Navvye/Olympiad-Problems-CSV huggingface dataset:
2020 Baltic Way
Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $\ell$ be the tangent line to $\omega$ at $A$. Let $X$ and $Y$ be the projections of $B$ onto lines $\ell$ and $AC$, respectively. Let $H$ be the orthocenter of $BXY$. Let $CH$ intersect $\ell$ at $D$. Prove that $BA$ bisects angle $CBD$.
2020 Baltic Way
An acute triangle $ABC$ is given and let $H$ be its orthocenter. Let $\omega$ be the circle through $B$, $C$ and $H$, and let $\Gamma$ be the circle with diameter $AH$. Let $X\neq H$ be the other intersection point of $\omega$ and $\Gamma$, and let $\gamma$ be the reflection of $\Gamma$ over $AX$.



Suppose $\gamma$ and $\omega$ intersect again at $Y\neq X$, and line $AH$ and $\omega$ intersect again at $Z \neq H$. Show that the circle through $A,Y,Z$ passes through the midpoint of segment $BC$.
2020 Baltic Way
On a plane, Bob chooses 3 points $A_0$, $B_0$, $C_0$ (not necessarily distinct) such that $A_0B_0+B_0C_0+C_0A_0=1$. Then he chooses points $A_1$, $B_1$, $C_1$ (not necessarily distinct) in such a way that $A_1B_1=A_0B_0$ and $B_1C_1=B_0C_0$.

Next he chooses points $A_2$, $B_2$, $C_2$ as a permutation of points $A_1$, $B_1$, $C_1$. Finally, Bob chooses points $A_3$, $B_3$, $C_3$ (not necessarily distinct) in such a way that $A_3B_3=A_2B_2$ and $B_3C_3=B_2C_2$. What are the smallest and the greatest possible values of $A_3B_3+B_3C_3+C_3A_3$ Bob can obtain?
2020 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?
2020 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 $f(p, n) \equiv c \pmod m$.
2020 Baltic Way
Let $n\geq 1$ be a positive integer. We say that an integer $k$ is a fan of $n$ if $0\leq k\leq n-1$ and there exist integers $x,y,z\in\mathbb{Z}$ such that

\begin{align*}

x^2+y^2+z^2 &\equiv 0 \pmod n;\\
xyz &\equiv k \pmod n.

\end{align*}Let $f(n)$ be the number of fans of $n$. Determine $f(2020)$.
2020 Baltic Way
Denote by $d(n)$ the number of positive divisors of a positive integer $n$.

Prove that there are infinitely many positive integers $n$ such that $\left\lfloor\sqrt{3}\cdot d(n)\right\rfloor$ divides $n$.
2020 Baltic Way
Let $A$ and $B$ be sets of positive integers with $|A|\ge 2$ and $|B|\ge 2$. Let $S$ be a set consisting of $|A|+|B|-1$ numbers of the form $ab$ where $a\in A$ and $b\in B$. Prove that there exist pairwise distinct $x,y,z\in S$ such that $x$ is a divisor of $yz$.
2019 Baltic Way
For all non-negative real numbers $x,y,z$ with $x \geq y$, prove the inequality

$$\frac{x^3-y^3+z^3+1}{6}\geq (x-y)\sqrt{xyz}.$$
2019 Baltic Way
Let $(F_n)$ be the sequence defined recursively by $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ for $n\geq 2$. Find all pairs of positive integers $(x,y)$ such that

$$5F_x-3F_y=1.$$
2019 Baltic Way
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that

$$f(xf(y)-y^2)=(y+1)f(x-y)$$holds for all $x,y\in\mathbb{R}$.
2019 Baltic Way
Determine all integers $n$ for which there exist an integer $k\geq 2$ and positive integers $x_1,x_2,\hdots,x_k$ so that

$$x_1x_2+x_2x_3+\hdots+x_{k-1}x_k=n\text{ and } x_1+x_2+\hdots+x_k=2019.$$
2019 Baltic Way
The $2m$ numbers

