VMO 2003

Let R be the reals and f: R → R a function such that f( cot x ) = cos 2x + sin 2x for all 0 < x < π. Define g(x) = f(x) f(1-x) for -1 ≤ x ≤ 1. Find the maximum and minimum values of g on the closed interval [-1, 1].

Let $S_n$ be the number of permutations ($a_1$, $a_2$, ... , $a_n$) of (1, 2, ... , n) such that 1 ≤ $|a_k$ - k | ≤ 2 for all k. Show that (7/4) $S_{n-1}$ < $S_n$ < 2 $S_{n-1}$ for n > 6.

Find the largest positive integer n such that the following equations have integer solutions in x, $y_1$, $y_2$, ... , $y_n$: (x + 1)$^2$ + $y_1^2$ = (x + 2)$^2$ + $y_2^2$ = ... = (x + n)$^2$ + $y_n^2$.

Define p(x) = $4x^3$ - $2x^2$ - 15x + 9, q(x) = $12x^3$ + $6x^2$ - 7x + 1. Show that each polynomial has just three distinct real roots. Let A be the largest root of p(x) and B the largest root of q(x). Show that $A^2$ + 3 $B^2$ = 4.

Let $R^+$ be the set of positive reals and let F be the set of all functions f : $R^+$ → $R^+$ such that f(3x) ≥ f( f(2x) ) + x for all x. Find the largest A such that f(x) ≥ A x for all f in F and all x in $R^+$.