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

## $show=home 1. Let $$A=\frac{1}{1^{2}}+\frac{1}{2^{3}}+\frac{1}{3^{4}}+\ldots+\frac{1}{2007^{2008}}.$$ Prove that$A$is not an integer. 2. Let$P(x)=x^{3}-7 x^{2}+14 x-8 .$Prove that for every natural number$n,$there exists a triple of distinct intergers$a_{1}$,$a_{2}$,$a_{3}$such that the following two conditions are satisfied •$\left|a_{i}-a_{j}\right|<5, \forall i, j \in\{1,2,3\}$•$P\left(a_{i}\right) \neq 0$and$5^{n} \mid P\left(a_{i}\right)$for all$i \in\{1,2,3\}$3. Find all pairs of intergers$x$,$y$such that $$\sqrt[n]{x+\sqrt[n]{x+\ldots+\sqrt[n]{x}}}=y$$ ($m$times) where$m$,$n$are positive intergers which are greater than$2$. 4. Given$x>2$. Prove the inequality $$\frac{x}{2}+\frac{8 x^{3}}{(x-2)(x+2)^{2}}>9.$$ 5. A pentagon$A B C D E$is inscribed in a circle with center at$O$and radius$R$such that$A B=C D=E A=R$. Let$M$,$N$be respectively the midpoints of$B C$and$D E$Prove that$A M N$is an equilateral triangle. 6. Do there exist two distinct positive integers$a$,$b$such that$b^{n}+n$is$a$multiple of$a^{n}+n$for every postive integer$n ?$. 7. Let$S$be the set of all pairs of real numbers$(\alpha, \beta)$such that the equation $$x^{3}-6 x^{2}+\alpha x-\beta=0$$ has three real roots (not necessarily distinct) and they are all greater than$1$. Find the maximum value of$T=8 \alpha-3 \beta$for$(\alpha, \beta) \in S$. 8. Let$A B C$be an acute triangle and denote by$Q$the center of its Euler's circle. The circumcircle of$A B C,$which has radius$R,$meets$A Q, B Q,$and$C Q$respectively at$M$,$N$and$P$Prove the inequality $$\frac{1}{Q M}+\frac{1}{Q N}+\frac{1}{Q P} \geq \frac{3}{R}.$$ 9. Find the interger part of $$A=\sqrt[n]{1-\frac{x}{2008}}+\sqrt[m]{1+\frac{x}{2008}}$$ where$x$is a real number in$[-2008 ; 2008]$, and$m$,$n$are natural numbers,$m \geq n \geq 2$. 10. Let$S$be denote the set of all$n$-tuples$(n>1)$of real numbers$\left(a_{1}, a_{2}, \ldots, a_{n}\right)$such that $$3\sum_{i=1}^{n} a_i^{2}=502.$$ a) Prove that$\displaystyle \min _{1 \leq \leq j \leq n}\left|a_{j}-a_{i}\right| \leq \sqrt{\frac{2008}{n\left(n^{2}-1\right)}}$. b) Give an example of an$n$-tuple$\left(a_{1}, a_{2}, \ldots, a_{n}\right)$such that the above condition holds and for which there is an equality in a). 11. A function$f(x),$whose domain is the interval$[1 ;+\infty),$has the following two properties $$f(1)=\frac{1}{2008},\quad f(x)+2007(f(x+1))^{2}=f(x+1),\,\forall x \in[1 ;+\infty).$$ Find the limit $$\lim_{n \rightarrow+\infty}\left(\frac{f(2)}{f(1)}+\frac{f(3)}{f(2)}+\ldots+\frac{f(n+1)}{f(n)}\right).$$ 12. The angle-bisectors$A A_{1}$,$B B_{1}$,$C C_{1}$of a triangle$A B C$with perimeter$p$meet$B_{1} C_{1}$,$C_{1} A_{1},$and$A_{1} B_{1}$respectively at$A_{2}$,$B_{2},$and$C_{2}$. The line through$A_{2}$and parallel to$B C$meets$A B$,$A C$at$A_{3}$,$A_{4}$. Construct the points$B_{3}$,$B_{4}$and$C_{3}$,$C_{4}$in a similar way. Prove the inequality $$A B_{4}+B C_{4}+C A_{4}+B A_{3}+C B_{3}+A C_{3} \leq p.$$ When does equality holds? ##$type=three$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 372
2008 Issue 372
Mathematics & Youth