$$1\cdot 2, 2\cdot 3, 3\cdot 4,\hdots,2m(2m+1)$$are written on a blackboard, where $m\geq 2$ is an integer. A move consists of choosing three numbers $a, b, c$, erasing them from the board and writing the single number

$$\frac{abc}{ab+bc+ca}$$After $m-1$ such moves, only two numbers will remain on the blackboard. Supposing one of these is $\tfrac{4}{3}$, show that the other is larger than $4$.
2019 Baltic Way
Alice and Bob play the following game. They write the expressions $x + y$, $x - y$, $x^2+xy+y^2$ and $x^2-xy+y^2$ each on a separate card. The four cards are shuffled and placed face down on a table. One of the cards is turned over, revealing the expression written on it, after which Alice chooses any two of the four cards, and gives the other two to Bob. All cards are then revealed. Now Alice picks one of the variables $x$ and $y$, assigns a real value to it, and tells Bob what value she assigned and to which variable. Then Bob assigns a real value to the other variable.



Finally, they both evaluate the product of the expressions on their two cards. Whoever gets the larger result, wins. Which player, if any, has a winning strategy?
2019 Baltic Way
Find the smallest integer $k \geq 2$ such that for every partition of the set $\{2, 3,\hdots, k\}$ into two parts, at least one of these parts contains (not necessarily distinct) numbers $a$, $b$ and $c$ with $ab = c$.
2019 Baltic Way
There are $2019$ cities in the country of Balticwayland. Some pairs of cities are connected by non-intersecting bidirectional roads, each road connecting exactly 2 cities. It is known that for every pair of cities $A$ and $B$ it is possible to drive from $A$ to $B$ using at most $2$ roads. There are $62$ cops trying to catch a robber. The cops and robber all know each others’ locations at all times. Each night, the robber can choose to stay in her current city or move to a neighbouring city via a direct road. Each day, each cop has the same choice of staying or moving, and they coordinate their actions. The robber is caught if she is in the same city as a cop at any time. Prove that the cops can always catch the robber
2019 Baltic Way
For a positive integer $n$, consider all nonincreasing functions $f : \{1,\hdots,n\}\to\{1,\hdots,n\}$. Some of them have a fixed point (i.e. a $c$ such that $f(c) = c$), some do not. Determine the difference between the sizes of the two sets of functions.



Remark. A function $f$ is nonincreasing if $f(x) \geq f(y)$ holds for all $x \leq y$
2019 Baltic Way
There are $2019$ points given in the plane. A child wants to draw $k$ (closed) discs in such a manner, that for any two distinct points there exists a disc that contains exactly one of these two points. What is the minimal $k$, such that for any initial configuration of points it is possible to draw $k$ discs with the above property?
2019 Baltic Way
Let $ABC$ be a triangle with $AB = AC$. Let $M$ be the midpoint of $BC$. Let the circles with diameters $AC$ and $BM$ intersect at points $M$ and $P$. Let $MP$ intersect $AB$ at $Q$. Let $R$ be a point on $AP$ such that $QR \parallel BP$. Prove that $CP$ bisects $\angle RCB$.
2019 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 $AE$.
2019 Baltic Way
Let $ABCDEF$ be a convex hexagon in which $AB=AF$, $BC=CD$, $DE=EF$ and $\angle ABC = \angle EFA = 90^{\circ}$. Prove that $AD\perp CE$.
2019 Baltic Way
Let $ABC$ be a triangle with $\angle ABC = 90^{\circ}$, and let $H$ be the foot of the altitude from $B$. The points $M$ and $N$ are the midpoints of the segments $AH$ and $CH$, respectively. Let $P$ and $Q$ be the second points of intersection of the circumcircle of the triangle $ABC$ with the lines $BM$ and $BN$, respectively. The segments $AQ$ and $CP$ intersect at the point $R$. Prove that the line $BR$ passes through the midpoint of the segment $MN$.
2019 Baltic Way
Let $n \geq 4$, and consider a (not necessarily convex) polygon $P_1P_2\hdots P_n$ in the plane. Suppose that, for each $P_k$, there is a unique vertex $Q_k\ne P_k$ among $P_1,\hdots, P_n$ that lies closest to it. The polygon is then said to be hostile if $Q_k\ne P_{k\pm 1}$ for all $k$ (where $P_0 = P_n$, $P_{n+1} = P_1$).



