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2009 Issue 387

  1. Prove that there exists a $2009$-digits multiple of $2007$ so that it is formed by four digits $0,2,7$ and $9$ and the sum of its digits is $7209$.
  2. Let $a_{1}, a_{2}, \ldots, a_{n}$ be $n$ odd integers $(n>2007)$ satisfying the following condition $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{2005}^{2}=a_{2006}^{2}+a_{2007}^{2}+\ldots+a_{n}^{2}.$$ Determine the least possible value of $n$ and construct an example of such a collection $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ for the smallest value found above.
  3. Determine the value of $S$, given that $$S=\frac{2^{3}-1}{2^{3}+1} \times \frac{3^{3}-1}{3^{3}+1} \times \ldots \times \frac{2009^{3}-1}{2009^{3}+1}$$
  4. Let $A B C$ be a right triangle with right angle at $A$ and $A C>A B$. Choose a poind $D$ on $A C$ such that $A B=A D .$ Let $E$ be the foot of the altitude from $D$ onto $B C .$ The line passing through $A$ and parallel to $B C$ meets $D E$ at $H .$ Prove that $$A H<\frac{\sqrt{2}}{2} A C.$$
  5. Let $A B$ be a fixed chord of a given circle $(O)$ and $E$ is a point moving on $A B$ (but distinct from $A$ and $B$). From $E$, draw another chord $C D$. $P$ and $Q$ are two points on the rays $D A$ and $D B$ respectively such that $P$ is the reflection of $Q$ through $E$. Prove that the circle $(I)$ passing through $C$ and touches $P Q$ at $E$ always passes through a fixed point.
  6. Of all pentagons whose sum of the squares of its diagonals equals $1,$ which one has the smallest possible sum of cube of its sides.
  7. Solve the system of equations $$\begin{cases}\sqrt{2 x}+2 \sqrt[4]{6-x}-y^{2} &=2 \sqrt{2} \\ \sqrt[4]{2 x}+2 \sqrt{6-x}+2 \sqrt{2} y &=8+\sqrt{2}\end{cases}$$
  8. Prove that $$\frac{\sin A}{\tan \frac{B}{2}}+\frac{\sin B}{\tan \frac{C}{2}}+\frac{\sin C}{\tan \frac{A}{2}} \geq \frac{9}{2}$$ where $A$, $B$, $C$ are the measures of the angles of a triangle. When does equality occur?.
  9. In a triangle $A B C$, let $P$ be a point such that $P A=P B+P C$. $R$ is the midpoint of the chord $A B$ (the one that contains $P$) of the circumcircle of the triangle $A B P$ and $S$ is the midpoint of the chord $A C$ (containing $P$) of the circumcircle of the triangle $A C P$. Prove that circumcircles of the triangles $B P S$ and $CPR$ touch each other.
  10. Does there exist a function $\mathbb{R} \rightarrow \mathbb{R}$ such that
    • $f$ is continuous on $\mathbb{R}$; and
    • $f(x+2008)(f(x)+\sqrt{2009})=-2010, \forall x \in \mathbb{R} . ?$
  11. Let $\left(u_{n}\right)(n=1,2, \ldots)$ be a sequence given by $$u_{1}=2,1;\quad  u_{n+1}=\frac{-2 u_{n}^{2}+5 u_{n}}{-u_{n}^{2}+2 u_{n}+1},\,\forall n=1,2, \ldots$$ Find $\displaystyle\lim_{n\to\infty} \left(u_{1}+u_{2}+\ldots+u_{n}\right)$.
  12. $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$ are two triangles in a given plane whose inradii are $r, r^{\prime}$ respectively. Let $R'$ be the radius of the circumcircle of the triangle $A^{\prime} B^{\prime} C^{\prime} .$ Prove that $$\left(\sin \frac{B^{\prime}}{2}+\sin \frac{C^{\prime}}{2}\right) P A +\left(\sin \frac{C^{\prime}}{2}+\sin \frac{A^{\prime}}{2}\right) P B +\left(\sin \frac{A^{\prime}}{2}+\sin \frac{B^{\prime}}{2}\right) P C \geq \frac{6 r r^{\prime}}{R^{\prime}}$$ for any point $P$ in the same plane. When does equalitiy occur?

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Mathematics & Youth: 2009 Issue 387
2009 Issue 387
Mathematics & Youth
https://www.molympiad.org/2020/09/2009-issue-387.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2009-issue-387.html
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