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

## $show=home 1. Without taking common denominator, find the integer$x$given that $$\left(\frac{2009}{2010}+\frac{2010}{2011}+\frac{2011}{2009}\right) \cdot(x-2011)>3 x-6033$$ 2. In a triangle$A B C$,$M$,$N$,$P$are midpoints of sides$B C$,$C A$and$A B$respectively. Choose the points$A_{1}$,$A_{2}$,$B_{1}$,$B_{2}$,$C_{1}$,$C_{2}$on the opposite rays of rays$B A$,$C A$,$C B$,$A B$,$A C$and$B C$respectively so that$B A_{1}=C A_{2}$,$C B_{1}=A B_{2}$,$A C_{1}=B C_{2} . A_{0}, B_{0}, C_{0}$are midpoints of$A_{1} A_{2}, B_{1} B_{2}, C_{1} C_{2}$respectively. Prove that the lines$A_{0} M, B_{0} N, C_{0} P$meet at a common point. 3. Let$b$be a positive integer with the following properties •$b$equals a sum of three squares. •$b$possess a divisor of the form$a=3 k^{2}+3 k+1(k \in \mathbb{N})$. Prove that$b^{n}$is also a sum of three squares for any positive integer$n$. 4. Given three non-negative numbers$a, b, c$, prove the inequality $$a+b+c \geq \frac{a-b}{b+2}+\frac{b-c}{c+2}+\frac{c-a}{a+2}.$$ When does equality hold? 5. Let$A B C$be a triangle with circumcircle$(O)$,$I$is the midpoint of side$B C$.$M$is chosen on$I C$(differ from both$C$and$I$).$A M$meets$(O)$at$D$. Point$E$is on$B D$such that$\widehat{B M E}=\widehat{M A I}$.$E M$and$D C$intersect at$F$. Prove that $$\frac{C F}{C D}=\frac{B E}{B D}$$ 6. Solve for$x$$$\sqrt{\frac{x+2}{2}}-1=\sqrt{3(x-3)^{2}}+\sqrt{9(x-3)}$$ 7. Triangle$A B C$is inscribed in a fixed circle$(O)$. The medians from$A$,$B$,$C$meets$(O)$at$A_{1}$,$B_{1}$,$C_{1}$respectively. Which triangle makes the value of the expression $$p=\frac{A A_{1}^{2}+B B_{1}^{2}+C C_{1}^{2}}{A B^{2}+B C^{2}+C A^{2}}$$ minimum possible? 8. In a triangle$A B C$, prove that $$\frac{\sin A \cdot \sin B}{\sin ^{2} \frac{C}{2}}+\frac{\sin B \cdot \sin C}{\sin ^{2} \frac{A}{2}}+\frac{\sin C \cdot \sin A}{\sin ^{2} \frac{B}{2}} \geq 9$$ 9. Let$A B C$be a triangle with points$A'$,$B'$,$C'$on sides$B C$,$C A$and$A B$respectively such that $$\frac{A^{\prime} B}{A^{\prime} C}=\frac{B^{\prime} C}{B^{\prime} A}=\frac{C^{\prime} A}{C^{\prime} B}.$$$A A^{\prime}$and$B B^{\prime}$meet at$D$,$B B$' meets$C C$' at$E$and$F$is the intersection of$C C'$and$A A '$. Parallel lines to$A A'$,$B B'$,$C C'$through point$O$in the interior of$A B C$meet$B C$,$C A$,$A B$at$A_{1}$,$B_{1}$,$C_{1}$respectively. Prove that for any point$M$$$A D\left(M A_{1}-O A_{1}\right)+B E\left(M B_{1}-O B_{1}\right)+C F\left(M C_{1}-O C_{1}\right) \geq 0$$ 10. Given$P=(n+1)^{7}-n^{7}-1(n \in \mathbb{N})$. Prove that there are infinitely many natural numbers$n$so that$P$is a perfect square. 11. Determine all functions$f: \mathbb{R} \rightarrow \mathbb{R}$, continuous on$\mathbb{R}$such that $$f(x y)+f(x+y)=f(x y+x)+f(y),\, \forall x, y \in \mathbb{R}.$$ 12.$f: \mathbb{R} \rightarrow \mathbb{R}$is a function with the following properties •$f(1)=2011$, •$f(x+1) f(x)=(f(x))^{2}+f(x)-1, \forall x \in \mathbb{R}$. Let$\displaystyle S_{1}=\sum_{i=1}^{n} \frac{1}{f(i)-1}$,$\displaystyle S_{2}=\sum_{i=1}^{n} \frac{1}{f(i)+1}$. Find$\displaystyle\lim_{n \rightarrow+\infty}\left(S_{1}+S_{2}\right)$. ##$type=three$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: 2011 Issue 409
2011 Issue 409
Mathematics & Youth