(a) Prove that no hostile polygon is convex.

(b) Find all $n \geq 4$ for which there exists a hostile $n$-gon.
2019 Baltic Way
For a positive integer $N$, let $f(N)$ be the number of ordered pairs of positive integers $(a,b)$ such that the number

$$\frac{ab}{a+b}$$is a divisor of $N$. Prove that $f(N)$ is always a perfect square.
2019 Baltic Way
Let $p$ be an odd prime. Show that for every integer $c$, there exists an integer $a$ such that

$$a^{\frac{p+1}{2}} + (a+c)^{\frac{p+1}{2}} \equiv c\pmod p.$$
2019 Baltic Way
Let $a,b$, and $c$ be odd positive integers such that $a$ is not a perfect square and

$$a^2+a+1 = 3(b^2+b+1)(c^2+c+1).$$Prove that at least one of the numbers $b^2+b+1$ and $c^2+c+1$ is composite.
2019 Baltic Way
Prove that the equation $7^x=1+y^2+z^2$ has no solutions over positive integers.
2019 Baltic Way
Let us consider a polynomial $P(x)$ with integers coefficients satisfying

$$P(-1)=-4,\ P(-3)=-40,\text{ and } P(-5)=-156.$$What is the largest possible number of integers $x$ satisfying

$$P(P(x))=x^2?$$
2018 Baltic Way
A finite collection of positive real numbers (not necessarily distinct) is balanced 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.
2018 Baltic Way
A $100 \times 100$ table is given. For each $k, 1 \le k \le 100$, the $k$-th row of the table contains the numbers $1,2,\dotsc,k$ in increasing order (from left to right) but not necessarily in consecutive cells; the remaining $100-k$ cells are filled with zeroes. Prove that there exist two columns such that the sum of the numbers in one of the columns is at least $19$ times as large as the sum of the numbers in the other column.
2018 Baltic Way
Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove the inequality

$$\frac{1}{\sqrt{a+2b+3c+10}}+\frac{1}{\sqrt{b+2c+3d+10}}+\frac{1}{\sqrt{c+2d+3a+10}}+\frac{1}{\sqrt{d+2a+3b+10}} \le 1.$$
2018 Baltic Way
Find all functions $f:[0, \infty) \to [0,\infty)$, such that for any positive integer $n$ and and for any non-negative real numbers $x_1,x_2,\dotsc,x_n$

$$f(x_1^2+\dotsc+x_n^2)=f(x_1)^2+\dots+f(x_n)^2.$$
2018 Baltic Way
A polynomial $f(x)$ with real coefficients is called generating, if for each polynomial $\varphi(x)$ with real coefficients there exists a positive integer $k$ and polynomials $g_1(x),\dotsc,g_k(x)$ with real coefficients such that

$$\varphi(x)=f(g_1(x))+\dotsc+f(g_k(x)).$$Find all generating polynomials.
2018 Baltic Way
Let $n$ be a positive integer. Elfie the Elf travels in $\mathbb{R}^3$. She starts at the origin: $(0,0,0)$. In each turn she can teleport to any point with integer coordinates which lies at distance exactly $\sqrt{n}$ from her current location. However, teleportation is a complicated procedure: Elfie starts off normal but she turns strange with her first teleportation. Next time she teleports she turns normal again, then strange again... etc.

For which $n$ can Elfie travel to any point with integer coordinates and be normal when she gets there?
2018 Baltic Way
On a  $16 \times 16$ torus as shown all $512$ edges are colored red or blue. A coloring is good if every vertex is an endpoint of an even number of red edges. A move consists of switching the color of each of the $4$ edges of an arbitrary cell. What is the largest number of good colorings so that none of them can be converted to another by a sequence of moves?
2018 Baltic Way
A graph has $N$ vertices. An invisible hare sits in one of the vertices. A group of hunters tries to kill the hare. In each move all of them shoot simultaneously: each hunter shoots at a single vertex, they choose the target vertices cooperatively. If the hare was in one of the target vertices during a shoot, the hunt is finished. Otherwise the hare can stay in its vertex or jump to one of the neighboring vertices.



