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

## $show=home 1. Find the integer value of the expression$f(x ; y)=\dfrac{x^{2}+x+2}{x y-1}$where$x, y$are positive integers. 2. Let$A B C$be an acute triangle which is not isosceles at$A$. The perpendicular bisectors of$A B$,$A C$cut the median$A M$at$E$,$F$respectively.$B E$and$C F$meet at$K .$Prove that$\widehat{A K B}=\widehat{A K C}$and$\widehat{M A B}=\widehat{K A C}$. 3. Find all triples of integers$(x ; y ; z)$such that $$2 x y+6 y z+3 z x-|x-2 y-z|=x^{2}+4 y^{2}+9 z^{2}-1.$$ 4. For each positive integer$n(n=1,2, \ldots),$put$a_{n}=\dfrac{4 n}{n^{4}+4} .$Prove that $$a_{1}+a_{2}+\ldots+a_{n}<\frac{3}{2}$$ 5. Let$A B C$be an acute triangle. The internal angle-bisector of angle$B A C$cuts$B C$at$D$.$E$,$F$are the orthogonal projections of point$D$on$A B$and$A C$respectively,$K$is the intersection of$C E$and$B F, H$is the intersection of$B F$with the circumcircle of triangle$A E K$. Prove that$D H$is perpendicular to$B F$6. Solve the system of equations $$\begin{cases} x+6 \sqrt{x y}-y &=6 \\ x+\dfrac{6\left(x^{3}+y^{3}\right)}{x^{2}+x y+y^{2}}-\sqrt{2\left(x^{2}+y^{2}\right)} &=3 \end{cases}.$$ 7. Let$a, b, c$be non-negative real numbers whose sum equals$1$. Prove that $$\left(1+a^{2}\right)\left(1+b^{2}\right)\left(1+c^{2}\right) \geq\left(\frac{10}{9}\right)^{3}$$ 8. Point$M$inside the triangle$A B C$with area$S$. Let$x, y, z$be distances of$M$to$A$,$B$,$C$respectively. Prove that $$(x+y+z)^{2} \geq 4 \sqrt{3} S.$$ When does the equality hold? 9. A nonempty set$S \subseteq \mathbb{Z}$posesses the following properties • There exist$a, b \in S$such that$(a, b)=(a-2 b-2)=1$, • If$x, y \in S$then$x^{2}-y \in S$($x$,$y$may be identical). Prove that$S=\mathbb{Z}$. ($(a, b)$is the greatest common divisor of two integers$a$and$b$.) 1. Find the greatest number$k$such that the inequality $$\sqrt{a+2 b+3 c}+\sqrt{b+2 c+3 a}+\sqrt{c+2 a+3 b} \geq k(\sqrt{a}+\sqrt{b}+\sqrt{c})$$ holds for all positive numbers$a, b, c$2. Let$\left(x_{n}\right)$be a sequence defined by $$x_{1}=\frac{1001}{1003} ,\quad x_{n+1}=x_{n}-x_{n}^{2}+x_{n}^{3}-x_{n}^{4}+\ldots+x_{n}^{2011}-x_{n}^{2012},\, \forall n \in \mathbb{N}.$$ Find$\displaystyle \lim _{n \rightarrow+\infty}\left(n x_{n}\right)$. 3. Given four distinct points$A$,$B$,$C$,$D$lying on a circle with center$O$. Let$I$,$J$be the feet of the perpendicular to$A B$and$A D$through$C$;$K$,$L$are the feet of the perpendicular to$B C$and$B A$through$D$;$N$is the midpoint of$C D$;$M$is the intersection of$I J$and$K L$.$I J$meets$O D$at$E$and$K L$meets$O C$at$F$. Prove that the five points$M$,$N$,$O$,$E$and$F$lie on the same circle. ##$type=three$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: 2012 Issue 420
2012 Issue 420
Mathematics & Youth