BALTIC-WAY 2016

Find all pairs of primes $(p, q)$ such that

$$p^{3}-q^{5}=(p+q)^{2} .$$

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.

For which integers $n=1, \ldots, 6$ does the equation

$$a^{n}+b^{n}=c^{n}+n$$

have a solution in integers?

Let $n$ be a positive integer and let $a, b, c, d$ be integers such that $n \mid a+b+c+d$ and $n \mid a^{2}+b^{2}+c^{2}+d^{2}$. Show that

$$n \mid a^{4}+b^{4}+c^{4}+d^{4}+4 a b c d .$$

Let $p>3$ be a prime such that $p \equiv 3(\bmod 4)$. Given a positive integer $a_{0}$, define the sequence $a_{0}, a_{1}, \ldots$ of integers by $a_{n}=a_{n-1}^{2^{n}}$ 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 $N$.

The set $\{1,2, \ldots, 10\}$ is partitioned into three subsets $A, B$ and $C$. For each subset the sum of its elements, the product of its elements and the sum of the digits of all its elements are calculated. Is it possible that $A$ alone has the largest sum of elements, $B$ alone has the largest product of elements, and $C$ alone has the largest sum of digits?

Find all positive integers $n$ for which

$$3 x^{n}+n(x+2)-3 \geq n x^{2}$$

holds for all real numbers $x$.

Find all real numbers a for which there exists a non-constant function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following two equations for all $x \in \mathbb{R}$ : i) $f(a x)=a^{2} f(x)$ and

ii) $f(f(x))=a f(x)$.

Find all quadruples $(a, b, c, d)$ of real numbers that simultaneously satisfy the following equations:

$$\left\{\begin{aligned} a^{3}+c^{3} & =2 \\ a^{2} b+c^{2} d & =0 \\ b^{3}+d^{3} & =1 \\ a b^{2}+c d^{2} & =-6 \end{aligned}\right.$$

Let $a_{0,1}, a_{0,2}, \ldots, a_{0,2016}$ be positive real numbers. For $n \geq 0$ and $1 \leq k<2016$ set

$$a_{n+1, k}=a_{n, k}+\frac{1}{2 a_{n, k+1}} \text { and } a_{n+1,2016}=a_{n, 2016}+\frac{1}{2 a_{n, 1}} .$$

Show that $\max _{1 \leq k \leq 2016} a_{2016, k}>44$.

The set $A$ consists of 2016 positive integers. All prime divisors of these numbers are smaller than 30. Prove that there are four distinct numbers $a, b, c$ and $d$ in $A$ such that abcd is a perfect square.

Does there exist a hexagon (not necessarily convex) with side lengths 1, 2, 3, 4, 5, 6 (not necessarily in this order) that can be tiled with a) 31 b) 32 equilateral triangles with side length 1 ?

Let $n$ numbers all equal to 1 be written on a blackboard. A move consists of replacing two numbers on

the board with two copies of their sum. It happens that after $h$ moves all $n$ numbers on the blackboard are equal to $m$. Prove that $h \leq \frac{1}{2} n \log _{2} m$.

A cube consists of $4^{3}$ unit cubes each containing an integer. At each move, you choose a unit cube and increase by 1 all the integers in the neighbouring cubes having a face in common with the chosen cube. Is it possible to reach a position where all the $4^{3}$ integers are divisible by 3 , no matter what the starting position is?

The Baltic Sea has 2016 harbours. There are two-way ferry connections between some of them. It is impossible to make a sequence of direct voyages $C_{1}-C_{2}-\cdots-C_{1062}$ where all the harbours $C_{1}, \ldots, C_{1062}$ are distinct. Prove that there exist two disjoint sets $A$ and $B$ of 477 harbours each, such that there is no harbour in $A$ with a direct ferry connection to a harbour in $B$.

In triangle $A B C$, the points $D$ and $E$ are the intersections of the angular bisectors from $C$ and $B$ with the sides $A B$ and $A C$, respectively. Points $F$ and $G$ on the extensions of $A B$ and $A C$ beyond $B$ and $C$, respectively, satisfy $B F=C G=B C$. Prove that $F G \| D E$.

Let $A B C D$ be a convex quadrilateral with $A B=A D$. Let $T$ be a point on the diagonal $A C$ such that $\angle A B T+\angle A D T=\angle B C D$. Prove that $A T+A C \geq A B+A D$.

Let $A B C D$ be a parallelogram such that $\angle B A D=60^{\circ}$. Let $K$ and $L$ be the midpoints of $B C$ and $C D$, respectively. Assuming that $A B K L$ is a cyclic quadrilateral, find $\angle A B D$.

Consider triangles in the plane where each vertex has integer coordinates. Such a triangle can be legally transformed by moving one vertex parallel to the opposite side to a different point with integer coordinates. Show that if two triangles have the same area, then there exists a series of legal transformations that transforms one to the other.

Let $A B C D$ be a cyclic quadrilateral with $A B$ and $C D$ not parallel. Let $M$ be the midpoint of $C D$. Let $P$ be a point inside $A B C D$ such that $P A=P B=C M$. Prove that $A B, C D$ and the perpendicular bisector of $M P$ are concurrent.