Olympiad problems

2007 JBMO Shortlist, 3

Tags: geometry , JBMO

Let the inscribed circle of the triangle $\vartriangle ABC$ touch side $BC$ at $M$ , side $CA$ at $N$ and side $AB$ at $P$ . Let $D$ be a point from $\left[ NP \right]$ such that $\frac{DP}{DN}=\frac{BD}{CD}$ . Show that $DM \perp PN$ .

2011 JBMO Shortlist, 2

Tags: geometry , JBMO

Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$.

2014 JBMO Shortlist, 1

Tags: JBMO , geometry

Let ${ABC}$ be a triangle with $m\left( \angle B \right)=m\left( \angle C \right)={{40}^{{}^\circ }}$ Line bisector of ${\angle{B}}$ intersects ${AC}$ at point ${D}$. Prove that $BD+DA=BC$.

2023 JBMO Shortlist, A6

Tags: JBMO , JBMO Shortlist , algebra

Find the maximum constant $C$ such that, whenever $\{a_n \}_{n=1}^{\infty}$ is a sequence of positive real numbers satisfying $a_{n+1}-a_n=a_n(a_n+1)(a_n+2)$, we have $$\frac{a_{2023}-a_{2020}}{a_{2022}-a_{2021}}>C.$$

2013 Junior Balkan MO, 1

Tags: JBMO , JBMO Shortlist , number theory

Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.

2009 JBMO Shortlist, 5

Tags: JBMO , number theory , system of equations

Show that there are infinitely many positive integers $c$, such that the following equations both have solutions in positive integers: $(x^2 - c)(y^2 -c) = z^2 -c$ and $(x^2 + c)(y^2 - c) = z^2 - c$.

2023 JBMO Shortlist, A7

Tags: Sequence , inequalities , JBMO , JBMO Shortlist , algebra

Let $a_1,a_2,a_3,\ldots,a_{250}$ be real numbers such that $a_1=2$ and $$a_{n+1}=a_n+\frac{1}{a_n^2}$$ for every $n=1,2, \ldots, 249$. Let $x$ be the greatest integer which is less than $$\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_{250}}$$ How many digits does $x$ have? [i]Proposed by Miroslav Marinov, Bulgaria[/i]

2008 Junior Balkan MO, 1

Tags: inequalities , algebra , polynomial , function , JBMO , Junior Balkan , algebra solved

Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\ ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]

2009 JBMO Shortlist, 3

Tags: JBMO , number theory

Find all pairs $(x,y)$ of integers which satisfy the equation $(x + y)^2(x^2 + y^2) = 2009^2$

2008 JBMO Shortlist, 9

Tags: JBMO , geometry

Let $O$ be a point inside the parallelogram $ABCD$ such that $\angle AOB + \angle COD = \angle BOC + \angle AOD$. Prove that there exists a circle $k$ tangent to the circumscribed circles of the triangles $\vartriangle AOB, \vartriangle BOC, \vartriangle COD$ and $\vartriangle DOA$.

JBMO Geometry Collection, 2012

Tags: geometry , JBMO

Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.

2008 JBMO Shortlist, 1

Tags: JBMO , algebra

If for the real numbers $x, y,z, k$ the following conditions are valid, $x \ne y \ne z \ne x$ and $x^3 +y^3 +k(x^2 +y^2) = y^3 +z^3 +k(y^2 +z^2) = z^3 +x^3 +k(z^2 +x^2) = 2008$, find the product $xyz$.

2007 JBMO Shortlist, 1

Tags: quadratics , inequalities , function , algebra , algebra solved , inequalities solved , JBMO

Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.

2011 JBMO Shortlist, 8

Tags: JBMO , algebra

Decipher the equality $(\overline{LARN} -\overline{ACA}) : (\overline{CYP} +\overline{RUS}) = C^{Y^P} \cdot R^{U^S} $ where different symbols correspond to different digits and equal symbols correspond to equal digits. It is also supposed that all these digits are different from $0$.

2023 JBMO Shortlist, G3

Tags: JBMO , JBMO Shortlist , geometry , circumcircle , tangent circles

Let $A,B,C,D$ and $E$ be five points lying in this order on a circle, such that $AD=BC$. The lines $AD$ and $BC$ meet at a point $F$. The circumcircles of the triangles $CEF$ and $ABF$ meet again at the point $P$. Prove that the circumcircles of triangles $BDF$ and $BEP$ are tangent to each other.

