Olympiad problems

2014 Balkan MO Shortlist, N1

$\boxed{N1}$Let $n$ be a positive integer,$g(n)$ be the number of positive divisors of $n$ of the form $6k+1$ and $h(n)$ be the number of positive divisors of $n$ of the form $6k-1,$where $k$ is a nonnegative integer.Find all positive integers $n$ such that $g(n)$ and $h(n)$ have different parity.

2019 Azerbaijan BMO TST, 1

Tags: number theory , Balkan MO Shortlist

For positive integers $m$ and $n$, let $d(m, n)$ be the number of distinct primes that divide both $m$ and $n$. For instance, $d(60, 126) = d(2^2 \cdot 3 \cdot 5, 2 \cdot 3^2 \cdot 7) = 2.$ Does there exist a sequence $(a_n)$ of positive integers such that: [list] [*] $a_1 \geq 2018^{2018};$ [*] $a_m \leq a_n$ whenever $m \leq n$; [*] $d(m, n) = d(a_m, a_n)$ for all positive integers $m\neq n$? [/list] [i](Dominic Yeo, United Kingdom)[/i]

2018 Balkan MO Shortlist, A1

Tags: algebra , Inequality , TST , Balkan MO Shortlist , AZE JBMO TST

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

2019 JBMO Shortlist, A2

Tags: algebra , Inequality , TST , Balkan MO Shortlist , AZE JBMO TST

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

2018 Azerbaijan JBMO TST, 1

Tags: algebra , Inequality , TST , Balkan MO Shortlist , AZE JBMO TST

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

2017 Balkan MO Shortlist, C6

Tags: combinatorics , combinatorial geometry , diagonals , minimum , Balkan MO Shortlist

What is the least positive integer $k$ such that, in every convex $101$-gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?

2020 Balkan MO Shortlist, A2

Tags: algebra , Inequality , Balkan MO Shortlist

Given are positive reals $a, b, c$, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$. Prove that $\frac{\sqrt{a+\frac{b}{c}}+\sqrt{b+\frac{c}{a}}+\sqrt{c+\frac{a}{b}}}{3}\leq \frac{a+b+c-1}{\sqrt{2}}$. [i]Albania[/i]

2018 Balkan MO Shortlist, N5

Tags: number theory , Balkan MO Shortlist

Let $x,y$ be positive integers. If for each positive integer $n$ we have that $$(ny)^2+1\mid x^{\varphi(n)}-1.$$ Prove that $x=1$. [i](Silouanos Brazitikos, Greece)[/i]

2018 Azerbaijan JBMO TST, 2

Tags: geometry , Balkan MO Shortlist , TST , AZE JBMO TST

Let $ABC$ be an acute triangle and let $M$ be the midpoint of side $BC$. Let $D,E$ be the excircles of triangles $AMB,AMC$ respectively, towards $M$. Circumcirscribed circle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. Circumcirscribed circles of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF=CG$. by Petru Braica, Romania

2018 Balkan MO Shortlist, G1

Tags: geometry , Balkan MO Shortlist , TST , AZE JBMO TST

Let $ABC$ be an acute triangle and let $M$ be the midpoint of side $BC$. Let $D,E$ be the excircles of triangles $AMB,AMC$ respectively, towards $M$. Circumcirscribed circle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. Circumcirscribed circles of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF=CG$. by Petru Braica, Romania

2020 Balkan MO Shortlist, G2

Tags: geometry , right angle , Balkan MO Shortlist , geometry solved , Balkan , Euler Line , Angle Chasing

Let $G, H$ be the centroid and orthocentre of $\vartriangle ABC$ which has an obtuse angle at $\angle B$. Let $\omega$ be the circle with diameter $AG$. $\omega$ intersects $\odot(ABC)$ again at $L \ne A$. The tangent to $\omega$ at $L$ intersects $\odot(ABC)$ at $K \ne L$. Given that $AG = GH$, prove $\angle HKG = 90^o$ . [i]Sam Bealing, United Kingdom[/i]

2018 Balkan MO Shortlist, N1

Tags: number theory , Balkan MO Shortlist

For positive integers $m$ and $n$, let $d(m, n)$ be the number of distinct primes that divide both $m$ and $n$. For instance, $d(60, 126) = d(2^2 \cdot 3 \cdot 5, 2 \cdot 3^2 \cdot 7) = 2.$ Does there exist a sequence $(a_n)$ of positive integers such that: [list] [*] $a_1 \geq 2018^{2018};$ [*] $a_m \leq a_n$ whenever $m \leq n$; [*] $d(m, n) = d(a_m, a_n)$ for all positive integers $m\neq n$? [/list] [i](Dominic Yeo, United Kingdom)[/i]

2023 Balkan MO Shortlist, G4

Tags: geometry , Balkan MO Shortlist , BMO Shortlist

Let $O$ and $H$ be the circumcenter and orthocenter of a scalene triangle $ABC$, respectively. Let $D$ be the intersection point of the lines $AH$ and $BC$. Suppose the line $OH$ meets the side $BC$ at $X$. Let $P$ and $Q$ be the second intersection points of the circumcircles of $\triangle BDH$ and $\triangle CDH$ with the circumcircle of $\triangle ABC$, respectively. Show that the four points $P, D, Q$ and $X$ lie on a circle.