Olympiad problems

2020 International Zhautykov Olympiad, 4

Tags: geometry

In a scalene triangle $ABC$ $I$ is the incentr and $CN$ is the bisector of angle $C$. The line $CN$ meets the circumcircle of $ABC$ again at $M$. The line $l$ is parallel to $AB$ and touches the incircle of $ABC$. The point $R$ on $l$ is such. That $CI \bot IR$. The circumcircle of $MNR$ meets the line $IR$ again at S. Prpve that $AS=BS$.

2019 Portugal MO, 1

Tags: geometry , square , area

In a square of side $10$ cm , the vertices are joined to the midpoints on the opposite sides, as shown in the figure. How much does the area of the colored region measure? [img]https://1.bp.blogspot.com/-bHrc1Nu0PQI/X4KaJysLAcI/AAAAAAAAMk0/LLGv1fotQO0Tk1AXqQymG_nNdpyWcbjyACLcBGAsYHQ/s109/2019%2BPortugal%2Bp1.png[/img]

2014 Regional Competition For Advanced Students, 4

Tags: geometry

For a point $P$ in the interior of a triangle $ABC$ let $D$ be the intersection of $AP$ with $BC$, let $E$ be the intersection of $BP$ with $AC$ and let $F$ be the intersection of $CP$ with $AB$.Furthermore let $Q$ and $R$ be the intersections of the parallel to $AB$ through $P$ with the sides $AC$ and $BC$, respectively. Likewise, let $S$ and $T$ be the intersections of the parallel to $BC$ through $P$ with the sides $AB$ and $AC$, respectively.In a given triangle $ABC$, determine all points $P$ for which the triangles $PRD$, $PEQ$and $PTE$ have the same area.

1999 Putnam, 2

Tags: polynomial , quadratic , derivative , algebra , calculus

Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P^{\prime\prime}(x)$, where $Q(x)$ is a quadratic polynomial and $P^{\prime\prime}(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.

2013 NIMO Problems, 5

Tags: geometry , trapezoid , parallelogram

In convex hexagon $AXBYCZ$, sides $AX$, $BY$ and $CZ$ are parallel to diagonals $BC$, $XC$ and $XY$, respectively. Prove that $\triangle ABC$ and $\triangle XYZ$ have the same area. [i]Proposed by Evan Chen[/i]

2013 Gheorghe Vranceanu, 2

Tags: number theory

Given a number $ a $ and natural number $ n\ge 3 $ having the property that $ x^n-x $ and $ x^2-x $ are integers, prove that $ x $ is integer.

2011 Dutch IMO TST, 5

Tags: coloring , combinatorics

Find all triples $(a, b, c)$ of positive integers with $a+b+c = 10$ such that there are $a$ red, $b$ blue and $c$ green points (all different) in the plane satisfying the following properties: $\bullet$ for each red point and each blue point we consider the distance between these two points, the sum of these distances is $37$, $\bullet$ for each green point and each red point we consider the distance between these two points, the sum of these distances is $30$, $\bullet$ for each blue point and each green point we consider the distance between these two points, the sum of these distances is $1$.

1999 Taiwan National Olympiad, 4

Tags: number theory

Let $P^{*}$ be the set of primes less than $10000$. Find all possible primes $p\in P^{*}$ such that for each subset $S=\{p_{1},p_{2},...,p_{k}\}$ of $P^{*}$ with $k\geq 2$ and each $p\not\in S$, there is a $q\in P^{*}-S$ such that $q+1$ divides $(p_{1}+1)(p_{2}+1)...(p_{k}+1)$.

2012 Turkey Team Selection Test, 3

Tags: combinatorics

Two players $A$ and $B$ play a game on a $1\times m$ board, using $2012$ pieces numbered from $1$ to $2012.$ At each turn, $A$ chooses a piece and $B$ places it to an empty place. After $k$ turns, if all pieces are placed on the board increasingly, then $B$ wins, otherwise $A$ wins. For which values of $(m,k)$ pairs can $B$ guarantee to win?

2016 BMT Spring, 2

Tags: geometry

Cyclic quadrilateral $ABCD$ has side lengths $AB = 6$, $BC = 7$, $CD = 7$, $DA = 6$. What is the area of $ABCD$?

2006 Purple Comet Problems, 3

Tags:

Find the sum of all the prime numbers less than $100$ which are one more than a multiple of six.

2004 USAMO, 6

Tags: geometry , trapezoid , trigonometry , inequalities

A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\omega$. Suppose that \[ (AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2. \] Prove that $ABCD$ is an isosceles trapezoid.

