CMO 1997

Determine the number of pairs of positive integers $x,y$ such that $x\le y$ , $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$ .

The closed interval $A = [0, 50]$ is the union of a finite number of closed intervals, each of length $1$ . Prove that some of the intervals can be removed so that those remaining are mutually disjoint and have total length greater than $25$ . Note: For reals $a\le b$ , the closed interval $[a, b] := \{x\in \mathbb{R}:a\le x\le b\}$ has length $b-a$ ; disjoint intervals have emptyintersection.

Prove that $\frac{1}{1999}< \prod_{i=1}^{999}{\frac{2i-1}{2i}}<\frac{1}{44}$ .

The point $O$ is situated inside the parallelogram $ABCD$ such that $\angle AOB+\angle COD=180^{\circ}$ . Prove that $\angle OBC=\angle ODC$ .

Write the sum $\sum_{i=0}^{n}{\frac{(-1)^i\cdot\binom{n}{i}}{i^3 +9i^2 +26i +24}}$ as the ratio of two explicitly defined polynomials with integer coefficients.