KURSCHAK 2012

Let $J_A$ and $J_B$ be the $A$ -excenter and $B$ -excenter of $\triangle ABC$ . Consider a chord $\overline{PQ}$ of circle $ABC$ which is parallel to $AB$ and intersects segments $\overline{AC}$ and $\overline{BC}$ . If lines $AB$ and $CP$ intersect at $R$ , prove that $$\angle J_AQJ_B+\angle J_ARJ_B=180^\circ.$$

Denote by $E(n)$ the number of $1$ 's in the binary representation of a positive integer $n$ . Call $n$ interesting if $E(n)$ divides $n$ . Prove that (a) there cannot be five consecutive interesting numbers, and (b) there are infinitely many positive integers $n$ such that $n$ , $n+1$ and $n+2$ are each interesting.

Consider $n$ events, each of which has probability $\frac12$ . We also know that the probability of any two both happening is $\frac14$ . Prove the following. (a) The probability that none of these events happen is at most $\frac1{n+1}$ . (b) We can reach equality in (a) for infinitely many $n$ .