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

## $show=home 1. Determine the sum of$2005$terms $$S=\frac{2^{2}}{1.3}+\frac{3^{2}}{2.4}+\frac{4^{2}}{3.5}+\ldots+\frac{2006^{2}}{2005.2007}$$ 2. Let$ABC$be an isosceles right triangle with right angle at$A .$Pick an arbitrary point$M$on$A C$. The foot of the altitude with$B C$through$M$is$H .$Let$I$be the midpoint of$BM$. Find the value of the angle$\angle H A I .$3. Prove that there exist infinitely many positive integers$n$such that$n !$is a multiple of$n^{2}+1$4. Let$a, b, c$be positive numbers such that$a+b+c=a b c .$Prove that $$\frac{a}{b^{3}}+\frac{b}{c^{3}}+\frac{c}{a^{3}} \geq 1.$$ 5. Solve the equation $$\frac{4}{x}+\sqrt{x-\frac{1}{x}}=x+\sqrt{2 x-\frac{5}{x}}$$ 6. Let$A B C$be a right triangle at$A$with$A B<A C$,$B C=2+2 \sqrt{3}$and its incircle has radius is$1 .$Determine the value of the angles$\angle B$and$\angle C$. 7. Let$A B C$be an acute triangle with orthocenter$H .$Let$A G$be the altitude through$A$and let$D$be the midpoint of$B C .$The circles, whose diameters are$B C$and$A D$respectively, meet at$E$and$F$. Prove that a) The circumcircles of$H, E$and$G$and of$H$,$F$and$G$are both tangent to the circle whose diameter is the side$B C$. b)$H, E$, and$F$lie on the same line. 8. Find the maximum value of the following expression $$x_{1}^{3} x_{2}^{2}+x_{2}^{3} x_{3}^{2}+\ldots+x_{n}^{3} x_{1}^{2}+n^{2(n-1)} x_{1}^{3} x_{2}^{3} \ldots x_{n}^{3}$$ where$x_{1}, x_{2}, \ldots, x_{n}$are non-negative numbers whose sum is$1(n \geq 2)$. 9. Solve the system of equations $$\begin{cases}20\left(x+\dfrac{1}{x}\right) &=11\left(y+\dfrac{1}{y}\right) &=2007\left(z+\dfrac{1}{z}\right) \\ x y+y z+z x &=1 \end{cases}$$ 10. Find the limit at$x=0$of the following function $$\frac{\sqrt{\frac{\cos 2 x+\sqrt{1+3 x}}{2}}-\sqrt{\frac{\cos 3 x+3 \cos x-\ln (1+x)^{4}}{4}}}{x}$$ 11. Let$\left(O_{1}\right)$be a circle inside a triangle$A B C$and touches both sides$A B$and$A C$. Construct a second circle$\left(O_{2}\right)$through$B$,$C$and touches outside of$\left(O_{1}\right)$at$T .$Prove that the angle bisector of$\widehat{B T C}$passes through the incenter of the triangle$A B C$. 12. Let$A_{1} A_{2} A_{3} A_{4}$be a tetrahedron with perpendicular opposite edges. Denote by$S_{i}$the area of the opposing faces of the vertices$A_{I}(i=1,2,3,4) .$Find all possible points$M$where the sum $$S_{1} \cdot M A_{1}+S_{2} \cdot M A_{2}+S_{3} \cdot M A_{3}+S_{4} \cdot M A_{4}$$ is smallest possible. ##$type=three$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 359
2007 Issue 359
Mathematics & Youth