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

## $show=home 1. Compare the following two fractions $$A=\frac{1010^{1010}}{2010^{2010}} \text { and } B=\frac{2010^{2010}}{3010^{3010}}$$ 2. Let$A B C$be a triangle with$\widehat{B A C} \geq 60^{\circ}$. Prove that$A B+A C \leq 2 B C$. 3. Find the remainder when dividing$3^{2^{n}}$by$2^{n+3}$where$n$is a positive integer. 4. Let$A B C$be a triangle. Construct a parallelogram$AMPN$so that the points$M$,$N$are in$A B$,$A C$respectively;$P$lies inside the triangle$A B C$. Let$Q$be the intersection of the line$A P$and$B C$. Prove that $$\frac{A M \cdot A N \cdot P Q}{A B \cdot A C \cdot A Q} \leq \frac{1}{27}.$$ Find the position of$P$when the equality occurs. 5. Solve the equation $$\left(x+\frac{5-x}{\sqrt{x}+1}\right)^{2}=\frac{-192(\sqrt{x}+1)}{5 \sqrt{x}-x \sqrt{x}}$$ 6. Two circles$\left(O_{1}\right)$and$\left(O_{2}\right)$meet at points$K$and$L$such that their centers$O_{1}$and$O_{2}$lie on the same side of the line$K L .$The tangent line to$\left(O_{1}\right)$at$K$meets$\left(O_{2}\right)$at$A .$The tangent line to$\left(O_{2}\right)$at$K$meets$\left(O_{1}\right)$at$B$. Find the area of the triangle$A K B,$given that$A L=3$,$B L=6$and$\tan \widehat{A K B}=-\dfrac{1}{2}$. 7. Let$a, b, c$be positive real numbers satisfying$a+b+c=3$. Prove the following inequality $$\sqrt{a}+\sqrt{b}+\sqrt{c}+5 \geq(a+b)(b+c)(c+a).$$ When does equality occur? 8. Solve the system of equations $$\begin{cases}x&=3 z^{3}+2 z^{2} \\ y&=3 x^{3}+2 x^{2} \\ z&=3 y^{3}+2 y^{2}\end{cases}$$ 9. Let$A B C$be an acute triangle. Let$a$,$b$,$c$be the side lengths of the triangle and$h_{a}$,$h_{b}$,$h_{c}$be the length of the corresponding altitudes. Let$r$,$R$be respectively the inradius and circumradius of this triangle. Prove the inequality $$\frac{9 R}{a^{2}+b^{2}+c^{2}} \leq \frac{1}{h_{a}+\sqrt{h_{b} h_{c}}}+\frac{1}{h_{b}+\sqrt{h_{c} h_{a}}}+\frac{1}{h_{c}+\sqrt{h_{a} h_{b}}} \leq \frac{1}{2 r}$$ 10. Let$A$be the set of$n$distinct points on the plane$(n \geq 2)$and$T(A)$be the set of vectors whose endpoints are in$A$. Find the maximum and minimum value of$|T(A)|$. (where$|T(A)|$denotes the cardinality of$T(A)$.) 11. Let$f$be a continuous function on$\mathbb{R}$satisfying the following two conditions $$f(2012)=2011,\quad f(x)f_{4}(x)=1,\, \forall x \in \mathbb{R}.$$ Denote$f_{n}(x)=\underbrace{f(f \ldots f(x)))}_{n \text { times } f}$. Find$f(2010)$12. Let$\left(x_{n}\right)(n=1,2, \ldots)$be a sequence given by $$x_{1}=2,1;\quad x_{n+1}=\frac{x_{n}-2+\sqrt{x_{n}^{2}+8 x_{n}-4}}{2}.$$ For each positive integer$n$, let$\displaystyle y_{n}=\sum_{i=1}^{n} \frac{1}{x_{i+1}^{2}-4}$. Find$\displaystyle\lim_{n \rightarrow+\infty} y_{n}$. ##$type=three$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: 2010 Issue 397
2010 Issue 397
Mathematics & Youth