r,a_i=b_i,a_r>b_r$ or $f(x)=g(x)$. Prove that: if the leading coefficients of $f$ and $g$ are positive, then $f(f(x))+g(g(x))\ge f(g(x))+g(f(x))$

2023 VN Math Olympiad For High School Students, Problem 11

Tags: geometry , ratio

Given a triangle $ABC$ inscribed in $(O)$ with $2$ symmedians $AD, CF(D,F$ are on the sides $BC, AB,$ respectively$).$ The ray $DF$ intersects $(O)$ at $P.$ The line passing through $P$ and perpendicular to $OA$ intersects $AB,AC$ at $Q,R,$ respectively$.$ Compute the ratio $\dfrac{PR}{PQ}.$

2007 Baltic Way, 12

Tags: circumcircle , geometric transformation , dilation , similar triangles , geometry

Let $M$ be a point on the arc $AB$ of the circumcircle of the triangle $ABC$ which does not contain $C$. Suppose that the projections of $M$ onto the lines $AB$ and $BC$ lie on the sides themselves, not on their extensions. Denote these projections by $X$ and $Y$, respectively. Let $K$ and $N$ be the midpoints of $AC$ and $XY$, respectively. Prove that $\angle MNK=90^{\circ}$ .

2023 Tuymaada Olympiad, 7

Tags: combinatorics

$3n$ people forming $n$ families of a mother, a father and a child, stand in a circle. Every two neighbours can exchange places except the case when a parent exchanges places with his/her child (this is forbidden). For what $n$ is it possible to obtain every arrangement of those people by such exchanges? The arrangements differing by a circular shift are considered distinct.

2014 IMAC Arhimede, 6

Tags: algebra , inequalities , cyclic inequality , four variables

If $a, b, c, d$ are positive numbers, prove that $$\sum_{cyclic}\frac{a-\sqrt[3]{bcd}}{a+3(b+c+d)}\ge 0$$

2022 OMpD, 4

Tags: perpendicular , cyclic quadrilateral , geometry

Let $ABCD$ be a cyclic quadrilateral and $M,N$ be the midpoints of $AB$, $CD$ respectively. The diagonals $AC$ and $BD$ intersect at $L$. Suppose that the circumcircle of $LMN$, with center $T$, intersects the circumcircle of $ABCD$ at two distinct points $X,Y$. If the line $MN$ intersects the line $XY$ at $S$ and the line $XM$ intersects the line $YN$ at $P$, prove that $PL$ is perpendicular to $ST$.

1996 India National Olympiad, 3

Tags: algebra

Solve the following system for real $a , b, c, d, e$: \[ \left\{ \begin{array}{ccc} 3a & = & ( b + c+ d)^3 \\ 3b & = & ( c + d +e ) ^3 \\ 3c & = & ( d + e +a )^3 \\ 3d & = & ( e + a +b )^3 \\ 3e &=& ( a + b +c)^3. \end{array}\right. \]

Estonia Open Senior - geometry, 1996.1.4

Tags: geometry , semicircle , minimum , maximum , area , circles

A unit square has a circle of radius $r$ with center at it's midpoint. The four quarter circles are centered on the vertices of the square and are tangent to the central circle (see figure). Find the maximum and minimum possible value of the area of the striped figure in the figure and the corresponding values of $r$ such these, the maximum and minimum are achieved. [img]https://2.bp.blogspot.com/-DOT4_B5Mx-8/XnmsTlWYfyI/AAAAAAAALgs/TVYkrhqHYGAeG8eFuqFxGDCTnogVbQFUwCK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs1.4.png[/img]