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## $show=home 1. Let $$S=\frac{1}{2^{2}}+\frac{2}{2^{3}}+\ldots+\frac{n}{2^{n+1}}+\ldots+\frac{2006}{2^{2007}}.$$ Compare$S$with$1 .$2. Let$A H$be the altitude of a right triangle$A B C,$right angle at$A .$Let$P$and$Q$be the incenters of$A B H$and$A C H$respectively.$P Q$meets with$A B$at$E$and with$A C$at$F .$Prove that$A E=A F$. 3. Find all possible pairs of positive integers$(x ; y)$such that$4 x^{2}+6 x+3$is a multiple of$2 x y-1$. 4. Solve the following equation $$\sqrt{2 x^{2}+4 x+7}=x^{4}+4 x^{3}+3 x^{2}-2 x-7.$$ 5. Find the maximum value of the following expression $$A=\frac{x}{x^{2}+y z}+\frac{y}{y^{2}+2 x}+\frac{z}{z^{2}+x y},$$ where$x, y, z$are positive real numbers such that$x^{2}+y^{2}+z^{2}=x y z$. 6.$(O)$is a point chosen arbitrarily inside a triangle$A B C$.$A O$,$B O$and$C O$meet$B C$,$C A$and$A B$at$M$,$N$and$P$respectively. Prove that the value of ratio $$\left(\frac{O A \cdot A P}{O P}\right)\left(\frac{O B \cdot B M}{O M}\right)\left(\frac{O C \cdot C N}{O N}\right)$$ does not depend on the position of the point$O$. 7. Let$M$be a point inside a circle$(O)$with center at$O$and radius$R .$Draw two chords$C D$and$E F$through$M$but not passing through$O$. The tangent lines with$(O)$at$C$and$D$intersects at$A$, and the tangent lines at$E$and$F$meets at$B$. Prove that$O M$and$A B$are orthogonal. 8. Let$p>2007$be a prime number and$n$is an integer exceeding$2006 p .$Prove that$\mathrm{C}_{n}^{2006 \mathrm{p}}-\mathrm{C}_{k}^{2006}$is a multiple of$p,$where$k=\left[\dfrac{n}{p}\right]$is the largest integer which is not larger than$\dfrac{n}{p}$9. Let$\left(u_{n}\right)$be a sequence given by$u_{1}$and the formula$u_{n+1}=\dfrac{k+u_{n}}{1-u_{n}},$where$k>0$where$n=1,2, \ldots .$Given that$u_{13}=u_{1},$find the value of$k$. 10. Suppose$a, b,$and$c$are the three lengths of sides of a triangle. Prove that $$\frac{a}{\sqrt{a^{2}+3 b c}}+\frac{b}{\sqrt{b^{2}+3 c a}}+\frac{c}{\sqrt{c^{2}+3 a b}} \geq \frac{3}{2}$$ 11. Let$P$be a arbitrary point in a quadrilateral$A B C D$such that$\widehat{P A B}$,$\widehat{P B A}$,$\widehat{P B C}$,$\widehat{P C B}$,$\widehat{P C D}$,$\widehat{P D C}$,$\widehat{P A D}$and$\widehat{P D A}$are all acute angles. Let$M$,$N$,$K$,$L$denote the feet of the altitude from$P$on$A B$,$B C$,$C D$and$D A$respectively. Find the smallest value of the sum $$\frac{A B}{P M}+\frac{B C}{P N}+\frac{C D}{P K}+\frac{D A}{P L}$$ 12. Let$A B C . A^{\prime} B^{\prime} C^{\prime}$be a triangular prism.$M$and$N$are two points, chosen on$A A^{\prime}$and$C B$respectively such that $$\frac{A M}{A A^{\prime}}=\frac{C N}{C B} .$$ Prove that the length of$M N$is no less than the distance from$A^{\prime}$to the straight line$C^{\prime} B$. ##$type=three$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: 2007 Issue 362
2007 Issue 362
Mathematics & Youth