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

## $show=home 1. Let$S$be the sum of$100$terms $$S=\frac{1}{1.1 .3}+\frac{1}{2.3 .5}+\frac{1}{3.5 .7}+\frac{1}{4.7 .9}+\ldots+\frac{1}{100.199 .201}.$$ Compare$S$with$\dfrac{2}{3}$. 2. Let$A B C$be an isosceles triangle$(A B=A C)$such that$\widehat{B A C}<90^{\circ} .$Let$B D$and$A H$be the altitudes. Choose a point$K$on$B D$such that$B K=B A$Find the measure of angle$HAK$. 3. Given$0<b<a \leq 4$and$2 a b \leq 3 a+4 b$. Find the maximum value of the expression$a^{2}+b^{2}$. 4. The equation $$5 x^{6}-16 x^{4}-33 x^{3}-40 x^{2}+8=0$$ has two roots which are reciprocal. Find these roots. 5. Let$A B C$be a right triangle with right angle at$A$and$A C>A B$. Let$O$be the midpoint of$B C,$and$I$be the incenter of the triangle$A B C .$Suppose that$\widehat{O I B}=90^{\circ},$find the ratio between three edges of the triangle$A B C$. 6. Find all pairs of positive integers$(x ; y)$such that $$y^{x}-1=(y-1) !$$ where$y$is a prime number. 7. Let$a, b, c$be non-negative real numbers. Prove the inequality $$\left(a^{2}+b^{2}+c^{2}\right)^{2} \geq 4(a-b)(b-c)(c-a)(a+b+c).$$ When does equality occur? 8. Let$A B C$be an isosceles triangle at vertex$A$and$B C \leq A C$. Choose a point$M$on$A B$(but not the vertices$A$or$B$) and a point$N$on$A C$such that$M N$touch the incircle of the triangle$A B C$. Find the maximum value of the ratio$\dfrac{A M}{B M \cdot C N}$when$M$moves along the edge$A B$. 9. Let$O$be the circumcenter of a triangle$A B C .$Choose a point$P,$different from$B$and$C$on the line connecting$B C$. The circumcircle of$A B C$meets$A P$at a point$N$and the circle whose diameter is$A P$at a point$E$($N$,$E$are both different from$A$).$B C$and$A E$intersect at$M .$Prove that$M N$always passes through a fixed point. 10. A sequence$\left(u_{n}\right)(n=1,2, \ldots)$is determined by the following recursive formula $$u_{1}=1,\quad u_{n+1}=\frac{16 u_{n}^{3}+27 u_{n}}{48 u_{n}^{2}+9}.$$ Find the largest integer which is smaller than the sum$S$of$2008$summands $$S=\frac{1}{4 u_{1}+3}+\frac{1}{4 u_{2}+3}+\ldots+\frac{1}{4 u_{2008}+3}$$ 11. Find the smallest value of the following expression $$P=\frac{1}{\cos ^{6} a}+\frac{1}{\cos ^{6} b}+\frac{1}{\cos ^{6} c}$$ where$a$,$b$and$c$form an arithmetic sequence whose common difference is$\dfrac{\pi}{3}$. 12. Let$f(x)$be a function defined on$[0 ; 1]$such that the following properties hold •$f(1)=1$•$f(x)=\dfrac{1}{3}\left(f\left(\dfrac{x}{3}\right)+f\left(\dfrac{x+1}{3}\right)+f\left(\dfrac{x+2}{3}\right)\right)$for all$x \in[0 ; 1]$• for every$\varepsilon$positive but can be arbitrarily small, there exists a positive number$\delta_{\varepsilon}\left(\delta_{\varepsilon}\right.$depends on$\varepsilon$) such that: For all$x, y \in[0 ; 1]$such that$|x-y|<\delta_{\varepsilon},$we have$|f(x)-f(y)|<\varepsilon$. Prove that$f(x)=1$for all$x \in[0 ; 1]$. ##$type=three$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: 2008 Issue 367
2008 Issue 367
Mathematics & Youth