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

## $show=home 1. Write the numbers$8^{2008}$and$125^{2008}$consecutively. What is the number of decimal digits of the resulting number? 2. Find a rational number$\dfrac{a}{b}$such that the following three conditions are satisfied •$-\dfrac{1}{2}<\dfrac{a}{b}<-\dfrac{2}{5}$•$11 a+5 b=26$•$200<|a|+|b|<230$3. Find all non-zero natural numbers$n$such that the number$A=\dfrac{1.3 .5 .7 \ldots(2 n-1)}{n^{n}}$is an integer, here the numerator of$A$is the product of the first$n$odd numbers. 4. Prove that $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{3}{2}(a+b+c-1)$$ where$a, b, c$are positive real numbers such that$a b c=1 .$When does equality hold?$?$5. In a right triangle$A B C$with right angle at$A,$the altitude$A H,$the median$B M,$and the angle-bisector$C D$meet at a common point. Determine the ratio$\dfrac{A B}{A C}$6. Solve for$x$$$\sqrt[3]{x+6}+\sqrt{x-1}=x^{2}-1$$ 7. Let$S$denote the area of a given triangle$A B C,$and denote$B C=a$,$C A=b$,$A B=c$. Prove the inequality $$a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2} \geq 16 S^{2}+\frac{1}{2} a^{2}(b-c)^{2}+\frac{1}{2} b^{2}(c-a)^{2}+\frac{1}{2} c^{2}(a-b)^{2}.$$ When does equality hold? 8. Given a triangle$A B C$with three sides$B C=a$,$A C=b$,$A B=c$such that$a+c=2 b$let$h_{a}$,$h_{c}$be the altitudes from$A$and$C$respectively; and let$r_{a}$,$r_{c}$denote the$A$-exradius and$C$-exradius respectively. Prove that $$\frac{1}{r_{a}}+\frac{1}{r_{c}}=\frac{1}{h_{a}}+\frac{1}{h_{c}}.$$ 9. The positive real numbers$a$,$b$,$c$,$x$,$y$and$z$are such that $$\begin{cases} c y+b z &=a \\ a z+c x &=b \\ b x+a y &=c\end{cases}.$$ Find the smallest possible value of the expression $$P=\frac{x^{2}}{1+x}+\frac{y^{2}}{1+y}+\frac{z^{2}}{1+z}.$$ 10. Let$f$be a continuous function on$\mathbb{R}$such that$f(2010)=2009$and$f(x) \cdot f_{4}(x)=1$for all$x \in \mathbb{R}$(where$\left.f_{4}(x)=f(f f(f(x)))\right)$. Determine the value of$f(2008)$. 11. Let$u_{1}, u_{2}, \ldots, u_{n}(n>2)$be a sequence of positive real numbers such that •$\dfrac{1004}{k}=u_{1} \geq u_{2} \geq \ldots \geq u_{n} \quad$for some positive integer$k$; •$u_{1}+u_{2}+\ldots+u_{n}=2008$. Show that it is possible to select$k$elements from the set$\left(u_{n}\right)$such that in this collection of$k$numbers, the smallest one is at least half of the largest. 12. Consider the circle$(O)$and three colinear points$X$,$Y$,$H$that are not on this circle such that$\overline{H X} \cdot \overline{H Y} \neq \mathscr{P}_{H/(O)} .$A straight line$d$through$H$meets$(O)$at two points$M$and$N$.$M X$and$N Y$intersect with$(O)$again at$P$and$Q$respectively. Show that as the line$d$through$H$varies, the line connecting$P$and$Q$always passes through a fixed point. ##$type=three$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 376
2008 Issue 376
Mathematics & Youth