CZ-SK-MO 2014

Let be $n$ a positive integer. Denote all its (positive) divisors as $1=d_1<d_2<\cdots<d_{k-1}<d_k=n$ . Find all values of $n$ satisfying $d_5-d_3=50$ and $11d_5+8d_7=3n$ .

(Day 1, 1st problem author: Matúš Harminc)

A segment $AB$ is given in (Euclidean) plane. Consider all triangles $XYZ$ such, that $X$ is an inner point of $AB$ , triangles $XBY$ and $XZA$ are similar (in this order of vertices), and points $A, B, Y, Z$ lie on a circle in this order. Find the locus of midpoints of all such segments $YZ$ .

(Day 1, 2nd problem authors: Michal Rolínek, Jaroslav Švrček)

Suppose we have a $8\times8$ chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into $1\times2$ domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers.

(Day 1, 3rd problem author: Michal Rolínek)

$234$ viewers came to the cinema. Determine for which $n \ge 4$ the viewers could be can be arranged in $n$ rows so that every viewer in $i$ -th row gets to know just $j$ viewers in $j$ -th row for any $i, j \in \{1, 2,... , n\}, i\ne j$ . (The relationship of acquaintance is mutual.)

(Tomáš Jurík)

Given is the acute triangle $ABC$ . Let us denote $k$ a circle with diameter $AB$ . Another circle, tangent to $AB$ at point $A$ and passing through point $C$ intersects the circle $k$ at point $P, P \ne A$ . Another circle which touches AB at point $B$ and passes point $C$ , intersects the circle $k$ at point $Q, Q \ne B$ . Prove that the intersection of the line $AQ$ and $BP$ lies on one of the sides of angle $ACB$ .

(Peter Novotný)

For arbitrary non-negative numbers $a$ and $b$ prove inequality $\frac{a}{\sqrt{b^2+1}}+\frac{b}{\sqrt{a^2+1}}\ge\frac{a+b}{\sqrt{ab+1}}$ , and find, where equality occurs.

(Day 2, 6th problem authors: Tomáš Jurík, Jaromír Šimša)