Olympiad problems

2023 JBMO Shortlist, G6

Let $ABC$ be an acute triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$ and let $M$ be the midpoint of $OD$. The points $O_b$ and $O_c$ are the circumcenters of triangles $AOC$ and $AOB$, respectively. If $AO=AD$, prove that points $A$, $O_b$, $M$ and $O_c$ are concyclic. [i]Marin Hristov and Bozhidar Dimitrov, Bulgaria[/i]

2014 Silk Road, 4

Tags: induction , inequalities , number theory

Find all $ f:N\rightarrow N$, such that $\forall m,n\in N $ $ 2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n) $

2020 Purple Comet Problems, 29

Tags: combinatorics

Find the number of distinguishable $2\times 2\times 2$ cubes that can be formed by gluing together two blue, two green, two red, and two yellow $1\times 1\times 1$ cubes. Two cubes are indistinguishable if one can be rotated so that the two cubes have identical coloring patterns.

1997 Slovenia Team Selection Test, 2

Tags: polynomial , functional , functional inequality , algebra , inequalities

Find all polynomials $p$ with real coefficients such that for all real $x$ , $xp(x)p(1-x)+x^3 +100 \ge 0$.

2018 Bangladesh Mathematical Olympiad, 6

Tags: algebra

Find all the pairs of integers $(m,n)$ satisfying the equality $3(m^2+n^2)-7(m+n)=-4$

2014 Thailand TSTST, 2

Tags: inequalities

Let $a, b, c$ be positive real numbers. Prove that $$\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}\geq\sqrt{\frac{2ab}{3a+b+2c}}+\sqrt{\frac{2bc}{3b+c+2a}}+\sqrt{\frac{2ca}{3c+a+2b}}.$$

2009 Puerto Rico Team Selection Test, 1

Tags: number theory

A positive integer is called [i]good [/i] if it can be written as the sum of two distinct integer squares. A positive integer is called [i]better [/i]if it can be written in at least two was as the sum of two integer squares. A positive integer is called [i]best [/i] if it can be written in at least four ways as the sum of two distinct integer squares. a) Prove that the product of two good numbers is good. b) Prove that $ 5$ is good, $ 2005$ is better, and $ 2005^2$ is best.

2010 Brazil Team Selection Test, 3

Tags: geometry , symmetry , triangle , parallelogram

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

2001 Hungary-Israel Binational, 1

Tags: diophantine equation , number theory

Find positive integers $x, y, z$ such that $x > z > 1999 \cdot 2000 \cdot 2001 > y$ and $2000x^{2}+y^{2}= 2001z^{2}.$

2010 National Olympiad First Round, 31

Tags:

For which pair $(A,B)$, \[ x^2+xy+y=A \\ \frac{y}{y-x}=B \] has no real roots? $ \textbf{(A)}\ (1/2,2) \qquad\textbf{(B)}\ (-1,1) \qquad\textbf{(C)}\ (\sqrt 2, \sqrt 2) \qquad\textbf{(D)}\ (1,1/2) \qquad\textbf{(E)}\ (2,2/3) $

2020 CMIMC Team, 14

Tags: team

Let $a_0=1$ and for all $n\ge 1$ let $a_n$ be the smaller root of the equation $$4^{-n}x^2-x+a_{n-1} = 0.$$ Given that $a_n$ approaches a value $L$ as $n$ goes to infinity, what is the value of $L$?

1993 IberoAmerican, 2

Tags: geometry , parallelogram , rotation , geometric transformation , reflection , combinatorics

Show that for every convex polygon whose area is less than or equal to $1$, there exists a parallelogram with area $2$ containing the polygon.

2004 National Olympiad First Round, 33

Tags: geometry , trapezoid , trigonometry , circumcircle , angle bisector , perpendicular bisector

Let $ABCD$ be a trapezoid such that $|AB|=9$, $|CD|=5$ and $BC\parallel AD$. Let the internal angle bisector of angle $D$ meet the internal angle bisectors of angles $A$ and $C$ at $M$ and $N$, respectively. Let the internal angle bisector of angle $B$ meet the internal angle bisectors of angles $A$ and $C$ at $L$ and $K$, respectively. If $K$ is on $[AD]$ and $\dfrac{|LM|}{|KN|} = \dfrac 37$, what is $\dfrac{|MN|}{|KL|}$? $ \textbf{(A)}\ \dfrac{62}{63} \qquad\textbf{(B)}\ \dfrac{27}{35} \qquad\textbf{(C)}\ \dfrac{2}{3} \qquad\textbf{(D)}\ \dfrac{5}{21} \qquad\textbf{(E)}\ \dfrac{24}{63} $

2013 Bangladesh Mathematical Olympiad, 7

Tags: number theory , prime number

Higher Secondary P7 If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.

2014 Sharygin Geometry Olympiad, 21

Tags: circumcircle , symmetry , incenter , power of a point , radical axis , geometry

Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$.

Novosibirsk Oral Geo Oly IX, 2021.5

Tags: geometry , pentagon , angle

The pentagon $ABCDE$ is inscribed in the circle. Line segments $AC$ and $BD$ intersect at point $K$. Line segment $CE$ touches the circumcircle of triangle $ABK$ at point $N$. Find the angle $CNK$ if $\angle ECD = 40^o.$

2001 National Olympiad First Round, 1

Tags: geometry , trigonometry

Let $A,B,C$ be points on $[OX$ and $D,E,F$ be points on $[OY$ such that $|OA|=|AB|=|BC|$ and $|OD|=|DE|=|EF|$. If $|OA|>|OD|$, which one below is true? $\textbf{(A)}$ For every $\widehat{XOY}$, $\text{ Area}(AEC)>\text{Area}(DBF)$ $\textbf{(B)}$ For every $\widehat{XOY}$, $\text{ Area}(AEC)=\text{Area}(DBF)$ $\textbf{(C)}$ For every $\widehat{XOY}$, $\text{ Area}(AEC)<\text{Area}(DBF)$ $\textbf{(D)}$ If $m(\widehat{XOY})<45^\circ$ then $\text{Area}(AEC)<\text{Area}(DBF)$, and if $45^\circ < m(\widehat{XOY})<90^\circ$ then $\text{Area}(AEC)>\text{Area}(DBF)$ $\textbf{(E)}$ None of above

2020 Bulgaria Team Selection Test, 3