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

## $show=home 1. Which number is greater?$A=\dfrac{1}{2006}$or $$B =\frac{1}{2008}+\left(\frac{1}{2008}+\frac{1}{2008^{2}}\right)^{2}+\ldots +\left(\frac{1}{2008}+\frac{1}{2008^{2}}+\ldots+\frac{1}{2008^{2007}}\right)^{2007}.$$ 2. In a triangle$A B C$with altitude$A D,$one has$A D=D C=3 B D$. Let$O$and$H$be the circumcenter and the orthocenter, respectively. Prove that$\dfrac{O H}{B C}=\dfrac{1}{4}$. 3. Find all pairs of natural numbers$x$and$y$such that $$x^{3}=y^{3}+2\left(x^{2}+y^{2}\right)+3 x y+17.$$ 4. Let$a, b, c$be positive real numbers. Prove the inequality $$\frac{a^{2}-b^{2}}{\sqrt{b+c}}+\frac{b^{2}-c^{2}}{\sqrt{c+a}}+\frac{c^{2}-a^{2}}{\sqrt{a+b}} \geq 0.$$ When does equality occur? 5. A circle$\left(S_{1}\right)$passing through the vertices$A$and$B$of a triangle$A B C$meets$B C$at another point$D .$Another circle,$\left(S_{2}\right),$passing through$B$and$C$meets$A B$at another point$E$and meets$\left(S_{1}\right)$at$F$. Prove that if all four points$A$,$C$,$D$and$E$lie on the same circle with center at$O$, then$\widehat{B F O}=90^{\circ}$. 6. Solve the system of equations $$\begin{cases}\dfrac{x}{a-30}+\dfrac{y}{a-4}+\dfrac{z}{a-14}+\dfrac{t}{a-10} &=1 \\ \dfrac{x}{b-30}+\dfrac{y}{b-4}+\dfrac{z}{b-14}+\dfrac{t}{b-10} &=1 \\ \dfrac{x}{c-30}+\dfrac{y}{c-4}+\dfrac{z}{c-14}+\dfrac{t}{c-10} &=1 \\ \dfrac{x}{d-30}+\dfrac{y}{d-4}+\dfrac{z}{d-14}+\dfrac{1}{d-10} &=1\end{cases}$$ where$a, b, c, d$are distinct numbers, none of which belong to the set$\{4 ; 10 ; 14 ; 30\} .$7. Let$a, b, c$be positive numbers. Prove that $$a^{b+c}+b^{c+a}+c^{a+b} \geq 1$$ 8. Choose six points$D$,$E$,$F$,$G$,$H$,$K$in that order, on a circle with radius$R$and center at$O$such that$D E=F G=H K=R$.$K D$and$E F$meet at$A$,$E F$and$G H$meet at$B$and$G H$meets$K D$at$C$. Prove that $$OA \cdot B C=O B \cdot C A=O C \cdot A B.$$ 9. Let$A B C D$be a cyclic quadrilateral, inscribed in a circle centered at$I .$Prove the following inequality $$(A I+D I)^{2}+(B I+C I)^{2} \leq(A B+C D)^{2}$$ and determine when equality occurs. 10. Consider the quadratic equation$x^{2}-a x-1=0$where$a$is a positive integer. Let$\alpha$be a positive root of this equation and construct a sequence$\left(x_{n}\right)$by the following recursive rule $$x_{0}=a, x_{n+1}=\left[\alpha x_{n}\right], n=0,1,2, \ldots$$ Prove that there exists infinitely many integer$n$such that$x_{n}$is a multiple of$a$. 11. Given a function$f: \mathrm{N} \rightarrow \mathrm{N}$such that for all$n \in \mathbb{N} $$$(f(2 n)+f(2 n+1)+1)(f(2 n+1)-f(2 n)-1)=3(1+2 f(n)),\quad f(2 n) \geq f(n).$$ Denote$M=\{m \in f(\mathrm{N}): m \leq 2008\} .$Find the number of elements of$M$. 12. For what kind of triangle$A B C$that the following relation among its sides and its angles holds $$\frac{b c}{b+c}(1+\cos A)+\frac{c a}{c+a}(1+\cos B)+\frac{a b}{a+b}(1+\cos C) \\ = \frac{3}{16}(a+b+c)^{2}+\cos ^{2} A+\cos ^{2} B+\cos ^{2} C.$$ ##$type=three$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 371
2008 Issue 371
Mathematics & Youth