2016 JBMO Shortlist, 5

Tags: geometry , JBMO

Let $ABC$ be an acute angled triangle with orthocenter ${H}$ and circumcenter ${O}$. Assume the circumcenter ${X}$ of ${BHC}$ lies on the circumcircle of ${ABC}$. Reflect $O$ across ${X}$ to obtain ${O'}$, and let the lines ${XH}$and ${O'A}$ meet at ${K}$. Let $L,M$ and $N$ be the midpoints of $\left[ XB \right],\left[ XC \right]$ and $\left[ BC \right]$, respectively. Prove that the points $K,L,M$ and ${N}$ are concyclic.

2003 Junior Balkan MO, 3

Tags: geometry , circumcircle , angle bisector , JBMO

Let $D$, $E$, $F$ be the midpoints of the arcs $BC$, $CA$, $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$, $B$, $C$, respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$, and let $M$ be the midpoint of the segment $GH$. Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$, and let $N$ be the midpoint of the segment $KJ$. a) Find the angles of triangle $DMN$; b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$, then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$.

2022 Azerbaijan JBMO TST, C4

Tags: combinatorics , Azerbaijan , Junior , JBMO , JBMO TST , Junior Balkan , table

$n$ is a natural number. Given $3n \cdot 3n$ table, the unit cells are colored white and black such that starting from the left up corner diagonals are colored in pure white or black in ratio of 2:1 respectively. ( See the picture below). In one step any chosen $2 \cdot 2$ square's white cells are colored orange, orange are colored black and black are colored white. Find all $n$ such that with finite steps, all the white cells in the table turns to black, and all black cells in the table turns to white. ( From starting point)

2008 JBMO Shortlist, 2

Tags: JBMO , number theory

Let $n \ge 2$ be a fixed positive integer. An integer will be called "$n$-free" if it is not a multiple of an $n$-th power of a prime. Let $M$ be an infinite set of rational numbers, such that the product of every $n$ elements of $M$ is an $n$-free integer. Prove that $M$ contains only integers.

2024 Junior Balkan MO, 3

Tags: number theory , Diophantine equation , exponent , JBMO , JBMO 2024

Find all triples of positive integers $(x, y, z)$ that satisfy the equation $$2020^x + 2^y = 2024^z.$$ [i]Proposed by Ognjen Tešić, Serbia[/i]

2011 JBMO Shortlist, 6

Tags: inequalities , algebra , JBMO

Let $\displaystyle {x_i> 1, \forall i \in \left \{1, 2, 3, \ldots, 2011 \right \}}$. Show that:$$\displaystyle{\frac{x^2_1}{x_2-1}+\frac{x^2_2}{x_3-1}+\frac{x^2_3}{x_4-1}+\ldots+\frac{x^2_{2010}}{x_{2011}-1}+\frac{x^2_{2011}}{x_1-1}\geq 8044}$$ When the equality holds?

2011 JBMO Shortlist, 2

Tags: JBMO , algebra , inequalities

Let $x, y, z$ be positive real numbers. Prove that: $\frac{x + 2y}{z + 2x + 3y}+\frac{y + 2z}{x + 2y + 3z}+\frac{z + 2x}{y + 2z + 3x} \le \frac{3}{2}$

2013 JBMO Shortlist, 3

Tags: JBMO , JBMO Shortlist , number theory

Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.

2013 JBMO Shortlist, 2

Tags: geometry , JBMO

Circles ${\omega_1}$ , ${\omega_2}$ are externally tangent at point M and tangent internally with circle ${\omega_3}$ at points ${K}$ and $L$ respectively. Let ${A}$ and ${B}$ be the points that their common tangent at point ${M}$ of circles ${\omega_1}$ and ${\omega_2}$ intersect with circle ${\omega_3.}$ Prove that if ${\angle KAB=\angle LAB}$ then the segment ${AB}$ is diameter of circle ${\omega_3.}$ Theoklitos Paragyiou (Cyprus)

2011 JBMO Shortlist, 3

Tags: JBMO , number theory

Find all positive integers $n$ such that the equation $y^2 + xy + 3x = n(x^2 + xy + 3y)$ has at least a solution $(x, y)$ in positive integers.