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

## $show=home 1. Does there exist a pair of integers$x$,$y$such that $$x^{3}-y^{3}=10 \times 10 \times 2010$$ 2. Let$n$be a natural number, greater than$1$. Prove that $$\frac{1+n}{1+n^{n+1}}>\left(\frac{1+n^{n}}{1+n^{n+1}}\right)^{n}$$ 3. Let$a, b, c, x, y, z$be positive integers satisfying the conditions $$\begin{cases}\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} &=2010 \\ a x^{3}=b y^{3}=c z^{3}\end{cases}.$$ Prove that $$x+y+z \geq \frac{3}{670}.$$ 4. Let$A B C$be a triangle whose sides satisfying the relation $$B C^{2}+A B \cdot A C-A B^{2}=0.$$ Determine the sum$\hat{A}+\dfrac{2}{3} \hat{B}$. 5. Two orthogonal diameters$A E$and$B F$of a circle center$O$radius$R$are given.$A$point$C$is chosen on the minor arc$E F$. The chord$A C$intersects$B F$at$P$and the chord$B C$meets$A E$at$Q$. Find the area of the quadrilateral$A P Q B$in term of$R$. 6. Solve the system of equations $$\begin{cases}\sqrt{y^{2}-8 x+9}-\sqrt{x y+12-6 x} &\leq 1 \\ \sqrt{2(x-y)^{2}+10 x-6 y+12}-\sqrt{y} &=\sqrt{x+2}\end{cases}.$$ 7. A quadrilateral$A B C D$is inscribed in a circle centered$I$and circumscribed another circle with center at$O$. The diagonals$A C$and$B D$intersect at$E$. Prove that$E$,$I$and$O$are collinear. 8. Let$\left(x_{n}\right)$be a sequence given by $$x_{1}=5, \quad x_{n+1}=x_{n}^{2}-2,\,\forall n \geq 1 .$$ Find a)$\displaystyle\lim _{n \rightarrow+\infty} \frac{x_{n+1}}{x_{1} x_{2} \ldots x_{n}}$. b)$\displaystyle\lim _{n \rightarrow+\infty}\left(\frac{1}{x_{1}}+\frac{1}{x_{1} x_{2}}+\ldots+\frac{1}{x_{1} x_{2} \ldots x_{n}}\right)$. 9. Prove that the equation$P(x)=2^{x}$where$P(x)$is a polynomial of degree$n,$has less than$n+1$roots. 10. Find all continuous functions$f: \mathbb{R} \rightarrow \mathbb{R}$satisfying the condition $$f(2010 x-f(y))=f(2009 x)-f(y)+x,\, \forall x, y \in \mathbb{R}$$ 11. Let$\left(a_{n}\right)$be a sequence with $$a_{1}=a_{2}=1,\quad a_{n+2}=a_{n+1}+a_{n},\,\forall n \geq 1.$$ Find all pair of positive integers$a$,$b$,$a<b$so that$a_{n}-2 n a^{n}$is a multible of$b$for any$n \geq 1$. 12.$I$and$R$are the incenter and the circumradius of a gven triangle$A B C$.$I A$,$I B$,$I C$intersect the circumcircle at$A_{1}$,$B_{1}$,$C_{1}$respectively. Prove the inequality $$2 R+\frac{L A+I B+I C}{3} \leq L A_{1}+I B_{1}+I C_{1} \leq \frac{5}{2} R+\frac{I A+I B+I C}{6}$$ ##$type=three$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: 2010 Issue 392
2010 Issue 392
Mathematics & Youth