Navvye/Olympiad-Problems-CSV huggingface dataset:
2009 Balkan MO Shortlist
Let $A_1, A_2, \ldots , A_m$ be subsets of the set $\{ 1,2, \ldots , n \}$, such that the cardinal of each subset $A_i$, such $1 \le i \le m$ is not divisible by $30$, while the cardinal of each of the subsets $A_i \cap A_j$ for $1 \le i,j \le m$, $i \neq j$ is divisible by $30$. Prove

\begin{align*} 2m - \left \lfloor \frac{m}{30} \right \rfloor  \le 3n \end{align*}
2009 Balkan MO Shortlist
Solve the given equation in integers

\begin{align*} y^3=8x^6+2x^3y-y^2 \end{align*}
2009 Balkan MO Shortlist
Determine all integers $1 \le m, 1 \le n \le 2009$, for which

\begin{align*} \prod_{i=1}^n \left( i^3 +1 \right) = m^2 \end{align*}
2008 Balkan MO Shortlist
For all $\alpha_1, \alpha_2,\alpha_3 \in \mathbb{R}^+$, Prove

\begin{align*} \sum \frac{1}{2\nu \alpha_1 +\alpha_2+\alpha_3} > \frac{2\nu}{2\nu +1} \left( \sum \frac{1}{\nu \alpha_1 + \nu \alpha_2 + \alpha_3} \right) \end{align*}for every positive real number $\nu$
2008 Balkan MO Shortlist
Let $(a_m)$ be a sequence satisfying $a_n \geq 0$, $n=0,1,2,\ldots$ Suppose there exists $A >0$, $a_m - a_{m+1}$ $\geq A a_m ^2$ for all $m \geq 0$. Prove that there exists $B>0$ such that

\begin{align*} a_n \le \frac{B}{n} \qquad \qquad \text{for }1 \le n \end{align*}
2008 Balkan MO Shortlist
We consider the set

\begin{align*} \mathbb{C}^{\nu} = \{ (z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C} \},\qquad  \nu \geq 2 \end{align*}and the function $\phi : \mathbb{C}^{\nu} \longrightarrow \mathbb{C}^{\nu}$ mapping every element $(z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C}^{\nu}$ to

\begin{align*}\phi ( z_1,z_2, \ldots , z_{\nu})= \left( z_1-z_2, z_2-z_3, \ldots, z_{\nu}-z_1 \right) \end{align*}We also consider the $\nu-$tuple $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$ $\in \mathbb{C}^{\nu}$ of the $n-$th roots of $-1$, where

\begin{align*} \omega_{\mu} = \cos \left( \frac{\pi + 2\mu \pi }{\nu} \right) + \iota \sin \left( \frac{\pi + 2\mu \pi}{\nu} \right) \qquad \mu =0,1, \ldots , \nu -1 \end{align*}Let after $\kappa$ (where $\kappa$ $\in$ $\mathbb{N}$ ), successive applications of $\phi$ to the element $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$, we obtain the element

\begin{align*} \phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right) =\left( Z_{\kappa 1}, Z_{\kappa 2}, \ldots , Z_{\kappa \nu } \right) \end{align*}Determine



the values of $\nu$ for which all coordinates of $\phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right) $ have measures less than or equal to $1$

for $\nu =4$, the minimal value of $\kappa \in \mathbb{N}$, for which

\begin{align*} \mid Z_{\kappa i} \mid \geq 2^{100} \qquad \qquad 1 \le i \le 4 \end{align*}
2008 Balkan MO Shortlist
Consider an integer $n \geq 1$, $a_1,a_2, \ldots , a_n$ real numbers in $[-1,1]$ satisfying

\begin{align*}a_1+a_2+\ldots +a_n=0 \end{align*}and a function $f: [-1,1] \mapsto \mathbb{R}$ such

\begin{align*} \mid f(x)-f(y) \mid \le \mid x-y \mid \end{align*}for every $x,y \in [-1,1]$. Prove

