Moldova Team Selection Test 2014

Find all pairs of non-negative integers $(x,y)$ such that \[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\] Let $a,b\in\mathbb{R}_+$ such that $a+b=1$....

  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 \[p(1)+p(2)+p(3)+...+p(999).\]
  5. Consider $n \geq 2 $ positive numbers $0<x_1 \leq x_2 \leq ... \leq x_n$, such that $x_1 + x_2 + ... + x_n = 1$. Prove that if $x_n \leq \dfrac{2}{3}$, then there exists a positive integer $1 \leq k \leq n$ such that $$\dfrac{1}{3} \leq x_1+x_2+...+x_k < \dfrac{2}{3}$$
  6. Let $a,b,c$ be positive real numbers such that $abc=1$. Determine the minimum value of $$E(a,b,c) = \sum \dfrac{a^3+5}{a^3(b+c)}$$
  7. Let $ABCD$ be a cyclic quadrilateral. The bisectors of angles $BAD$ and $BCD$ intersect in point $K$ such that $K \in BD$. Let $M$ be the midpoint of $BD$. A line passing through point $C$ and parallel to $AD$ intersects $AM$ in point $P$. Prove that triangle $\triangle DPC$ is isosceles.
  8. Consider $n \geq 2$ distinct points in the plane $A_1,A_2,...,A_n$ . Color the midpoints of the segments determined by each pair of points in red. What is the minimum number of distinct red points?
  9. Prove that there do not exist $4$ points in the plane such that the distances between any pair of them is an odd integer.
  10. Find all functions $f:R \rightarrow R$, which satisfy the equality for any $x,y \in R$ $$f(xf(y)+y)+f(xy+x)=f(x+y)+2xy$$
  11. Let $\triangle ABC$ be a triangle with $\angle A$-acute. Let $P$ be a point inside $\triangle ABC$ such that $\angle BAP = \angle ACP$ and $\angle CAP =\angle ABP$. Let $M, N$ be the centers of the incircle of $\triangle ABP$ and $\triangle ACP$, and $R$ the radius of the circumscribed circle of $\triangle AMN$. Prove that $$\displaystyle \frac{1}{R}=\frac{1}{AB}+\frac{1}{AC}+\frac{1}{AP}. $$
  12. On a circle $n \geq 1$ real numbers are written, their sum is $n-1$. Prove that one can denote these numbers as $x_1, x_2, ..., x_n$ consecutively, starting from a number and moving clockwise, such that for any $k$ ($1\leq k \leq n$) $$ x_1 + x_2+...+x_k \geq k-1$$



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MATHEMATICAL OLYMPIAD PROBLEMS: Moldova Team Selection Test 2014
Moldova Team Selection Test 2014
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