CZ-SK-MO 2013

Find all pairs of integers $a, b$ for which equality holds $\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}$

Each of the thieves in the $n$ -member party ( $n \ge 3$ ) charged a certain number of coins. All the coins were $100n$ . Thieves decided to share their prey as follows: at each step, one of the bandits puts one coin to the other two. Find them all natural numbers $n \ge 3$ for which after a finite number of steps each outlaw can have $100$ coins no matter how many coins each thug has charged.

In the parallelolgram A $BCD$ with the center $S$ , let $O$ be the center of the circle of the inscribed triangle $ABD$ and let $T$ be the touch point with the diagonal $BD$ . Prove that the lines $OS$ and $CT$ are parallel.

On the board is written in decimal the integer positive number $N$ . If it is not a single digit number, wipe its last digit $c$ and replace the number $m$ that remains on the board with a number $m -3c$ . (For example, if $N = 1,204$ on the board, $120 - 3 \cdot 4 = 108$ .) Find all the natural numbers $N$ , by repeating the adjustment described eventually we get the number $0$ .

Given the parallelogram $ABCD$ such that the feet $K, L$ of the perpendiculars from point $D$ on the sides $AB, BC$ respectively are internal points. Prove that $KL \parallel AC$ when $|\angle BCA| + |\angle ABD| = |\angle BDA| + |\angle ACD|$ .

Find all positive real numbers $p$ such that $\sqrt{a^2 + pb^2} +\sqrt{b^2 + pa^2} \ge a + b + (p - 1) \sqrt{ab}$ holds for any pair of positive real numbers $a, b$ .