\begin{align*} \left| f(x) - \frac{f(a_1) +f(a_2) + \ldots + f(a_n)}{n}  \right| \le 1 \end{align*}for every $x$ $\in [-1,1]$. For a given sequence $a_1,a_2, \ldots ,a_n$, Find $f$ and $x$ so hat the equality holds.
2008 Balkan MO Shortlist
Prove that if $x,y,z \in \mathbb{R}^+$ such that $xy,yz,zx$ are sidelengths of a triangle and $k$ $\in$ $[-1,1]$, then

\begin{align*} \sum \frac{\sqrt{xy}}{\sqrt{xz+yz+kxy}} \geq 2 \sqrt{1-k} \end{align*}Determine the equality condition too.
2008 Balkan MO Shortlist
Let $x,y,z,t \in \mathbb{R}_{\geq 0}$. Show

\begin{align*} \sqrt{xy}+\sqrt{xz}+\sqrt{xt}+\sqrt{yz}+\sqrt{yt}+\sqrt{zt} \geq 3 \sqrt[3]{xyz+xyt+xzt+yzt} \end{align*}and determine the equality cases.
2008 Balkan MO Shortlist
All $n+3$ offices of University of Somewhere are numbered with numbers $0,1,2, \ldots ,$ $n+1,$ $n+2$ for some $n \in \mathbb{N}$. One day, Professor $D$ came up with a polynomial with real coefficients and power $n$. Then, on the door of every office he wrote the value of that polynomial evaluated in the number assigned to that office. On the $i$th office, for $i$ $\in$ $\{0,1, \ldots, n+1 \}$ he wrote $2^i$ and on the $(n+2)$th office he wrote $2^{n+2}$ $-n-3$.



Prove that Professor D made a calculation error

Assuming that Professor D made a calculation error, what is the smallest number of errors he made? Prove that in this case the errors are uniquely determined, find them and correct them.
2008 Balkan MO Shortlist
In one of the countries, there are $n \geq 5$ cities operated by two airline companies. Every two cities are operated in both directions by at most one of the companies. The government introduced a restriction that all round trips that a company can offer should have atleast six cities. Prove that there are no more than $\lfloor \tfrac{n^2}{3} \rfloor$ flights offered by these companies.
2008 Balkan MO Shortlist
An array $n \times n$ is given, consisting of $n^2$ unit squares. A pawn is placed arbitrarily on a unit square. A move of the pawn means a jump from a square of the $k$th column to any square of the $k$th row. Show that there exists a sequence of $n^2$ moves of the pawn so that all the unit squares of the array are visited once and, in the end, the pawn returns to the original position.
2008 Balkan MO Shortlist
In acute angled triangle $ABC$ we denote by $a,b,c$ the side lengths, by $m_a,m_b,m_c$ the median lengths and by $r_{b}c,r_{ca},r_{ab}$ the radii of the circles tangents to two sides and to circumscribed circle of the triangle, respectively. Prove that

