# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 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 $$\begin{cases}\sqrt{2 x}+2 \sqrt{6-x}-y^{2} &=2 \sqrt{2} \\ \sqrt{2 x}+2 \sqrt{6-x}+2 \sqrt{2} y &=8+\sqrt{2}\end{cases}$$ 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? ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
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Mathematics & Youth: 2009 Issue 387
2009 Issue 387
Mathematics & Youth