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

## $show=home 1. Find the last two decimal digits of the sum$2008^{2009}+2009^{2008}$2. Let$ABC$be an isosceles right triangle with the right angle at$A$. Let$G$be a point on$A B$such that$A G=\dfrac{1}{3} A B$, let$M$be the midpoint of$B C$and$E$be the foot of the altitude from$M$to$CG$. The two lines$M G$and$A C$meet at$D$. Prove that$D E=B C$. 3. Find all integer solutions of the equation $$4 x^{4}+2\left(x^{2}+y^{2}\right)^{2}+x y(x+y)^{2}=132$$ 4. Let$a$,$b$and$c$satisfy the conditions$a \leq b \leq c$and $$a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$ Find the least value of the expression$P=a b^{2} c^{3}$. 5. A triangle$A B C$inscribed in a circle centered at$O$, radius$R$,$A D$is its anglebisector. Let$E$,$F$be the circumeenters pf the triangles$A B D$and$A C D$. respectively. Given that$B C=a$. Determine the area of the quadrilateral$A E O F$. 6. Let$A B C$be a triangle with incenter$I$such that its centroid$G$lies inside$(I)$. Let$a$,$b$,$c$be the lengths of the sides$B C$,$A C$,$A B$, respectively. Find the greatest and least value of the following expression $$P=\frac{a^{2}+b^{2}+c^{2}}{a b+b c+c a}$$ 7. Let$x$,$y$,$z$be positive numbers satisfying $$x^{2}+y^{2}+z^{2}=\frac{1-16 x y z}{4}.$$ Find the least value of the expression $$S=\frac{x+y+z+4 x y z}{1+4 x y+4 y z+4 z x}.$$ 8. Solve for$x$$$2^{x}+5^{x}=2-\frac{x}{3}+44 \log _{2}\left(2+\frac{131 x}{3}-5^{x}\right)$$ 9. Let$A B C$be a right triangle, right angle at$A$,$M$is the midpoint of$B C$. Construct a right angle$P M Q$with$P \in A B$,$Q \in A C$. Prove that $$P Q^{2} \geq A P \cdot C Q+A Q \cdot B P$$ 10. Let$a_{n}$be the last non-zero digit (counting from left to right) when expressing$n !$in the decimal number system. Is the sequence$\left(a_{n}\right)$for$n=1.2 .3 \ldots$periodic? (That is, there exist the positive integers$T$and$N$such that$a_{i+T}=a_{i} \forall i \geq N$). 11. Given$n$non-negative numbers$a_{1}, a_{2}, \ldots, a_{n}(n \geq 3)$satisfying $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}=1.$$ Prove that $$\frac{1}{\sqrt{3}}\left(a_{1}+a_{2}+\ldots+a_{n}\right) \geq a_{1} a_{2}+a_{2} a_{3}+\ldots+a_{n} a_{1}.$$ When does equality occur? 12. Let$\left(x_{n}\right)(n=1,2, \ldots)$be a sequence given by $$x_{1}=a \ (a>1),\, x_{2}=1,\quad x_{n+2}=x_{n}-\ln x_{n},\,\forall n \in \mathbb{N}^{*}.$$ Put$\displaystyle S_{n}=\sum_{k=5}^{n-1}(n-k) \ln \sqrt{x_{2 k-1}}(n \geq 2)$. Find$\displaystyle\lim_{n \rightarrow \infty}\left(\frac{S_{n}}{n}\right)$. ##$type=three$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: 2009 Issue 386
2009 Issue 386
Mathematics & Youth