$$\frac{m_a^2}{r_{bc}}+\frac{m_b^2}{r_{ab}}+\frac{m_c^2}{r_{ab}} \ge \frac{27\sqrt3}{8}\sqrt[3]{abc}$$
2008 Balkan MO Shortlist
We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.
2008 Balkan MO Shortlist
A triangle $ABC$ is given with barycentre $G$ and circumcentre $O$. The perpendicular bisectors of $GA, GB$ meet at $C_1$,of $GB,GC$ meet at $A _1$, and $GC,GA$ meet at $B_1$. Prove that $O$ is the barycenter of the triangle $A_1B_1C_1$.
2008 Balkan MO Shortlist
The circle $k_a$ touches the extensions of sides $AB$ and $BC$, as well as the circumscribed circle of the triangle $ABC$ (from the outside). We denote the intersection of $k_a$ with the circumscribed circle of the triangle $ABC$ by $A'$. Analogously, we define points $B'$ and $C'$. Prove that the lines $AA',BB'$ and $CC'$ intersect in one point.
2008 Balkan MO Shortlist
On triangle $ABC$ the $AM$ ($M\in BC$) is median and $BB_1$ and $CC_1$ ($B_1 \in AC,C_1 \in AB$) are altitudes. The stright line $d$ is perpendicular to $AM$ at the point $A$ and intersect the lines $BB_1$ and $CC_1$ at the points $E$ and $F$ respectively. Let denoted with $\omega$ the circle passing through the points $E, M$ and $F$ and with $\omega_1$ and with $\omega_2$ the circles that are tangent to segment $EF$ and with $\omega$ at the arc $EF$ which is not contain the point $M$. If the points $P$ and $Q$ are intersections points for $\omega_1$ and $\omega_2$ then prove that the points $P, Q$ and $M$ are collinear.
2008 Balkan MO Shortlist
In the non-isosceles triangle $ABC$ consider the points $X$ on $[AB]$ and $Y$ on $[AC]$ such that $[BX]=[CY]$, $M$ and $N$ are the midpoints of the segments $[BC]$, respectively $[XY]$, and the straight lines $XY$ and $BC$ meet in $K$. Prove that the circumcircle of triangle $KMN$ contains a point, different from $M$ , which is independent of the position of the points $X$ and $Y$.
2008 Balkan MO Shortlist
Let $P$ be a point in the interior of a triangle $ABC$ and let $d_a,d_b,d_c$ be its distances to $BC,CA,AB$ respectively. Prove that  max $(AP, BP, CP) \ge \sqrt{d_a^2+d_b^2+d_c^2}$
2008 Balkan MO Shortlist
Prove that for every natural number $a$, there exists a natural number that has the number $a$ (the sequence of digits that constitute $a$) at its beginning, and which decreases $a$ times when $a$ is moved from its beginning to it end (any number zeros that appear in the beginning of the number obtained in this way are to be removed).



Example



$a=4$, then $\underline{4}10256= 4 \cdot 10256\underline{4}$

$a=46$, then $\underline{46}0100021743857360295716= 46 \cdot 100021743857360295716\underline{46}$
2008 Balkan MO Shortlist
The sequence $(\chi_n) _{n=1}^{\infty}$ is defined as follows

\begin{align*} \chi_{n+1}=\chi_n  + \chi _{\lceil \frac{n}{2} \rceil} ~, \chi_1 =1 \end{align*}Prove that none of the terms of this sequence are divisible by $4$
2008 Balkan MO Shortlist
Solve the given equation in primes

\begin{align*} xyz=1 +2^{y^2+1} \end{align*}
2008 Balkan MO Shortlist
Let $(a_n)$ be a sequence with $a_1=0$ and $a_{n+1}=2+a_n$ for odd $n$ and $a_{n+1}=2a_n$ for even $n$. Prove that for each prime $p>3$, the number

\begin{align*} b=\frac{2^{2p}-1}{3} \mid a_n \end{align*}for infinitely many values of $n$
2008 Balkan MO Shortlist
Let $(x_n)$, $n=1,2, \ldots $ be a sequence defined by $x_1=2008$ and

\begin{align*} x_1 +x_2 + \ldots + x_{n-1} = \left( n^2-1 \right) x_n \qquad ~ ~ ~ \forall n \geq 2 \end{align*}Let the sequence $a_n=x_n + \frac{1}{n} S_n$, $n=1,2,3, \ldots $ where $S_n$ $=$ $x_1+x_2 +\ldots +x_n$. Determine the values of $n$ for which the terms of the sequence $a_n$ are perfect squares of an integer.
2007 Balkan MO Shortlist
Find the minimum and maximum value of the function

\begin{align*} f(x,y)=ax^2+cy^2 \end{align*}Under the condition $ax^2-bxy+cy^2=d$, where $a,b,c,d$ are positive real numbers such that $b^2 -4ac <0$
2007 Balkan MO Shortlist
Find all values of $a \in \mathbb{R}$ for which the polynomial

\begin{align*} f(x)=x^4-2x^3 + \left(5-6a^2 \right)x^2 + \left(2a^2-4 \right)x + \left(a^2 -2 \right)^2 \end{align*}has exactly three real roots.
2007 Balkan MO Shortlist
Show that the sequence

\begin{align*} a_n = \left \lfloor \left( \sqrt[3]{n-2} + \sqrt[3]{n+3} \right)^3 \right \rfloor \end{align*}contains infinitely many terms of the form $a_n^{a_n}$
2007 Balkan MO Shortlist
find all the function $f,g:R\rightarrow R$ such that

