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

## $show=home 1. Let$a_{1}, a_{2}, \ldots, a_{2010}$be natural numbers such that $$\frac{1}{a_{1}^{11}}+\frac{1}{a_{2}^{11}}+\frac{1}{a_{3}^{11}}+\ldots+\frac{1}{a_{2010}^{11}}=\frac{1005}{1024}.$$ Determine the value of the following expression $$A=\frac{a_{2010}^{6}}{a_{1}^{5}}+\frac{a_{2009}^{6}}{a_{2}^{5}}+\frac{a_{2008}^{6}}{a_{3}^{5}}+\ldots+\frac{a_{1}^{6}}{a_{2010}^{5}}$$ 2. In a triangle$A B C$where$B$and$C$are acute angles, let$B D$and$A H$be respectively the angle bisector and the altitude. Given that$\widehat{A D B}=\widehat{A H D}=\alpha,$find the measure of$\alpha$. 3. Find all integer solutions of the equation $$y^{3}=x^{6}+2 x^{4}-1000$$ 4. Find the minimum value of the expression $$P=\frac{a}{b+c+d-a}+\frac{b}{c+d+a-b}+\frac{c}{d+a+b-c}+\frac{d}{a+b+c-d}$$ where$a, b, c, d$are the length of 4 sides of a convex quadrilateral. 5. Let$A B C$be an isosceles triangle with vertex at$C$. Let$O$,$I$be its circumcenter and incenter respctively.$D$is a point chosen on the side$B C$so that$D O$is perpendicular to$B I$. Prove that$D I$is parallel to$A C$. 6. Let$a, b, c$be positive real numbers satisfying the condition $$\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1} \geq 1.$$ Prove that $$a+b+c=a b+b c+c a$$ 7. Let$A B C D E$be a cyclic pentagon with$A C \parallel D E$and$\widehat{A M B}=\widehat{B M C}$where$M$is the midpoint of$B D$. Prove that the line$B E$passes through the midpoint of$A C$. 8. Let$O A B C$be a tetrahedron with right trihedral angle at vertex$O$. Prove that $$\cot A B \cdot \cot B C+\cot B C \cdot \cot C A+\cot C A \cdot \cot A B \leq \frac{3}{2}$$ where cot$A B$is the cotangent of the dihedral angle of side$A B$. 9. Let$A B C$be an acute triangle. The altitudes$B K$and$C L$meet at$H$. The line passing through$H$meets$A B$,$A C$at$P$,$Q$respectively. Prove that$H P=H Q$if and only if$M P=M Q$where$M$is the midpoint of$B C$10. Find all real numbers$k$and$m$such that $$k\left(x^{3}+y^{3}+z^{3}\right)+m x y z \geq(x+y+z)^{3}$$ for any non-negative numbers$x, y, z$11. Let$\left(x_{n}\right)$be a sequence of real numbers,$n=1,2, \ldots$satisfying $$\ln \left(1+x_{n}^{2}\right)+n x_{n}=1$$ for any positive integers$n$. Find $$\lim _{n \rightarrow+\infty} \frac{n\left(1+n x_{n}\right)}{x_{n}}$$ 12. Find all continuous functions$f: \mathbb{R} \rightarrow \mathbb{R}$such that $$f(x+f(y))=2 y+f(x),\, \forall x, y \in \mathbb{R}$$ ##$type=three$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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Mathematics & Youth: 2010 Issue 393
2010 Issue 393
Mathematics & Youth