VMO 1968 — Division A

Let $a,b$ satisfy $a\ge b\gt 0$ and $a+b=1$.

(a) Prove that if $m,n$ are positive integers satisfying $m\gt n$, then $a^m-a^n\ge b^m-b^n\gt 0$.

(b) For every positive integer $n$, form the quadratic
$$f_n(x)=x^2-b^nx-a^n.$$
Prove that $f_n(x)$ has two distinct roots lying between $-1$ and $1$.

Editorial notice: The original assertions are retained verbatim in mathematical content. They contain defects; see the solution discussion.

A fixed circle has center $O$ and radius $r$. A variable triangle $ABC$ always circumscribes this circle, with vertex $A$ on a fixed line $x$ and the other two vertices on a fixed line $y$ parallel to $x$.

(a) Given the circle $O$, the lines $x,y$, and the angle $A$, construct triangle $ABC$. Discuss.

(b) Calculate the angles of triangle $ABC$ in terms of angle $A$, radius $r$, and the distance $h$ between $x,y$. Discuss.

(c) What relationship do you notice between $IB$ and $IC$ ($I$ is the tangency point of circle $O$ with line $y$)? Is there a way to guess this relationship without proving anything yet? Prove the conjecture.

From the relationship between $IB$ and $IC$, try to discover other noteworthy relationships (the more you discover, the better).