Toán học, Olympic toán toàn quốc - Việt nam 2001

A1. 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.

A2. Let N = 6n, 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 $12^{n}$ .

A3. 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.

B1. Find the maximum value of $1/{\sqrt {x}}+2/{\sqrt {y}}+3/{\sqrt {z}}$ where x, y, z are positive reals satisfying $1/{\sqrt {2}}\leq z\leq min(x{\sqrt {2}},y{\sqrt {3}}),X+z{\sqrt {3}}\geq {\sqrt {6}},y{\sqrt {3}}+z{\sqrt {10}}\geq 2{\sqrt {5}}$ .

B2. 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.

B3. $a_{1},a_{2},...,a_{{2n}}$ is a permutation of 1, 2, ... , 2n such that $|a_{i}-a_{{i+1}}|\neq |a_{j}-a_{{j+1}}|$ for i $\neq$ j. Show that $a_{1}=a_{2}n+n$ iff 1 $\leq$ $a_{{2i}}$ $\leq$ n for i = 1, 2, ... n.