(1)for every $x,y\in R$  we have  $f(xg(y+1))+y=xf(y)+f(x+g(y))$

(2)$f(0)+g(0)=0$
2007 Balkan MO Shortlist
Let $c>2$ and $a_0,a_1, \ldots$ be a sequence of real numbers such that

\begin{align*} a_n = a_{n-1}^2 - a_{n-1} < \frac{1}{\sqrt{cn}} \end{align*}for any $n$ $\in$ $\mathbb{N}$. Prove, $a_1=0$
2007 Balkan MO Shortlist
Let $\omega$ be a circle with center $O$ and let $A$ be a point outside $\omega$. The tangents from $A$ touch $\omega$ at points $B$, and $C$. Let $D$ be the point at which the line $AO$ intersects the circle such that $O$ is between $A$ and $D$. Denote by $X$ the orthogonal projection of $B$ onto $CD$, by $Y$ the midpoint of the segment $BX$ and by $Z$ the second point of intersection of the line $DY$ with $\omega$. Prove that $ZA$ and $ZC$ are perpendicular to each other.
2007 Balkan MO Shortlist
Points $M,N$ and $P$ on the sides $BC, CA$ and $AB$ of  $\vartriangle  ABC$ are such that $\vartriangle MNP$ is acute. Denote by $h$ and $H$ the lengths of the shortest altitude of $\vartriangle ABC$ and the longest altitude of $\vartriangle MNP$. Prove that $h  \le 2H$.
2007 Balkan MO Shortlist
Solve the given system in prime numbers

\begin{align*} x^2+yu = (x+u)^v \end{align*}\begin{align*} x^2+yz=u^4 \end{align*}
2007 Balkan MO Shortlist
Prove that there are no distinct positive integers $x$ and $y$ such that



$x^{2007} + y! = y^{2007} + x! $
2007 Balkan MO Shortlist
i thought that this problem was in mathlinks but when i searched i didn't find it.so here it is:

Find all positive integers m for which for all $\alpha,\beta \in \mathbb{Z}-\{0\}$

$$ \frac{2^m \alpha^m-(\alpha+\beta)^m-(\alpha-\beta)^m}{3 \alpha^2+\beta^2} \in \mathbb{Z} $$
2007 Balkan MO Shortlist
Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$,

the $p$-th term of the progression is also prime.
2007 Balkan MO Shortlist
Let $p \geq 5$ be a prime and let

\begin{align*} (p-1)^p +1 = \prod _{i=1}^n q_i^{\beta_i} \end{align*}where $q_i$ are primes. Prove,

\begin{align*} \sum_{i=1}^n q_i \beta_i >p^2 \end{align*}
2021 Baltic Way
Let $n$ be a positive integer. Find all functions $f\colon \mathbb{R}\rightarrow \mathbb{R}$ that satisfy the equation

$$

    (f(x))^n f(x+y) = (f(x))^{n+1} + x^n f(y)

$$for all $x ,y \in \mathbb{R}$.
2021 Baltic Way
Let $a$, $b$, $c$ be the side lengths of a triangle. Prove that

$$

    \sqrt[3]{(a^2+bc)(b^2+ca)(c^2+ab)} > \frac{a^2+b^2+c^2}{2}.

$$
2021 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 $n \ge 1$.
2021 Baltic Way
Let $\Gamma$ be a circle in the plane and $S$ be a point on $\Gamma$. Mario and Luigi drive around the circle $\Gamma$ with their go-karts. They both start at $S$ at the same time. They both drive for exactly $6$ minutes at constant speed counterclockwise around the circle. During these $6$ minutes, Luigi makes exactly one lap around $\Gamma$ while Mario, who is three times as fast, makes three laps.



