VMO 2001

A circle center O meets a circle center O' at A and B. The line TT' touches the first circle at T and the second at T'. The perpendiculars from T and T' meet the line OO' at S and S'. The ray AS meets the first circle again at R, and the ray AS' meets the second circle again at R'. Show that R, B and R' are collinear.

Let N = $6^n$, where n is a positive integer, and let M = $a^N$ + $b^N$, where a and b are relatively prime integers greater than 1. M has at least two odd divisors greater than 1. Find the residue of M mod 6 $12^n$.

For real a, b define the sequence $x_0$, $x_1$, $x_2$, ... by $x_0$ = a, $x_{n+1}$ = $x_n$ + b sin $x_n$. If b = 1, show that the sequence converges to a finite limit for all a. If b > 2, show that the sequence diverges for some a.

Find the maximum value of 1/√x + 2/√y + 3/√z, where x, y, z are positive reals satisfying 1/√2 ≤ z ≤ min(x√2, y√3), x + z√3 ≥ √6, y√3 + z√10 ≥= 2√5.

Find all real-valued continuous functions defined on the interval (-1, 1) such that (1 - $x^2$) f(2x/(1 + $x^2$) ) = (1 + $x^2$)$^2$ f(x) for all x.

$a_1$, $a_2$, ... , $a_{2n}$ is a permutation of 1, 2, ... , 2n such that $|a_i$ - $a_{i+1}$| ≠ $|a_j$ - $a_{j+1}$| for i ≠ j. Show that $a_1$ = $a_{2n}$ + n iff 1 ≤ $a_{2i}$ ≤ n for i = 1, 2, ... n.