Olympiad problems

2014 Baltic Way, 3

Positive real numbers $a, b, c$ satisfy $\frac{1}{a} +\frac{1}{b} +\frac{1}{c} = 3.$ Prove the inequality \[\frac{1}{\sqrt{a^3+ b}}+\frac{1}{\sqrt{b^3 + c}}+\frac{1}{\sqrt{c^3 + a}}\leq \frac{3}{\sqrt{2}}.\]

2018 Baltic Way, 3

Tags: Inequality , Baltic Way , algebra

Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove the inequality \[\frac{1}{\sqrt{a+2b+3c+10}}+\frac{1}{\sqrt{b+2c+3d+10}}+\frac{1}{\sqrt{c+2d+3a+10}}+\frac{1}{\sqrt{d+2a+3b+10}} \le 1.\]

2017 Baltic Way, 17

Tags: number theory , Baltic Way

Determine whether the equation $$x^4+y^3=z!+7$$ has an infinite number of solutions in positive integers.

2016 Baltic Way, 1

Tags: number theory , Baltic Way

Find all pairs of primes $(p, q)$ such that $$p^3 - q^5 = (p + q)^2.$$

2020 Baltic Way, 16

Tags: number theory , Baltic Way

Richard and Kaarel are taking turns to choose numbers from the set $\{1,\dots,p-1\}$ where $p > 3$ is a prime. Richard is the first one to choose. A number which has been chosen by one of the players cannot be chosen again by either of the players. Every number chosen by Richard is multiplied with the next number chosen by Kaarel. Kaarel wins the game if at any moment after his turn the sum of all of the products calculated so far is divisible by $p$. Richard wins if this does not happen, i.e. the players run out of numbers before any of the sums is divisible by $p$. Can either of the players guarantee their victory regardless of their opponent's moves and if so, which one?

2018 Baltic Way, 18

Tags: number theory , Baltic Way

Let $n \ge 3$ be an integer such that $4n+1$ is a prime number. Prove that $4n+1$ divides $n^{2n}-1$.

2014 Baltic Way, 4

Tags: functional equation , algebra , Baltic Way

Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$

2015 Baltic Way, 5

Tags: algebra , functional equation , Baltic Way

Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the equation \[|x|f(y)+yf(x)=f(xy)+f(x^2)+f(f(y))\] for all real numbers $x$ and $y$.

2017 Baltic Way, 12

Tags: geometry , Baltic Way

Line $\ell$ touches circle $S_1$ in the point $X$ and circle $S_2$ in the point $Y$. We draw a line $m$ which is parallel to $\ell$ and intersects $S_1$ in a point $P$ and $S_2$ in a point $Q$. Prove that the ratio $XP/YQ$ does not depend on the choice of $m$.

2019 Baltic Way, 12

Tags: geometry , Baltic Way

Let $ABC$ be a triangle and $H$ its orthocenter. Let $D$ be a point lying on the segment $AC$ and let $E$ be the point on the line $BC$ such that $BC\perp DE$. Prove that $EH\perp BD$ if and only if $BD$ bisects $AE$.

2016 Baltic Way, 18

Tags: geometry , Baltic Way

Let $ABCD$ be a parallelogram such that $\angle BAD = 60^{\circ}.$ Let $K$ and $L$ be the midpoints of $BC$ and $CD,$ respectively. Assuming that $ABKL$ is a cyclic quadrilateral, find $\angle ABD.$

2012 Baltic Way, 20

Tags: number theory , Baltic Way

Find all integer solutions of the equation $2x^6 + y^7 = 11$.

2016 Baltic Way, 9

Tags: algebra , system of equations , Baltic Way

Find all quadruples $(a, b, c, d)$ of real numbers that simultaneously satisfy the following equations: $$\begin{cases} a^3 + c^3 = 2 \\ a^2b + c^2d = 0 \\ b^3 + d^3 = 1 \\ ab^2 + cd^2 = -6.\end{cases}$$

2015 Baltic Way, 2

Tags: inequalities , Baltic Way , n-variable inequality

Let $n$ be a positive integer and let $a_1,\cdots ,a_n$ be real numbers satisfying $0\le a_i\le 1$ for $i=1,\cdots ,n.$ Prove the inequality \[(1-{a_i}^n)(1-{a_2}^n)\cdots (1-{a_n}^n)\le (1-a_1a_2\cdots a_n)^n.\]

2021 Baltic Way, 13

Tags: geometry , Baltic Way

Let $D$ be the foot of the $A$-altitude of an acute triangle $ABC$. The internal bisector of the angle $DAC$ intersects $BC$ at $K$. Let $L$ be the projection of $K$ onto $AC$. Let $M$ be the intersection point of $BL$ and $AD$. Let $P$ be the intersection point of $MC$ and $DL$. Prove that $PK \perp AB$.

