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2009 Issue 385

  1. Let $S(n)$ be denote the sum of the digits of $n$. Find a positive integer $n$ such that $S(n)=n^{2}-2009 n+11$.
  2. Inside a square $A B C D$ thoose two points $P$, $Q$ such that $B P$ and $D Q$ are parallel and $B P^{2}+D O^{2}=P Q^{2}$. Find the measure of the angle $P A Q$.
  3. Compare $2008$ with the sum $S$ of $2009$ terms $$S=\frac{2008+2007}{2009+2008}+\frac{2008^{2}+2007^{2}}{2009^{2}+2008^{2}}+\ldots +\frac{2008^{2009}+2007^{2009}}{2009^{2009}+2008^{2009}}.$$
  4. Let $a$, $b$, $c$ be three positive numbers such that $a+b+c=1$. Find the least value of the expression $$P=\frac{9}{1-2(a b+b c+c a)}+\frac{2}{a b c}.$$
  5. Let $B$, $C$ be two fixed points on the circle $(\omega)$ such that $B C$ does not pass through the center of $(\omega) .$ On the major arc $B C$, choose a point $A$ differs from $B$ and $C$. Another point $M$ moves on the line segment $B C$. The lines passing through $M$ and parallel to $A B$, $A C$ intersect $A C$ and $A B$ at $F$ and $E$, respectively. When $M$ moves on the line segment $B C$ and for each point $A$, let $x$ be the least possible length of $EF$. Find the positions of $A$ and $M$ such that $x$ is greatest possible.
  6. Find the greatest and the least value of the expression $$A=\frac{y^{2}}{25}+\frac{t^{2}}{144}$$ where $x$, $y$, $z$, $t$ satisfy the system of equations $$\begin{cases}x^{2}+y^{2}+2 x+4 y-20 &=0 \\ t^{2}+z^{2}-2 t-143 &= 0\\ x t+y z-x+t+2 z-61 &\geq 0\end{cases}$$
  7. Consider the sequence $\left(u_{n}\right)$ defined as follow $$u_{0}=9,\, u_{1}=161,\quad u_{n}=18 u_{n-1}-u_{n-2},\,\forall n=2,3, \ldots$$ Prove that for any $n$, $\dfrac{u_{n}^{2}-1}{5}$ is always a perfect square.
  8. Let $P$ be an arbitrary point inside a given triangle $A B C$. Let $A'$, $B'$, $C'$ be the orthogonal projection of $P$ on $B C$, $C A$, $A B$ respectively. Let $I$ be the incenter and $r$ be the inradius of the triangle $A B C$. Find the least value of the expression $$P A^{\prime}+P B^{\prime}+P C^{\prime}+\frac{P I^{2}}{2 r}$$

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Mathematics & Youth: 2009 Issue 385
2009 Issue 385
Mathematics & Youth
https://www.molympiad.org/2020/09/2009-issue-385.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2009-issue-385.html
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