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

## $show=home 1. Find all natural numbers$a$such that both$a+593$and$a-159$are perfect squares. 2. Let$A B C$be a right triangle, with right angle at$A$and$\widehat{A C B}=15^{\circ}$. Let$B C=a$,$A C=b$,$A B=c .$Prove that$a^{2}=4 b c$. 3. Find all intergers$x, y, z, t$such that $$x^{2008}+y^{2008}+z^{2008}=2007 . t^{2008}.$$ 4. Prove the inequality $$\left(\frac{4}{a^{2}+b^{2}}+1\right)\left(\frac{4}{b^{2}+c^{2}}+1\right)\left(\frac{4}{c^{2}+a^{2}}+1\right) \geq 3(a+b+c)^{2}$$ where$a, b, c$are positive numbers such that$a^{2}+b^{2}+c^{2}=3$. 5. Let$A B C$be an acute triangle. Choose a point$D,$different from$B$and$C,$on the side$B C .$Prove that the vertex$A$and the centers of the circumcircles of the triangles$A B D$,$A C D$and$A B C$lie on the same circle. 6. Find the coefficient of$x^{2}$in the expansion of $$\left(\left(\ldots\left((x-2)^{2}-2\right)^{2}-\ldots-2\right)^{2}-2\right)^{2}$$ given that the number 2 occurs 1004 times in the expression above and there are 2008 round brackets. 7. Find the smallest value of the following expression $$\frac{\sqrt{a_{1}+2008}+\sqrt{a_{2}+2008}+\ldots+\sqrt{a_{n}+2008}}{\sqrt{a_{1}}+\sqrt{a_{2}}+\ldots+\sqrt{a_{n}}}$$ where$n$is a given positive natural number and$a_{1}, a_{2}, \ldots, a_{n}$are non-negative real numbers such that$a_{1}+a_{2}+\ldots+a_{n}=n$8. Let$(O)$be a circle centered at$O$and fixed diameter$A B .$Let$\Delta$be a straight line which touches$(O)$at$A .$Choose a point$M$on the circle$(O),$different from$A$and$B .$The tangent line with$(O)$through$M$meets$\Delta$at$C$. Let$(I)$be the circle through$M$and touches$\Delta$at$C$. Let$C D$be the diameter of$(I)$. Prove that a)$D O C$is an isosceles triangle. b) The line through$D$and perpendicular to$B C$always passes through a fixed point when$M$moves on the circle$(O)$. 9. Does there exist a sequence of positive integers$a_{2003}>a_{2002}>\ldots>a_{2}>a_{1}$with$a_{1}=2003$such that the following two conditions are satisfied • All integers in the interval$\left(2003 ; a_{2 \times 13}\right)$are either a member of this sequence or a non-prime. •$A=\dfrac{2004}{a_{1}}+\dfrac{2004}{a_{2}}+\ldots+\dfrac{2004}{a_{2003}}$is an integer?. 1. Find all continuous functions$f, g, h$on$\mathbb{R}$such that $$f(x+y)=g(x)+h(y)$$ for all real numbers$x$,$y$. 2. Let$H$and$O$denote the orthocenter and circumcenter respectively of a triangle$A B C$. Prove the inequality $$3 R-2 O H \leq H A+H B+H C \leq 3 R+O H$$ where$R$is its circumradius. 3. Let$A B C D . A_{1} B_{1} C_{1} D_{1}$be a cubic whose side is a. Let$N$,$M$be two points on$A B_{1}$and$B C_{1}$respectively such that the angle between$M N$and the plane$(A B C D)$is$60^{\circ}$. Prove that $$M N \geq 2 a(\sqrt{3}-\sqrt{2}).$$ When does equality 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: 2008 Issue 378
2008 Issue 378
Mathematics & Youth