While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana.



How many points in the plane, apart from $S$, are marked with a banana by the end of the $6$ minutes.
2021 Baltic Way
Let $x,y\in\mathbb{R}$ be such that $x = y(3-y)^2$ and $y = x(3-x)^2$. Find all possible values of $x+y$.
2021 Baltic Way
Let $n$ be a positive integer and $t$ be a non-zero real number. Let $a_1, a_2, \ldots, a_{2n-1}$ be real numbers (not necessarily distinct). Prove that there exist distinct indices $i_1, i_2, \ldots, i_n$ such that, for all $1 \le k, l \le n$, we have $a_{i_k} - a_{i_l} \neq t$.
2021 Baltic Way
Let $n>2$ be an integer. Anna, Edda and Magni play a game on a hexagonal board tiled with regular hexagons, with $n$ tiles on each side. The figure shows a board with 5 tiles on each side. The central tile is marked.

[asy]unitsize(.25cm);

real s3=1.73205081;

pair[] points={(-4,4*s3),(-2,4*s3),(0,4*s3),(2,4*s3),(4,4*s3),(-5,3*s3), (-3,3*s3), (-1,3*s3), (1,3*s3), (3,3*s3), (5,3*s3), (-6,2*s3),(-4,2*s3), (-2,2*s3), (0,2*s3), (2,2*s3), (4,2*s3),(6,2*s3),(-7,s3), (-5,s3), (-3,s3), (-1,s3), (1,s3), (3,s3), (5,s3),(7,s3),(-8,0), (-6,0), (-4,0), (-2,0), (0,0), (2,0), (4,0), (6,0), (8,0),(-7,-s3),(-5,-s3), (-3,-s3), (-1,-s3), (1,-s3), (3,-s3), (5,-s3), (7,-s3), (-6,-2*s3), (-4,-2*s3), (-2,-2*s3), (0,-2*s3), (2,-2*s3), (4,-2*s3), (6,-2*s3),  (-5,-3*s3),  (-3,-3*s3), (-1,-3*s3), (1,-3*s3), (3,-3*s3), (5,-3*s3), (-4,-4*s3), (-2,-4*s3), (0,-4*s3), (2,-4*s3), (4,-4*s3)};



void draw_hexagon(pair p)

{

draw(shift(p)*scale(2/s3)*(dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--dir(30)));

}

{for (int i=0;i<61;++i){draw_hexagon(points[i]);}}

label((0,0), "\Large $*$");

[/asy]

The game begins with a stone on a tile in one corner of the board. Edda and Magni are on the same team, playing against Anna, and they win if the stone is on the central tile at the end of any player's turn. Anna, Edda and Magni take turns moving the stone: Anna begins, then Edda, then Magni, then Anna, and so on.



The rules for each player's turn are:



Anna has to move the stone to an adjacent tile, in any direction.

Edda has to move the stone straight by two tiles in any of the $6$ possible directions.

Magni has a choice of passing his turn, or moving the stone straight by three tiles in any of the $6$ possible directions.



Find all $n$ for which Edda and Magni have a winning strategy.
2021 Baltic Way
We are given a collection of $2^{2^k}$ coins, where $k$ is a non-negative integer. Exactly one coin is fake.

We have an unlimited number of service dogs. One dog is sick but we do not know which one.

A test consists of three steps: select some coins from the collection of all coins; choose a service dog; the dog smells all of the selected coins at once.

A healthy dog will bark if and only if the fake coin is amongst them. Whether the sick dog will bark or not is random.

