[Solutions] Baltic Way Mathematical Competition 2016

1. Find all pairs of primes $(p, q)$ such that $$p^3 - q^5 = (p + q)^2.$$
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.
3. For which integers $n = 1, \ldots , 6$ does the equation $$a^n + b^n = c^n + n$$have a solution in integers?
4. 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 $$n | a^4 + b^4 + c^4 + d^4 + 4abcd.$$
5. 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 $N.$
6. The set $\{1, 2, . . . , 10\}$ is partitioned to 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?
7. Find all positive integers $n$ for which $$3x^n + n(x + 2) - 3 \geq nx^2$$holds for all real numbers $x.$
8. Find all real numbers $a$ for which there exists a non-constant function $f :\Bbb R \to \Bbb R$ satisfying the following two equations for all $x\in \Bbb R$:  $f(ax) = a^2f(x)$ and $f(f(x)) = a f(x).$
9. Find all quadruples $(a, b, c, d)$ of real numbers that simultaneously satisfy the following equations: $$\begin{cases} a^3 + c^3 = 2 \\ a^2b + c^2d = 0 \\ b^3 + d^3 = 1 \\ ab^2 + cd^2 = -6.\end{cases}$$
10. Let $a_{0,1}, a_{0,2}, . . . , a_{0, 2016}$ be positive real numbers. For $n\geq 0$ and $1 \leq k < 2016$ set $$a_{n+1,k} = a_{n,k} +\frac{1}{2a_{n,k+1}}, \ \ a_{n+1,2016} = a_{n,2016} +\frac{1}{2a_{n,1}}.$$Show that $\max_{1\leq k \leq 2016} a_{2016,k} > 44.$
11. 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.
12. 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 $1?$
13. 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 $h \leq \frac{1}{2}n \log_2 m.$
14. 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?
15. 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 $C_1 - C_2 - ... - C_{1062}$ where all the harbours $C_1, . . . , C_{1062}$ 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 $B.$
16. In triangle $ABC,$ the points $D$ and $E$ are the intersections of the angular bisectors from $C$ and $B$ with the sides $AB$ and $AC,$ respectively. Points $F$ and $G$ on the extensions of $AB$ and $AC$ beyond $B$ and $C,$ respectively, satisfy $BF = CG = BC.$ Prove that $F G \parallel DE.$
17. Let $ABCD$ be a convex quadrilateral with $AB = AD.$ Let $T$ be a point on the diagonal $AC$ such that $\angle ABT + \angle ADT = \angle BCD.$ Prove that $AT + AC \geq AB + AD.$
18. 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 $\angle ABD.$
19. 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.
20. Let $ABCD$ be a cyclic quadrilateral with $AB$ and $CD$ not parallel. Let $M$ be the midpoint of $CD.$ Let $P$ be a point inside $ABCD$ such that $P A = P B = CM.$ Prove that $AB, CD$ and the perpendicular bisector of $MP$ are concurrent.

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