2023 Baltic Way, 1

Tags: algebra , Baltic Way

Find all strictly increasing sequences of positive integers $a_1, a_2, \ldots$ with $a_1=1$, satisfying $$3(a_1+a_2+\ldots+a_n)=a_{n+1}+\ldots+a_{2n}$$ for all positive integers $n$.

2020 Baltic Way, 17

Tags: number theory , Baltic Way

For a prime number $p$ and a positive integer $n$, denote by $f(p, n)$ the largest integer $k$ such that $p^k \mid n!$. Let $p$ be a given prime number and let $m$ and $c$ be given positive integers. Prove that there exist infinitely many positive integers $n$ such that $f(p, n) \equiv c \pmod m$.

2021 Baltic Way, 3

Tags: algebra , algebra proposed , Sequence , Baltic Way

Determine all infinite sequences $(a_1,a_2,\dots)$ of positive integers satisfying \[a_{n+1}^2=1+(n+2021)a_n\] for all $n \ge 1$.

2018 Baltic Way, 12

Tags: geometry , Baltic Way

The altitudes $BB_1$ and $CC_1$ of an acute triangle $ABC$ intersect in point $H$. Let $B_2$ and $C_2$ be points on the segments $BH$ and $CH$, respectively, such that $BB_2=B_1H$ and $CC_2=C_1H$. The circumcircle of the triangle $B_2HC_2$ intersects the circumcircle of the triangle $ABC$ in points $D$ and $E$. Prove that the triangle $DEH$ is right-angled.

2012 Baltic Way, 11

Tags: geometry , Baltic Way

Let $ABC$ be a triangle with $\angle A = 60^\circ$. The point $T$ lies inside the triangle in such a way that $\angle ATB = \angle BTC = \angle CTA = 120^\circ$. Let $M$ be the midpoint of $BC$. Prove that $TA + TB + TC = 2AM$.

2018 Baltic Way, 1

Tags: algebra , Baltic Way

A finite collection of positive real numbers (not necessarily distinct) is [i]balanced [/i]if each number is less than the sum of the others. Find all $m \ge 3$ such that every balanced finite collection of $m$ numbers can be split into three parts with the property that the sum of the numbers in each part is less than the sum of the numbers in the two other parts.

2017 Baltic Way, 13

Tags: geometry , Baltic Way

Let $ABC$ be a triangle in which $\angle ABC = 60^{\circ}$. Let $I$ and $O$ be the incentre and circumcentre of $ABC$, respectively. Let $M$ be the midpoint of the arc $BC$ of the circumcircle of $ABC$, which does not contain the point $A$. Determine $\angle BAC$ given that $MB = OI$.

2015 Baltic Way, 15

Tags: geometry , Baltic Way

In triangle $ABC$, the interior and exterior angle bisectors of $ \angle BAC$ intersect the line $BC$ in $D $ and $E$, respectively. Let $F$ be the second point of intersection of the line $AD$ with the circumcircle of the triangle $ ABC$. Let $O$ be the circumcentre of the triangle $ ABC $and let $D'$ be the reflection of $D$ in $O$. Prove that $ \angle D'FE =90.$

2020 Baltic Way, 12

Tags: geometry , Baltic Way

Let $ABC$ be a triangle with circumcircle $\omega$. The internal angle bisectors of $\angle ABC$ and $\angle ACB$ intersect $\omega$ at $X\neq B$ and $Y\neq C$, respectively. Let $K$ be a point on $CX$ such that $\angle KAC = 90^\circ$. Similarly, let $L$ be a point on $BY$ such that $\angle LAB = 90^\circ$. Let $S$ be the midpoint of arc $CAB$ of $\omega$. Prove that $SK=SL$.

2020 Baltic Way, 4

Tags: algebra , Baltic Way , functional equation

Find all functions $f:\mathbb{R} \to \mathbb{R}$ so that \[f(f(x)+x+y) = f(x+y) + y f(y)\] for all real numbers $x, y$.