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2012 Issue 420

  1. Find the integer value of the expression $f(x ; y)=\dfrac{x^{2}+x+2}{x y-1}$ where $x, y$ are positive integers.
  2. Let $A B C$ be an acute triangle which is not isosceles at $A$. The perpendicular bisectors of $A B$, $A C$ cut the median $A M$ at $E$, $F$ respectively. $B E$ and $C F$ meet at $K .$ Prove that $\widehat{A K B}=\widehat{A K C}$ and $\widehat{M A B}=\widehat{K A C}$.
  3. Find all triples of integers $(x ; y ; z)$ such that $$2 x y+6 y z+3 z x-|x-2 y-z|=x^{2}+4 y^{2}+9 z^{2}-1.$$
  4. For each positive integer $n(n=1,2, \ldots),$ put $a_{n}=\dfrac{4 n}{n^{4}+4} .$ Prove that $$a_{1}+a_{2}+\ldots+a_{n}<\frac{3}{2}$$
  5. Let $A B C$ be an acute triangle. The internal angle-bisector of angle $B A C$ cuts $B C$ at $D$. $E$, $F$ are the orthogonal projections of point $D$ on $A B$ and $A C$ respectively, $K$ is the intersection of $C E$ and $B F, H$ is the intersection of $B F$ with the circumcircle of triangle $A E K$. Prove that $D H$ is perpendicular to $B F$
  6. Solve the system of equations $$\begin{cases} x+6 \sqrt{x y}-y &=6 \\ x+\dfrac{6\left(x^{3}+y^{3}\right)}{x^{2}+x y+y^{2}}-\sqrt{2\left(x^{2}+y^{2}\right)} &=3 \end{cases}.$$
  7. Let $a, b, c$ be non-negative real numbers whose sum equals $1$. Prove that $$\left(1+a^{2}\right)\left(1+b^{2}\right)\left(1+c^{2}\right) \geq\left(\frac{10}{9}\right)^{3}$$
  8. Point $M$ inside the triangle $A B C$ with area $S$. Let $x, y, z$ be distances of $M$ to $A$, $B$, $C$ respectively. Prove that $$(x+y+z)^{2} \geq 4 \sqrt{3} S.$$ When does the equality hold?
  9. A nonempty set $S \subseteq \mathbb{Z}$ posesses the following properties
    • There exist $a, b \in S$ such that $(a, b)=(a-2 b-2)=1$,
    • If $x, y \in S$ then $x^{2}-y \in S$ ($x$, $y$ may be identical).
      Prove that $S=\mathbb{Z}$. ($(a, b)$ is the greatest common divisor of two integers $a$ and $b$.)
    1. Find the greatest number $k$ such that the inequality $$\sqrt{a+2 b+3 c}+\sqrt{b+2 c+3 a}+\sqrt{c+2 a+3 b} \geq k(\sqrt{a}+\sqrt{b}+\sqrt{c})$$ holds for all positive numbers $a, b, c$
    2. Let $\left(x_{n}\right)$ be a sequence defined by $$x_{1}=\frac{1001}{1003} ,\quad x_{n+1}=x_{n}-x_{n}^{2}+x_{n}^{3}-x_{n}^{4}+\ldots+x_{n}^{2011}-x_{n}^{2012},\, \forall n \in \mathbb{N}.$$ Find $\displaystyle \lim _{n \rightarrow+\infty}\left(n x_{n}\right)$.
    3. Given four distinct points $A$, $B$, $C$, $D$ lying on a circle with center $O$. Let $I$, $J$ be the feet of the perpendicular to $A B$ and $A D$ through $C$; $K$, $L$ are the feet of the perpendicular to $B C$ and $B A$ through $D$; $N$ is the midpoint of $C D$; $M$ is the intersection of $I J$ and $K L$. $I J$ meets $O D$ at $E$ and $K L$ meets $O C$ at $F$. Prove that the five points $M$, $N$, $O$, $E$ and $F$ lie on the same circle.

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    Mathematics & Youth: 2012 Issue 420
    2012 Issue 420
    Mathematics & Youth
    https://www.molympiad.org/2020/09/2012-issue-420.html
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