MK-MO 2000

Let $AB$ be a diameter of a circle with centre $O$ , and $CD$ be a chord perpendicular to $AB$ . A chord $AE$ intersects $CO$ at $M$ , while $DE$ and $BC$ intersect at $N$ . Prove that $CM:CO=CN:CB$ .

If $a_1,a_2,a_3\ldots a_n$ are positive numbers, find the maximum value of $$\frac{a_1a_2\ldots a_{n-1}a_n}{(1+a_1)(a_1+a_2)\ldots (a_{n-1}+a_n)(a_n+2^{n+1})}$$

In a triangle with sides $a,b,c,t_a,t_b,t_c$ are the corresponding medians and $D$ the diameter of the circumcircle. Prove that $$\frac{a^2+b^2}{t_c}+\frac{b^2+c^2}{t_a}+\frac{c^2+a^2}{t_b}\le 6D$$

Let $a,b$ be coprime positive integers. Show that the number of positive integers $n$ for which the equation $ax+by=n$ has no positive integer solutions is equal to $\frac{(a-1)(b-1)}{2}-1$ .