The hunters know an algorithm that allows them to kill the hare in at most $N!$ moves. Prove that then there exists an algorithm that allows them to kill the hare in at most $2^N$ moves.
2018 Baltic Way
Olga and Sasha play a game on an infinite hexagonal grid. They take turns in placing a stone on a free hexagon of their choice. Olga starts the game. Just before the $2018$th stone is placed, a new rule comes into play. A stone may now be placed only on those free hexagons having at least two occupied neighbors.

A player loses when she or he either is unable to make a move, or makes a move such that a pattern of the rhomboid shape as shown (rotated in any possible way) appears. Determine which player, if any, possesses a winning strategy.
2018 Baltic Way
The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers.

What is th smallest value of $n$ for which the second player can ensure that he competes both his moves?
2018 Baltic Way
The points $A,B,C,D$ lie, in this order, on a circle $\omega$, where $AD$ is a diameter of $\omega$. Furthermore, $AB=BC=a$ and $CD=c$ for some relatively prime integers $a$ and $c$. Show that if the diameter $d$ of $\omega$ is also an integer, then either $d$ or $2d$ is a perfect square.
2018 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.
2018 Baltic Way
The bisector of the angle $A$ of a triangle $ABC$ intersects $BC$ in a point $D$ and intersects the circumcircle of the triangle $ABC$ in a point $E$. Let $K,L,M$ and $N$ be the midpoints of the segments $AB,BD,CD$ and $AC$, respectively. Let $P$  be the circumcenter of the triangle $EKL$, and $Q$ be the circumcenter of the triangle $EMN$. Prove that $\angle PEQ=\angle BAC$.
2018 Baltic Way
A quadrilateral $ABCD$ is circumscribed about a circle $\omega$. The intersection point of $\omega$ and the diagonal $AC$, closest to $A$, is $E$. The point $F$ is diametrally opposite to the point $E$ on the circle $\omega$. The tangent to $\omega$ at the point $F$ intersects lines $AB$ and $BC$ in points $A_1$ and $C_1$, and lines $AD$ and $CD$ in points $A_2$ and $C_2$, respectively. Prove that $A_1C_1=A_2C_2$.
2018 Baltic Way
Two circles in the plane do not intersect and do not lie inside each other. We choose diameters $A_1B_1$ and $A_2B_2$ of these circles such that the segments $A_1A_2$ and $B_1B_2'$ intersect. Let $A$ and $B$ be the midpoints of the segments $A_1A_2$ and $B_1B_2$, and $C$ be the intersection point of these segments. Prove that the orthocenter of the triangle $ABC$ belongs to a fixed line that does not depend on the choice of diameters.
2018 Baltic Way
Let $p$ be an odd prime. Find all positive integers $n$ for which $\sqrt{n^2-np}$ is a positive integer.
2018 Baltic Way
Prove that for any positive integers $p,q$ such that $\sqrt{11}>\frac{p}{q}$, the following inequality holds:

$$\sqrt{11}-\frac{p}{q}>\frac{1}{2pq}.$$
2018 Baltic Way
Let $n \ge 3$ be an integer such that $4n+1$ is a prime number. Prove that $4n+1$ divides $n^{2n}-1$.
2018 Baltic Way
An infinite set $B$ consisting of positive integers has the following property. For each $a,b \in B$ with $a>b$ the number $\frac{a-b}{(a,b)}$ belongs to $B$. Prove that $B$ contains all positive integers. Here, $(a,b)$ is the greatest common divisor of numbers $a$ and $b$.
2018 Baltic Way
Find all the triples of positive integers $(a,b,c)$ for which the number