Devise a strategy to find the fake coin, using at most $2^k+k+2$ tests, and prove that it works.
2021 Baltic Way
We are given $2021$ points on a plane, no three of which are collinear. Among any $5$ of these points, at least $4$ lie on the same circle. Is it necessarily true that at least $2020$ of the points lie on the same circle?
2021 Baltic Way
John has a string of paper where $n$ real numbers  $a_i \in [0, 1]$, for all $i \in \{1, \ldots, n\}$, are written in a row.

Show that for any given $k < n$, he can cut the string of paper into non-empty $k$ pieces, between adjacent numbers, in such a way that the sum of the numbers on each piece does not differ from any other sum by more than $1$.
2021 Baltic Way
A point $P$ lies inside a triangle $ABC$. The points $K$ and $L$ are the projections of $P$ onto $AB$ and $AC$, respectively. The point $M$ lies on the line $BC$ so that $KM = LM$, and the point $P'$ is symmetric to $P$ with respect to $M$. Prove that $\angle BAP = \angle P'AC$.
2021 Baltic Way
Let $I$ be the incentre of a triangle $ABC$. Let $F$ and $G$ be the projections of $A$ onto the lines $BI$ and $CI$, respectively. Rays $AF$ and $AG$ intersect the circumcircles of the triangles $CFI$ and $BGI$ for the second time at points $K$ and $L$, respectively. Prove that the line $AI$ bisects the segment $KL$.
2021 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 $PK \perp AB$.
2021 Baltic Way
Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. Denote by $M$ the midpoint of $BC$. The point $D$ is the reflection of $A$ over $BC$, and the point $E$ is the intersection of $\Gamma$ and the ray $MD$. Let $S$ be the circumcentre of the triangle $ADE$. Prove that the points $A$, $E$, $M$, $O$, and $S$ lie on the same circle.
2021 Baltic Way
For which positive integers $n\geq4$ does there exist a convex $n$-gon with side lengths $1, 2, \dots, n$ (in some order) and with all of its sides tangent to the same circle?
2021 Baltic Way
Show that no non-zero integers $a$, $b$, $x$, $y$ satisfy

$$

    \begin{cases}

    a x - b y = 16,\\
    a y + b x = 1.

    \end{cases}

$$
2021 Baltic Way
Distinct positive integers $a, b, c, d$ satisfy

$$\begin{cases} a \mid b^2 + c^2 + d^2,\\
 b\mid a^2 + c^2 + d^2,\\
c \mid a^2 + b^2 + d^2,\\
d \mid a^2 + b^2 + c^2,\end{cases}$$and none of them is larger than the product of the three others. What is the largest possible number of primes among them?
2021 Baltic Way
Find all integer triples $(a, b, c)$ satisfying the equation

$$

    5 a^2 + 9 b^2 = 13 c^2.

$$
2021 Baltic Way
Find all polynomials $p$ with integer coefficients such that the number $p(a) - p(b)$ is divisible by $a + b$ for all integers $a, b$, provided that $a + b \neq 0$.
2021 Baltic Way
Let $n\ge 2$ be an integer. Given numbers $a_1, a_2, \ldots, a_n \in \{1,2,3,\ldots,2n\}$ such that $\operatorname{lcm}(a_i,a_j)>2n$ for all $1\le i<j\le n$, prove that

$$a_1a_2\ldots a_n \mid (n+1)(n+2)\ldots (2n-1)(2n).$$
2020 Baltic Way
Let $a_0>0$ be a real number, and let

$$a_n=\frac{a_{n-1}}{\sqrt{1+2020\cdot a_{n-1}^2}}, \quad \textrm{for } n=1,2,\ldots ,2020.$$Show that $a_{2020}<\frac1{2020}$.
2020 Baltic Way
Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that

$$\frac{1}{a\sqrt{c^2 + 1}} + \frac{1}{b\sqrt{a^2 + 1}} + \frac{1}{c\sqrt{b^2+1}} > 2.$$
2020 Baltic Way
A real sequence $(a_n)_{n=0}^\infty$ is defined recursively by $a_0 = 2$ and the recursion formula

$$

a_{n} = 

\begin{dcases}

a_{n-1}^2 & \text{if $a_{n-1}<\sqrt3$} \\
\frac{a_{n-1}^2}{3} & \text{if $a_{n-1}\geq\sqrt 3$.}

