CMO 2014

Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers whose product is 1 . Show that the sum $\frac{a_{1}}{1+a_{1}}+\frac{a_{2}}{\left(1+a_{1}\right)\left(1+a_{2}\right)}+\frac{a_{3}}{\left(1+a_{1}\right)\left(1+a_{2}\right)\left(1+a_{3}\right)}+\cdots+\frac{a_{n}}{\left(1+a_{1}\right)\left(1+a_{2}\right) \cdots\left(1+a_{n}\right)}$ is greater than or equal to $\frac{2^{n}-1}{2^{n}}$.

Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.

Let $p$ be a fixed odd prime. A $p$-tuple $\left(a_{1}, a_{2}, a_{3}, \ldots, a_{p}\right)$ of integers is said to be good if

(i) $0 \leq a_{i} \leq p-1$ for all $i$, and

(ii) $a_{1}+a_{2}+a_{3}+\cdots+a_{p}$ is not divisible by $p$, and

(iii) $a_{1} a_{2}+a_{2} a_{3}+a_{3} a_{4}+\cdots+a_{p} a_{1}$ is divisible by $p$.

Determine the number of good $p$-tuples.

The quadrilateral $A B C D$ is inscribed in a circle. The point $P$ lies in the interior of $A B C D$, and $\angle P A B=\angle P B C=\angle P C D=\angle P D A$. The lines $A D$ and $B C$ meet at $Q$, and the lines $A B$ and $C D$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $A B C D$.

Fix positive integers $n$ and $k \geq 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add 1 to all of them or subtract 1 from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.