IMO 1970

$M$ is any point on the side $AB$ of the triangle $ABC$ . $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$ . $q$ is the radius of the circle on the opposite side of $AB$ to $C$ , touching the three sides of $AB$ and the extensions of $CA$ and $CB$ . Similarly, $q_1$ and $q_2$ . Prove that $r_1r_2q=rq_1q_2$ .

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$ .

a.) Prove that $0\le b_n<2$ .

b.) Given $c$ satisfying $0\le c<2$ , prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$ .

Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.

In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$ . Prove that: $$(AB+BC+CA)^2\le6(AD^2+BD^2+CD^2).$$ When do we have equality?