Olympiad problems

2013 Argentina National Olympiad, 2

In a convex quadrilateral $ABCD$ the angles $\angle A$ and $\angle C$ are equal and the bisector of $\angle B$ passes through the midpoint of the side $CD$. If it is known that $CD = 3AD$, calculate $\frac{AB}{BC}$.

2005 Bundeswettbewerb Mathematik, 3

Tags: geometric transformation , rotation , ratio , geometry

Two circles $k_1$ and $k_2$ intersect at two points $A$ and $B$. Some line through the point $B$ meets the circle $k_1$ at a point $C$ (apart from $B$), and the circle $k_2$ at a point $E$ (apart from $B$). Another line through the point $B$ meets the circle $k_1$ at a point $D$ (apart from $B$), and the circle $k_2$ at a point $F$ (apart from $B$). Assume that the point $B$ lies between the points $C$ and $E$ and between the points $D$ and $F$. Finally, let $M$ and $N$ be the midpoints of the segments $CE$ and $DF$. Prove that the triangles $ACD$, $AEF$ and $AMN$ are similar to each other.

2011 Vietnam Team Selection Test, 4

Tags: floor function , induction , number theory

Let $\langle a_n\rangle_{n\ge 0}$ be a sequence of integers satisfying $a_0=1, a_1=3$ and $a_{n+2}=1+\left\lfloor \frac{a_{n+1}^2}{a_n}\right\rfloor \ \ \forall n\ge0.$ Prove that $a_n\cdot a_{n+2}-a_{n+1}^2=2^n$ for every natural number $n.$

2024 Moldova EGMO TST, 1

Tags: parabola , algebra , parameterization

Let $P$ be the set of all parabolas with the equation of the form $$y=(a-1)x^2-2(a+2)x+a+1$$ where $a$ is a real parameter and $a\neq1$. Prove that there exists an unique point $M$ such that all parabolas in $P$ pass through $M$.

2006 Turkey MO (2nd round), 1

Tags: circumcircle , power of a point , radical axis , cyclic quadrilateral , geometry

Points $P$ and $Q$ on side $AB$ of a convex quadrilateral $ABCD$ are given such that $AP = BQ.$ The circumcircles of triangles $APD$ and $BQD$ meet again at $K$ and those of $APC$ and $BQC$ meet again at $L$. Show that the points $D,C,K,L$ lie on a circle.

2002 Mongolian Mathematical Olympiad, Problem 5

Tags: ratio , geometry

Let $A$ be the ratio of the product of sides to the product of diagonals in a circumscribed pentagon. Find the maximum possible value of $A$.

JOM 2015 Shortlist, G1

Tags: geometry , parallelogram

Given a triangle $ABC$, and let $ E $ and $ F $ be the feet of altitudes from vertices $ B $ and $ C $ to the opposite sides. Denote $ O $ and $ H $ be the circumcenter and orthocenter of triangle $ ABC $. Given that $ FA=FC $, prove that $ OEHF $ is a parallelogram.

2023 Kyiv City MO Round 1, Problem 2

Tags: algebra

You are given $n\geq 4$ positive real numbers. Consider all $\frac{n(n-1)}{2}$ pairwise sums of these numbers. Show that some two of these sums differ in at most $\sqrt[n-2]{2}$ times. [i]Proposed by Anton Trygub[/i]

2003 May Olympiad, 2

Tags: geometry , rectangle , perpendicular

Let $ABCD$ be a rectangle of sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from $A$ cuts $BD$ at point $H$. We call $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the measure of the segment $MN$.

2003 Croatia National Olympiad, Problem 2

Tags: algebra , floor function , function

For every integer $n>2$, prove the equality $$\left\lfloor\frac{n(n+1)}{4n-2}\right\rfloor=\left\lfloor\frac{n+1}4\right\rfloor.$$

2022 Bulgaria JBMO TST, 3

Tags: combinatorics , counting , inclusion-exclusion

For a positive integer $n$ let $t_n$ be the number of unordered triples of non-empty and pairwise disjoint subsets of a given set with $n$ elements. For example, $t_3 = 1$. Find a closed form formula for $t_n$ and determine the last digit of $t_{2022}$. (I also give here that $t_4 = 10$, for a reader to check his/her understanding of the problem statement.)

2007 ITest, 55

Tags: floor function , algebra

Let $T=\text{TNFTPP}$, and let $R=T-914$. Let $x$ be the smallest real solution of \[3x^2+Rx+R=90x\sqrt{x+1}.\] Find the value of $\lfloor x\rfloor$.

2009 India National Olympiad, 1

Tags: rectangle , parallelogram , trapezoid , trigonometry , perpendicular bisector , geometry

Let $ ABC$ be a tringle and let $ P$ be an interior point such that $ \angle BPC \equal{} 90 ,\angle BAP \equal{} \angle BCP$.Let $ M,N$ be the mid points of $ AC,BC$ respectively.Suppose $ BP \equal{} 2PM$.Prove that $ A,P,N$ are collinear.