$$\frac{(a+b)^4}{c}+\frac{(b+c)^4}{a}+\frac{(c+a)^4}{b}$$is an integer and $a+b+c$ is a prime.
2017 Baltic Way
Let $a_0,a_1,a_2,...$ be an infinite sequence of real numbers satisfying $\frac{a_{n-1}+a_{n+1}}{2}\geq a_n$ for all positive integers $n$. Show that $$\frac{a_0+a_{n+1}}{2}\geq \frac{a_1+a_2+...+a_n}{n}$$holds for all positive integers $n$.
2017 Baltic Way
Does there exist a finite set of real numbers such that their sum equals $2$, the sum of their squares equals $3$, the sum of their cubes equals $4$, ..., and the sum of their ninth powers equals $10$?
2017 Baltic Way
Positive integers $x_1,...,x_m$ (not necessarily distinct) are written on a blackboard. It is known that each of the numbers $F_1,...,F_{2018}$ can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of $m$?

(Here $F_1,...,F_{2018}$ are the first $2018$ Fibonacci numbers: $F_1=F_2=1, F_{k+1}=F_k+F_{k-1}$ for $k>1$.)
2017 Baltic Way
A linear form in $k$ variables is an expression of the form $P(x_1,...,x_k)=a_1x_1+...+a_kx_k$ with real constants $a_1,...,a_k$. Prove that there exist a positive integer $n$ and linear forms $P_1,...,P_n$ in $2017$ variables such that the equation $$x_1\cdot x_2\cdot ... \cdot x_{2017}=P_1(x_1,...,x_{2017})^{2017}+...+P_n(x_1,...,x_{2017})^{2017}$$holds for all real numbers $x_1,...,x_{2017}$.
2017 Baltic Way
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x^2y)=f(xy)+yf(f(x)+y)$$for all real numbers $x$ and $y$.
2017 Baltic Way
Fifteen stones are placed on a $4 \times 4$ board, one in each cell, the remaining cell being empty. Whenever two stones are on neighbouring cells (having a common side), one may jump over the other to the opposite neighbouring cell, provided this cell is empty. The stone jumped over is removed from the board.



For which initial positions of the empty cell is it possible to end up with exactly one stone on the board?
2017 Baltic Way
Each edge of a complete graph on $30$ vertices is coloured either red or blue. It is allowed to choose a non-monochromatic triangle and change the colour of the two edges of the same colour to make the triangle monochromatic.

Prove that by using this operation repeatedly it is possible to make the entire graph monochromatic.

(A complete graph is a graph where any two vertices are connected by an edge.)
2017 Baltic Way
A chess knight has injured his leg and is limping. He alternates between a normal move and a short move where he moves to any diagonally neighbouring cell.

The limping knight moves on a $5 \times 6$ cell chessboard starting with a normal move. What is the largest number of moves he can make if he is starting from a cell of his own choice and is not allowed to visit any cell (including the initial cell) more than once?
2017 Baltic Way
A positive integer $n$ is Danish if a regular hexagon can be partitioned into $n$ congruent polygons. Prove that there are infinitely many positive integers $n$ such that both $n$ and $2^n+n$ are Danish.
2017 Baltic Way
Maker and Breaker are building a wall. Maker has a supply of green cubical building blocks, and Breaker has a supply of red ones, all of the same size. On the ground, a row of $m$ squares has been marked in chalk as place-holders. Maker and Breaker now take turns in placing a block either directly on one of these squares, or on top of another block already in place, in such a way that the height of each column never exceeds $n$. Maker places the first block.

Maker bets that he can form a green row, i.e. all $m$ blocks at a certain height are green. Breaker bets that he can prevent Maker from achieving this. Determine all pairs $(m,n)$ of positive integers for which Maker can make sure he wins the bet.
2017 Baltic Way
Let $H$ and $I$ be the orthocenter and incenter, respectively, of an acute-angled triangle $ABC$. The circumcircle of the triangle $BCI$ intersects the segment $AB$ at the point $P$ different from $B$. Let $K$ be the projection of $H$ onto $AI$ and $Q$ the reflection of $P$ in $K$. Show that $B$, $H$ and $Q$ are collinear.



