KURSCHAK 2015

In fencing, you win a round if you are the first to reach $15$ points. Suppose that when $A$ plays against $B$ , at any point during the round, $A$ scores the next point with probability $p$ and $B$ scores the next point with probability $q=1-p$ . (However, they never can both score a point at the same time.)

Suppose that in this round, $A$ already has $14-k$ points, and $B$ has $14-\ell$ (where $0\le k,\ell\le 14$ ). By how much will the probability that $A$ wins the round increase if $A$ scores the next point?

Consider a triangle $ABC$ and a point $D$ on its side $\overline{AB}$ . Let $I$ be a point inside $\triangle ABC$ on the angle bisector of $ACB$ . The second intersections of lines $AI$ and $CI$ with circle $ACD$ are $P$ and $Q$ , respectively. Similarly, the second intersection of lines $BI$ and $CI$ with circle $BCD$ are $R$ and $S$ , respectively. Show that if $P\neq Q$ and $R\neq S$ , then lines $AB$ , $PQ$ and $RS$ pass through a point or are parallel.

Let $Q=\{0,1\}^n$ , and let $A$ be a subset of $Q$ with $2^{n-1}$ elements. Prove that there are at least $2^{n-1}$ pairs $(a,b)\in A\times (Q\setminus A)$ for which sequences $a$ and $b$ differ in only one term.