TUYMAADA 2011

Each real number greater than $1$ is coloured red or blue with both colours being used. Prove that there exist real numbers $a$ and $b$ such that the numbers $a+b$ and $ab$ are of different colours.

An excircle of triangle $ABC$ touches the side $AB$ at $P$ and the extensions of sides $AC$ and $BC$ at $Q$ and $R$ , respectively. Prove that if the midpoint of $PQ$ lies on the circumcircle of $ABC$ , then the midpoint of $PR$ also lies on that circumcircle.

Each real number greater than 1 is colored red or blue with both colors being used. Prove that there exist real numbers $a$ and $b$ such that the numbers $a+\frac1b$ and $b+\frac1a$ are different colors.

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$ , and $M$ is the midpoint of $AB$ . Points $S_1$ and $S_2$ lie on the line $AB$ (but not between $A$ and $B$ ). The tangents drawn from $S_1$ to $\omega_1$ touch it at $X_1$ and $Y_1$ , and the tangents drawn from $S_2$ to $\omega_2$ touch it at $X_2$ and $Y_2$ . Prove that if the line $X_1X_2$ passes through $M$ , then line $Y_1Y_2$ also passes through $M$ .