\end{dcases} 

$$Another real sequence $(b_n)_{n=1}^\infty$ is defined in terms of the first by the formula

$$

b_{n} =

\begin{dcases}

0 & \text{if $a_{n-1}<\sqrt3$} \\
\frac{1}{2^{n}} & \text{if $a_{n-1}\geq\sqrt 3$,}

\end{dcases}

$$valid for each $n\geq 1$. Prove that

$$

b_1 + b_2 + \cdots + b_{2020} < \frac23.

$$
2020 Baltic Way
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 $x, y$.
2020 Baltic Way
Find all real numbers $x,y,z$ so that

\begin{align*}

    x^2 y + y^2 z + z^2 &= 0 \\
    z^3 + z^2 y + z y^3 + x^2 y &= \frac{1}{4}(x^4 + y^4).

\end{align*}
2020 Baltic Way
Let $n>2$ be a given positive integer. There are $n$ guests at Georg's bachelor party and each guest is friends with at least one other guest. Georg organizes a party game among the guests. Each guest receives a jug of water such that there are no two guests with the same amount of water in their jugs. All guests now proceed simultaneously as follows. Every guest takes one cup for each of his friends at the party and distributes all the water from his jug evenly in the cups. He then passes a cup to each of his friends. Each guest having received a cup of water from each of his friends pours the water he has received into his jug. What is the smallest possible number of guests that do not have the same amount of water as they started with?
2020 Baltic Way
A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$. She also has a box with dimensions $10\times11\times14$. The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out.

Find all possible values of the total number of bricks that she can pack.
2020 Baltic Way
Let $n$ be a given positive integer.

A restaurant offers a choice of $n$ starters, $n$ main dishes, $n$ desserts and $n$ wines.

A merry company dines at the restaurant, with each guest choosing a starter, a main dish, a dessert and a wine.

No two people place exactly the same order.

It turns out that there is no collection of $n$ guests such that their orders coincide in three of these aspects,

but in the fourth one they all differ. (For example, there are no $n$ people that order exactly the same three courses of food, but $n$ different wines.) What is the maximal number of guests?
2020 Baltic Way
Each vertex $v$ and each edge $e$ of a graph $G$ are assigned numbers $f(v)\in\{1,2\}$ and $f(e)\in\{1,2,3\}$, respectively.

Let $S(v)$ be the sum of numbers assigned to the edges incident to $v$ plus the number $f(v)$.

We say that an assignment $f$ is cool if  $S(u) \ne S(v)$ for every pair $(u,v)$ of adjacent (i.e. connected by an edge) vertices in $G$.

Prove that for every graph there exists a cool assignment.
2020 Baltic Way
Alice and Bob are playing hide and seek. Initially, Bob chooses a secret fixed point $B$ in the unit square. Then Alice chooses a sequence of points $P_0, P_1, \ldots, P_N$ in the plane. After choosing $P_k$ (but before choosing $P_{k+1}$) for $k \geq 1$, Bob tells "warmer'' if $P_k$ is closer to $B$ than $P_{k-1}$, otherwise he says "colder''. After Alice has chosen $P_N$ and heard Bob's answer, Alice chooses a final point $A$. Alice wins if the distance $AB$ is at most $\frac 1 {2020}$, otherwise Bob wins. Show that if $N=18$, Alice cannot guarantee a win.
2020 Baltic Way
Let $ABC$ be a triangle with $AB > AC$. The internal angle bisector of $\angle BAC$ intersects the side $BC$ at $D$. The circles with diameters $BD$ and $CD$ intersect the circumcircle of $\triangle ABC$ a second time at $P \not= B$ and $Q \not= C$, respectively. The lines $PQ$ and $BC$ intersect at $X$. Prove that $AX$ is tangent to the circumcircle of $\triangle ABC$.
2020 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 $SK=SL$.