Proposed by Mads Christensen, Denmark
2017 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 $m$.
2017 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 $MB = OI$.
2017 Baltic Way
Let $P$ be a point inside the acute angle $\angle BAC$. Suppose that $\angle ABP = \angle ACP = 90^{\circ}$. The points $D$ and $E$ are on the segments $BA$ and $CA$, respectively, such that $BD = BP$ and $CP = CE$. The points $F$ and $G$ are on the segments $AC$ and $AB$, respectively, such that $DF$ is perpendicular to $AB$ and $EG$ is perpendicular to $AC$. Show that $PF = PG$.
2017 Baltic Way
Let $n \ge 3$ be an integer. What is the largest possible number of interior angles greater than $180^\circ$ in an $n$-gon in the plane, given that the $n$-gon does not intersect itself and all its sides have the same length?
2017 Baltic Way
Is it possible for any finite group of people to choose a positive integer $N$ and assign a positive integer to each person in the group such that the product of two persons' number is divisible by $N$ if and only if they are friends?
2017 Baltic Way
Determine whether the equation $$x^4+y^3=z!+7$$has an infinite number of solutions in positive integers.
2017 Baltic Way
Let $p>3$ be a prime and let $a_1,a_2,...,a_{\frac{p-1}{2}}$ be a permutation of $1,2,...,\frac{p-1}{2}$. For which $p$ is it always possible to determine the sequence $a_1,a_2,...,a_{\frac{p-1}{2}}$ if it for all $i,j\in\{1,2,...,\frac{p-1}{2}\}$ with $i\not=j$ the residue of $a_ia_j$ modulo $p$ is known?
2017 Baltic Way
For an integer $n\geq 1$ let $a(n)$ denote the total number of carries which arise when adding $2017$ and $n\cdot 2017$. The first few values are given by $a(1)=1$, $a(2)=1$, $a(3)=0$, which can be seen from the following:

\begin{align*}

001  &&001 && 000 \\
2017 &&4034 &&6051 \\
+2017 &&+2017 &&+2017\\
=4034 &&=6051 &&=8068\\
\end{align*}Prove that

$$a(1)+a(2)+...+a(10^{2017}-1)=10\cdot\frac{10^{2017}-1}{9}.$$
2017 Baltic Way
Let $S$ be the set of all ordered pairs $(a,b)$ of integers with $0<2a<2b<2017$ such that $a^2+b^2$ is a multiple of $2017$. Prove that $$\sum_{(a,b)\in S}a=\frac{1}{2}\sum_{(a,b)\in S}b.$$

Proposed by Uwe Leck, Germany
2016 Baltic Way
Find all pairs of primes $(p, q)$ such that $$p^3 - q^5 = (p + q)^2.$$
2016 Baltic Way
Prove or disprove the following hypotheses.

a) For all $k \geq  2,$ each sequence of $k$ consecutive positive integers contains a number that is not divisible by any prime number less than $k.$

b) For all $k\geq 2,$ each sequence of $k$ consecutive positive integers contains a number that is relatively prime to all other members of the sequence.
2016 Baltic Way
For which integers $n = 1, \ldots  , 6$ does the equation $$a^n + b^n = c^n + n$$have a solution in integers?
2016 Baltic Way
Let $n$ be a positive integer and let $a, b, c, d$ be integers such that $n | a + b + c + d$ and $n | a^2 + b^2 + c^2 + d^2. $

Show that $$n | a^4 + b^4 + c^4 + d^4 + 4abcd.$$
2016 Baltic Way
Let $p > 3$ be a prime such that $p\equiv  3 \pmod 4.$ Given a positive integer $a_0$ define the sequence $a_0, a_1, \ldots $ of integers by $a_n = a^{2^n}_{n-1}$ for all $n = 1, 2,\ldots.$ Prove that it is possible to choose $a_0$ such that the subsequence $a_N , a_{N+1}, a_{N+2}, \ldots $  is not constant modulo $p$ for any positive integer $N.$