test de selectie "seniori" Moldova 2014
Scris: Mar Feb 03, 2015 8:06 pm
1)Find all pairs of non-negative integers $(x,y)$ such that
$\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.$
2)Let $a,b\in\mathbb{R}_+$ such that $a+b=1$. Find the minimum value of the following expression:
$E(a,b)=3\sqrt{1+2a^2}+2\sqrt{40+9b^2}.$
3)Let $\triangle ABC$ be an acute triangle and AD the bisector of the angle $\angle BAC$ with $D\in(BC)$. Let E and F denote feet of perpendiculars from D to AB and AC respectively. If $BF\cap CE=K$ and $\odot AKE\cap BF=L$ prove that $DL\perp BF$.
4)Define $p(n)$to be th product of all non-zero digits of n. For instance $p(5)=5, p(27)=14, p(101)=1$ and so on. Find the greatest prime divisor of the following expression:
$p(1)+p(2)+p(3)+...+p(999)$.
$\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.$
2)Let $a,b\in\mathbb{R}_+$ such that $a+b=1$. Find the minimum value of the following expression:
$E(a,b)=3\sqrt{1+2a^2}+2\sqrt{40+9b^2}.$
3)Let $\triangle ABC$ be an acute triangle and AD the bisector of the angle $\angle BAC$ with $D\in(BC)$. Let E and F denote feet of perpendiculars from D to AB and AC respectively. If $BF\cap CE=K$ and $\odot AKE\cap BF=L$ prove that $DL\perp BF$.
4)Define $p(n)$to be th product of all non-zero digits of n. For instance $p(5)=5, p(27)=14, p(101)=1$ and so on. Find the greatest prime divisor of the following expression:
$p(1)+p(2)+p(3)+...+p(999)$.