BALTIC-WAY 2015

For $n \geq 2$, an equilateral triangle is divided into $n^{2}$ congruent smaller equilateral triangles. Determine all ways in which real numbers can be assigned to the $\frac{(n+1)(n+2)}{2}$ vertices so that three such numbers sum to zero whenever the three vertices form a triangle with edges parallel to the sides of the big triangle.

Let $n$ be a positive integer and let $a_{1}, \ldots, a_{n}$ be real numbers satisfying $0 \leq a_{i} \leq 1$ for $i=1, \ldots, n$. Prove the inequality

$$\left(1-a_{1}^{n}\right)\left(1-a_{2}^{n}\right) \cdots\left(1-a_{n}^{n}\right) \leq\left(1-a_{1} a_{2} \cdots a_{n}\right)^{n} .$$

Let $n>1$ be an integer. Find all non-constant real polynomials $P(x)$ satisfying, for any real $x$, the identity

$$P(x) P\left(x^{2}\right) P\left(x^{3}\right) \cdots P\left(x^{n}\right)=P\left(x^{\frac{n(n+1)}{2}}\right) .$$

A family wears clothes of three colours: red, blue and green, with a separate, identical laundry bin for each colour. At the beginning of the first week, all bins are empty. Each week, the family generates a total of $10 \mathrm{~kg}$ of laundry (the proportion of each colour is subject to variation). The laundry is sorted by colour and placed in the bins. Next, the heaviest bin (only one of them, if there are several that are heaviest) is emptied and its contents washed. What is the minimal possible storing capacity required of the laundry bins in order for them never to overflow?

Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfying the equation

$$|x| f(y)+y f(x)=f(x y)+f\left(x^{2}\right)+f(f(y))$$

for all real numbers $x$ and $y$.

Two players take alternate turns in the following game. At the outset there are two piles, containing 10,000 and 20,000 tokens, respectively. A move consists of removing any positive number of tokens from a single pile or removing $x>0$ tokens from one pile and $y>0$ tokens from the other, where $x+y$ is divisible by 2015. The player who cannot make a move loses. Which player has a winning strategy?

There are 100 members in a ladies' club. Each lady has had tea (in private) with exactly 56 of the other members of the club. The Board, consisting of the 50 most distinguished ladies, have all had tea with one another. Prove that the entire club may be split into two groups in such a way that, within each group, any lady has had tea with any other.

With inspiration drawn from the rectilinear network of streets in New York, the Manhattan distance between two points $(a, b)$ and $(c, d)$ in the plane is defined to be

$$|a-c|+|b-d| \text {. }$$

Suppose only two distinct Manhattan distances occur between all pairs of distinct points of some point set. What is the maximal number of points in such a set?

Let $n>2$ be an integer. A deck contains $\frac{n(n-1)}{2}$ cards, numbered

$$1,2,3, \ldots, \frac{n(n-1)}{2}$$

Two cards form a magic pair if their numbers are consecutive, or if their numbers are 1 and $\frac{n(n-1)}{2}$.

For which $n$ is it possible to distribute the cards into $n$ stacks in such a manner that, among the cards in any two stacks, there is exactly one magic pair?

A subset $S$ of $\{1,2, \ldots, n\}$ is called balanced if for every $a \in S$ there exists some $b \in S, b \neq a$, such that $\frac{a+b}{2} \in S$ as well.

(a) Let $k>1$ be an integer and let $n=2^{k}$. Show that every subset $S$ of $\{1,2, \ldots, n\}$ with $|S|>\frac{3 n}{4}$ is balanced.

(b) Does there exist an $n=2^{k}$, with $k>1$ an integer, for which every subset $S$ of $\{1,2, \ldots, n\}$ with $|S|>\frac{2 n}{3}$ is balanced?

Solution of part (a). Let $m=n-|S|$, thus $m<\frac{n}{4}$ and (as $n$ is a multiple of 4 ) $m \leq \frac{n}{4}-1$. Let $a \in S$. There are $\frac{n}{2}-1$ elements in $\{1,2, \ldots, n\}$ distinct from $a$ and with the same parity as $a$. At most $m$ of those elements are not in $S$, hence at least $\frac{n}{2}-1-m \geq \frac{n}{4}$ of them are in $S$. For each such $b$, the number $\frac{a+b}{2}$ is an integer, and all of these at least $\frac{n}{4}$ numbers are distinct. But at most $m<\frac{n}{4}$ of them are not in $S$, so at least one is a member of $S$. Hence $S$ is balanced.

The diagonals of the parallelogram $A B C D$ intersect at $E$. The bisectors of $\angle D A E$ and $\angle E B C$ intersect at $F$. Assume that $E C F D$ is a parallelogram. Determine the ratio $A B: A D$.

A circle passes through vertex $B$ of the triangle $A B C$, intersects its sides $A B$ and $B C$ at points $K$ and $L$, respectively, and touches the side $A C$ at its midpoint $M$. The point $N$ on the $\operatorname{arc} B L$ (which does not contain $K)$ is such that $\angle L K N=\angle A C B$. Find $\angle B A C$ given that the triangle $C K N$ is equilateral.

Let $D$ be the footpoint of the altitude from $B$ in the triangle $A B C$, where $A B=1$. The incentre of triangle $B C D$ coincides with the centroid of triangle $A B C$. Find the lengths of $A C$ and $B C$.

In the non-isosceles triangle $A B C$ the altitude from $A$ meets side $B C$ in $D$. Let $M$ be the midpoint of $B C$ and let $N$ be the reflection of $M$ in $D$. The circumcircle of the triangle $A M N$ intersects the side $A B$ in $P \neq A$ and the side $A C$ in $Q \neq A$. Prove that $A N, B Q$ and $C P$ are concurrent.

In triangle $A B C$, the interior and exterior angle bisectors of $\angle B A C$ intersect the line $B C$ in $D$ and $E$, respectively. Let $F$ be the second point of intersection of the line $A D$ with the circumcircle of the triangle $A B C$. Let $O$ be the circumcentre of the triangle $A B C$ and let $D^{\prime}$ be the reflection of $D$ in $O$. Prove that $\angle D^{\prime} F E=90^{\circ}$.

Denote by $P(n)$ the greatest prime divisor of $n$. Find all integers $n \geq 2$ for which

$$P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor .$$

(Note: $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.)

Find all positive integers $n$ for which $n^{n-1}-1$ is divisible by $2^{2015}$, but not by $2^{2016}$.

Let $f(x)=x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}$ be a polynomial of degree $n \geq 1$ with $n$ (not necessarily distinct) integer roots. Assume that there exist distinct primes $p_{0}, p_{1}, \ldots, p_{n-1}$ such that $a_{i}>1$ is a power of $p_{i}$, for all $i=0, \ldots, n-1$. Find all possible values of $n$.

Three pairwise distinct positive integers $a, b, c$, with $\operatorname{gcd}(a, b, c)=1$, satisfy

$$a\left|(b-c)^{2}, b\right|(c-a)^{2} \text { and } c \mid(a-b)^{2} \text {. }$$

Prove that there does not exist a non-degenerate triangle with side lengths $a, b, c$.

For any integer $n \geq 2$, we define $A_{n}$ to be the number of positive integers $m$ with the following property: the distance from $n$ to the nearest non-negative multiple of $m$ is equal to the distance from $n^{3}$ to the nearest non-negative multiple of $m$. Find all integers $n \geq 2$ for which $A_{n}$ is odd.

(Note: The distance between two integers $a$ and $b$ is defined as $|a-b|$.)