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

## $show=home 1. Prove the inequality $$\frac{3}{2}+\frac{7}{4}+\frac{11}{8}+\frac{15}{16}+\ldots+\frac{4 n-1}{2^{n}}<7$$ where$n$is an arbitrary positive integer. 2. Let$A B C$be a right triangle with right angle at$A$. Suppose$A B=5cm$and$I C=6cm$where$I$is the incenter of$A B C$. Determine the length of$B C$. 3. Find all positive integers$x, y, z$such that the following equality holds $$5 x y z=x+5 y+7 z+10.$$ 4. Solve for$x$$$x^{4}-2 x^{2}-16 x+1=0.$$ 5. Let$A B C$be an acute triangle whose altitudes$A A^{\prime}$,$B B^{\prime}$,$C C^{\prime}$meet at$H$Denote by$A_{1}$,$B_{1},$and$C_{1}$the othocenters of the triangles$A B^{\prime} C^{\prime}$,$B C^{\prime} A^{\prime}$and$C A^{\prime} B^{\prime}$respectively. Suppose that$H$is the incenter of the triangle$A_{1} B_{1} C_{1}$. Prove that$A B C$is an equilateral triangle. 6. Let$A B C$be an isosceles triangle with$A B=B C=a$and$\widehat{A B C}=140^{\circ} .$Let$A N$and$A H$respectively be the anglebisector and the altitude from$A$. Prove that$2 B H \cdot C N=a^{2}$7. Find the minimum value of the function $$f(x)=\left(32 x^{5}-40 x^{3}+10 x-1\right)^{2006}+\left(16 x^{3}-12 x+\sqrt{5}-1\right)^{2008}.$$ 8. Prove that the following equation $$A \cdot a^{x}+B \cdot b^{x}=A+B$$ where$a>1$,$0<b<1$,$A, B \in \mathbb{R}$has at most two solutions. 9. Let$a_{1}=\dfrac{1}{2}$and for each$n$greater than$1,$let $$a_{n}=\frac{1}{d_{1}+1}+\frac{1}{d_{2}+1}+\ldots+\frac{1}{d_{k}+1}$$ where$d_{1}, d_{2}, \ldots, d_{k}$is the collection of all distinct positive divisors of$n .$Prove the inequality $$n-\ln n<a_{1}+a_{2}+\ldots+a_{n}<n.$$ 10. Given$n$numbers$a_{1}, a_{2}, \ldots, a_{n}$in$[-1 ; 2]$such that their total sum is 0. Let$U_{k}=\dfrac{a_{k} \sqrt{4 k-1}}{(4 k-3)(4 k+1)}$for$k=1,2, \ldots, n .$Prove that $$\left|U_{1}+U_{2}+\ldots+U_{n}\right|<\frac{\sqrt{n}}{2}$$ 11. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$such that $$f(x+y)=x^{2} f\left(\frac{1}{x}\right)+y^{2} f\left(\frac{1}{y}\right),\,\forall x, y \in \mathbb{R}^{*}.$$ 12. Let$P$be a point on the insphere of a tetrahedron$A B C D$and let$G_{a}$,$G_{b}$,$G_{c}$,$G_{d}$be the centroids of the tetrahedra$P B C D$,$PCDA$,$PDAB$,$PABC$, respectively. Prove that$A G_{a}$,$B G_{b}$,$C G_{c}$and$D G_{d}$pass through a common point; and find the orbit of this common point when$P$moves on the insphere of the given tetrahedron$A B C D$. ##$type=three$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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2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,4,Anniversary,4,
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Mathematics & Youth: 2008 Issue 370
2008 Issue 370
Mathematics & Youth