SMO 2002

In the plane, $\Gamma$ is a circle with centre $O$ and radius $r, P$ and $Q$ are distinct points on $\Gamma , A$ is a point outside $\Gamma , M$ and $N$ are the midpoints of $PQ$ and $AO$ respectively. Suppose $OA = 2a$ and $\angle PAQ$ is a right angle. Find the length of $MN$ in terms of $r$ and $a$ . Express your answer in its simplest form, and justify your answer.

Let $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$ be real numbers between $1001$ and $2002$ inclusive. Suppose $\sum_{i=1}^n a_i^2= \sum_{i=1}^n b_i^2$ . Prove that $$\sum_{i=1}^n\frac{a_i^3}{b_i} \le \frac{17}{10} \sum_{i=1}^n a_i^2$$ Determine when equality holds.

Let $n$ be a positive integer. Determine the smallest value of the sum $a_1b_1+a_2b_2+...+a_{2n+2}b_{2n+2}$ where $(a_1,a_2,...,a_{2n+2})$ and $(b_1,b_2,...,b_{2n+2})$ are rearrangements of the binomial coefficients $2n+1 \choose 0$ , $2n+1 \choose 1$ ,..., $2n+1 \choose 2n+1$ . Justify your answer

Find all real-valued functions $f : Q \to R$ defined on the set of all rational numbers $Q$ satisfying the conditions $f(x + y) = f(x) + f(y) + 2xy$ for all $x, y$ in $Q$ and $f(1) = 2002.$ Justify your answers.