Bosnia and Herzegovina TST 2012
Day 1
$\boxed{1}$ Let $D$ be the midpoint of the arc $ B-A-C $ of the circumcircle of $ \triangle ABC (AB<AC) $. Let $E$ be the foot of perpendicular from $D$ to $AC$. Prove that $ |CE|=\frac{|BA|+|AC|}{2} $.
$\boxed{2}$ Prove for all positive real numbers $a,b,c$, such that $a^2+b^2+c^2=1$:
\[\frac{a^3}{b^2+c}+\frac{b^3}{c^2+a}+\frac{c^3}{a^2+b}\ge \frac{\sqrt{3}}{1+\sqrt{3}}.\]
$\boxed{3}$ Prove that for all odd prime numbers $p$ there exist a natural number $m<p$ and integers $x_1, x_2, x_3$ such that:
\[mp=x_1^2+x_2^2+x_3^2.\]
AoPS
Bài viết đã được chỉnh sửa nội dung bởi nsthanh: 22-05-2012 - 07:32