2022 Assara - South Russian Girl's MO, 8

Tags: tangent circle , geometry

In parallelogram $ABCD$, angle $A$ is acute. Let $X$ be a point, symmetrical to point $C$ wrt to straight line $AD$, $Y$ is a point symmetrical to the point $C$ wrt point $D$, and $M$ is the intersection point of $AC$ and $BD$. It turned out, that the circumcircles of triangles $BMC$ and $AXY$ are tangent internally. Prove that $AM = AB$.

2008 Peru IMO TST, 1

Tags: geometry , reflection , geometric transformation

Let $ ABC$ be a triangle and let $ I$ be the incenter. $ Ia$ $ Ib$ and $ Ic$ are the excenters opposite to points $ A$ $ B$ and $ C$ respectively. Let $ La$ be the line joining the orthocenters of triangles $ IBC$ and $ IaBC$. Define $ Lb$ and $ Lc$ in the same way. Prove that $ La$ $ Lb$ and $ Lc$ are concurrent. Daniel

2018 India PRMO, 24

Tags: angle , geometry , integer , combinatorics

If $N$ is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (in degrees), what is $\frac{N}{100}$?

2008 Croatia Team Selection Test, 1

Tags: inequalities

Let $ x$, $ y$, $ z$ be positive numbers. Find the minimum value of: $ (a)\quad \frac{x^2 \plus{} y^2 \plus{} z^2}{xy \plus{} yz}$ $ (b)\quad \frac{x^2 \plus{} y^2 \plus{} 2z^2}{xy \plus{} yz}$

1989 Bulgaria National Olympiad, Problem 2

Tags: series , floor function , limit , sequence , algebra

Prove that the sequence $(a_n)$, where $$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$.

1999 CentroAmerican, 5

Tags: modular arithmetic , number theory

Let $a$ be an odd positive integer greater than 17 such that $3a-2$ is a perfect square. Show that there exist distinct positive integers $b$ and $c$ such that $a+b,a+c,b+c$ and $a+b+c$ are four perfect squares.

1984 Bulgaria National Olympiad, Problem 2

Tags: geometry , trapezoid

The diagonals of a trapezoid $ABCD$ with bases $AB$ and $CD$ intersect in a point $O$, and $AB/CD=k>1$. The bisectors of the angles $AOB,BOC,COD,DOA$ intersect $AB,BC,CD,DA$ respectively at $K,L,M,N$. The lines $KL$ and $MN$ meet at $P$, and the lines $KN$ and $LM$ meet at $Q$. If the areas of $ABCD$ and $OPQ$ are equal, find the value of $k$.

1985 IMO Longlists, 26

Tags: dissection , geometry , square

Let $K$ and $K'$ be two squares in the same plane, their sides of equal length. Is it possible to decompose $K$ into a finite number of triangles $T_1, T_2, \ldots, T_p$ with mutually disjoint interiors and find translations $t_1, t_2, \ldots, t_p$ such that \[K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? \]

2024 Argentina Iberoamerican TST, 6

Tags: number theory , combinatorics

Uri has $99$ empty bags and an unlimited number of balls. The weight of each ball is a number of the form $3^n$ where $n$ is an integer that can vary from ball to ball (negative integer exponents are allowed, such as $3^{-4}=\dfrac{1}{81}$, and the exponent $0$, where $3^0=1$). Uri chose a finite number of balls and distributed them into the bags so that all the bags had the same total weight and there were no balls left over. It is known that Uri chose at most $k$ balls of the same weight. Determine the smallest possible value of $k$.

2018 BMT Spring, 6

Tags:

Let $x,y,z \in \mathbb{R}$ and $7x^2 + 7y^2 + 7z^2 + 9xyz = 12$. The minimum value of $x^2 + y^2 + z^2$ can be expressed as $\dfrac{a}{b}$ where $a,b \in \mathbb{Z}, \gcd(a,b) = 1$. What is $a + b$?

1980 IMO Shortlist, 5

Tags: analytic geometry , geometry , algebra , polynomial

In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.

2010 Finnish National High School Mathematics Competition, 1

Tags: geometry

Let $ABC$ be right angled triangle with sides $s_1,s_2,s_3$ medians $m_1,m_2,m_3$. Prove that $m_1^2+m_2^2+m_3^2=\frac{3}{4}(s_1^2+s_2^2+s_3^2)$.

2016 ASDAN Math Tournament, 1

Tags:

Moor owns $3$ shirts, one each of black, red, and green. Moor also owns $3$ pairs of pants, one each of white, red, and green. Being stylish, he decides to wear an outfit consisting of one shirt and one pair of pants that are different colors. How many combinations of shirts and pants can Moor choose?