2008 Singapore Senior Math Olympiad, 4

Tags: combinatorics

There are $11$ committees in a club. Each committee has $5$ members and every two committees have a member in common. Show that there is a member who belongs to $4$ committees.

1988 Polish MO Finals, 2

Tags: induction , pigeonhole principle , modular arithmetic , number theory

The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = a_2 = a_3 = 1$, $a_{n+3} = a_{n+2}a_{n+1} + a_n$. Show that for any positive integer $r$ we can find $s$ such that $a_s$ is a multiple of $r$.

2019 Turkey Team SeIection Test, 9

Tags: inequalities

Let $x, y, z$ be real numbers such that $y\geq 2z \geq 4x$ and $$ 2(x^3+y^3+z^3)+15(xy^2+yz^2+zx^2)\geq 16(x^2y+y^2z+z^2x)+2xyz.$$ Prove that: $4x+y\geq 4z$

2011 Pre - Vietnam Mathematical Olympiad, 1

Tags: induction , algebra

Let a sequence $\left\{ {{x_n}} \right\}$ defined by: \[\left\{ \begin{array}{l} {x_0} = - 2 \\ {x_n} = \frac{{1 - \sqrt {1 - 4{x_{n - 1}}} }}{2},\forall n \ge 1 \\ \end{array} \right.\] Denote $u_n=n.x_n$ and ${v_n} = \prod\limits_{i = 0}^n {\left( {1 + x_i^2} \right)} $. Prove that $\left\{ {{u_n}} \right\}$, $\left\{ {{v_n}} \right\}$ have finite limit.

2013 NIMO Problems, 7

Tags: modular arithmetic

Let $p$ be the largest prime less than $2013$ for which \[ N = 20 + p^{p^{p+1}-13} \] is also prime. Find the remainder when $N$ is divided by $10^4$. [i]Proposed by Evan Chen and Lewis Chen[/i]

2009 District Olympiad, 2

Tags: abstract algebra , algebra , polynomial , function , superior algebra

Prove that in an abelian ring $ A $ in which $ 1\neq 0, $ every element is idempotent if and only if the number of polynomial functions from $ A $ to $ A $ is equal to the square of the cardinal of $ A. $

2024 Malaysian APMO Camp Selection Test, 4

Tags: combinatorics

Ivan has a $n \times n$ board. He colors some of the squares black such that every black square has exactly two neighbouring square that are also black. Let $d_n$ be the maximum number of black squares possible, prove that there exist some real constants $a$, $b$, $c\ge 0$ such that; $$an^2-bn\le d_n\le an^2+cn.$$ [i]Proposed by Ivan Chan Kai Chin[/i]

2024 European Mathematical Cup, 1

Tags: graph theory , number theory

We call a pair of distinct numbers $(a, b)$ a [i]binary pair[/i] if $ab+1$ is a power of two. Given a set $S$ of $n$ positive integers, what is the maximum possible numbers of binary pairs in S?

2014 Online Math Open Problems, 22

Tags: floor function

Find the smallest positive integer $c$ for which the following statement holds: Let $k$ and $n$ be positive integers. Suppose there exist pairwise distinct subsets $S_1$, $S_2$, $\dots$, $S_{2k}$ of $\{1, 2, \dots, n\}$, such that $S_i \cap S_j \neq \varnothing$ and $S_i \cap S_{j+k} \neq \varnothing$ for all $1 \le i,j \le k$. Then $1000k \le c \cdot 2^n$. [i]Proposed by Yang Liu[/i]

2007 Bulgarian Autumn Math Competition, Problem 12.4

Tags: prime number , number theory , linear recurrence

Let $p$ and $q$ be prime numbers and $\{a_{n}\}_{n=1}^{\infty}$ be a sequence of integers defined by: \[a_{0}=0, a_{1}=1, a_{n+2}=pa_{n+1}-qa_{n}\quad\forall n\geq 0\] Find $p$ and $q$ if there exists an integer $k$ such that $a_{3k}=-3$.

Fractal Edition 1, P3

Tags: inequalities

Let $ a $, $ b $, and $ c $ be three positive real numbers that satisfy $ ab + bc + ca = 1 $. Show that: \[ \frac{a}{a^2+1} + \frac{b}{b^2+1} + \frac{c}{c^2+1} \le \frac{1}{4abc} \]

2022 Baltic Way, 1

Tags: function , algebra

Let $\mathbb{R^+}$ denote the set of positive real numbers. Assume that $f:\mathbb{R^+} \to \mathbb{R^+}$ is a function satisfying the equations: $$ f(x^3)=f(x)^3 \quad \text{and} \quad f(2x)=f(x) $$ for all $x \in \mathbb{R^+}$. Find all possible values of $f(\sqrt[2022]{2})$.