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

## $show=home 1. Pick$n$numbers$(n \geq 2)$from the first hundred natural numbers (from$1$to$100$) so that the sum of any two distinct numbers is a multiple of$6 .$What is the largest possible number$n$so that this can be done? 2. Given$A=\dfrac{5^{a}}{5^{b+c}}$and$B=\dfrac{5^{a}+2011}{5^{b+c}+2011}$where$a, b, c$are the side lengths of a triangle. Compare$A$and$B$. 3. Do there exists three integers$x$,$y$and$z$such that $$|x-2005 y|+|y-2007 z|+|z-2009 x|=2011^{x}+2013^{y}+2015^{z} ?$$ 4. Determine the following sum of$2011$terms $$S=\frac{1}{1^{4}+1^{2}+1}+\ldots+\frac{2011}{2011^{4}+2011^{2}+1}$$ 5. Given a circle$(O),$a chord$B C$($B C$is not a diameter) and point$A$moving on the major arc$B C$. Draw a circle$\left(O_{1}\right)$passing through$B$and touches$A C$at$A$, another circle$\left(O_{2}\right)$passing through$C$and touches$A B$at$A .\left(O_{1}\right)$meets$\left(O_{2}\right)$at a second point$D$, different from$A$. Prove that line$A D$always passes through a fixed point. 6. A quadrilateral$A B C D$with$A C \perp B D$is inscribed in a fixed circle$(O ; R)$. Let$p$be the perimeter of$A B C D$. Prove that $$\frac{A B^{2}}{p-A B}+\frac{B C^{2}}{p-B C}+\frac{C D^{2}}{p-C D}+\frac{D A^{2}}{p-D A} \geq \frac{4 R \sqrt{2}}{3}$$ 7. Solve the system of equations $$\begin{cases}(17-3 x) \sqrt{5-x}+(3 y-14) \sqrt{4-y} &=0 \\ 2 \sqrt{2 x+y+5}+3 \sqrt{3 x+2 y+11} &=x^{2}+6 x+13\end{cases}$$ 8. Prove that the following inequality holds for any triangles$A B C$$$\cos ^{2} \frac{A-B}{2}+\cos ^{2} \frac{B-C}{2}+\cos ^{2} \frac{C-A}{2} \geq 24 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}.$$ 9. Two circles$\left(C_{1}\right),\left(C_{2}\right)$are given such that the center$O$of$\left(C_{2}\right)$lies on$\left(C_{1}\right) .$Let$C$,$D$be their intersection points. Points$A$and$B$on$\left(C_{1}\right)$and$\left(C_{2}\right)$respectively such that$A C$touches$\left(C_{2}\right)$at$C$and$B C$touches$\left(C_{1}\right)$at$C$. The line$A B$intersects$\left(C_{2}\right)$at$E$and$\left(C_{1}\right)$at$F$.$C E$meets$\left(C_{1}\right)$at$G, C F$meets$G D$at$H$. Prove that$G O$intersects$E H$at the circumcenter of triangle$D E F$. 10. Let$a_{1}, a_{2} \ldots, a_{n}$be$n$positive real numbers such that $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{k}^{2} \leq \frac{k(2 k-1)(2 k+1)}{3}$$ where any$k=\overline{1, n}$. Find the largest possible value of the expression $$P=a_{1}+2 a_{2}+\ldots+n a_{n}.$$ 11. Given a sequence$\left(x_{n}\right)$such that $$x_{n}=2 n+a \sqrt{8 n^{3}+1}, \forall n=1,2, \ldots$$ where$a$is any real number. a) For what values of$a$does the sequence has finite limit? b) Find$a$such that$\left(x_{n}\right)$is eventually increasing. 12. Find all function$f: \mathbb{R} \rightarrow \mathbb{R}$such that $$f\left(x^{3}+y^{3}\right)=x^{2} f(x)+y^{2} f(y)$$ where$x, y \in \mathbb{R}$##$type=three$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 412
2011 Issue 412
Mathematics & Youth