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

## $show=home 1. Let$A=14916 . . .4040100$be the number obtained by writing the perfect squares$1^{2}, 2^{2}, \ldots, 2010^{2}$consecutively. Let$B+C$be the sum obtained by putting the sign "$+$" in between certain two digits of$A$. Is$B+C$divisible by$9$? Explain your reasoning? 2. Let$A B C$be a triangle with the altitude$A H$satisfying$B C=A H \sqrt{2}$. Compute the measure of the angle$\widehat{A C B}$, given that$\widehat{A B C}=67^{\circ} 30^{\prime}$. 3. Let$n_{1}, n_{2}, \ldots, n_{m}$be a sequence of strictly decreasing natural numbers. For each natural number$n$, put $$P_{n}=2\left(3^{n}+3^{n_{1}}+3^{n_{2}}+\ldots+3^{n_{m}}\right).$$ Does there exist an$n$with$n>n_{1}$such that$P_{n}$is a perfect square? 4. Let$x, y, z$be real numbers satisfying$x \geq 2$,$y \geq 9$,$z \geq 1945$,$x+y+z=2010$. Find the greatest value of the product$x y z$. 5. Let$A B C$be a triangle. Let$M$,$N$,$P$be the points of contact of its incircle$(I)$with the sides$A B$,$A C$,$B C$respectively and let$M D$,$N E$,$P F$be the altitudes of the triangle$M N P$. Prove that $$D A \cdot F B \cdot E C=E A \cdot D B \cdot F C.$$ 6. Let$a, b, c$be positive real numbers satisfying$a^{2}+b^{2}+c^{2}=1$. Prove the inequality $$\frac{1}{1-a b}+\frac{1}{1-b c}+\frac{1}{1-c a} \leq \frac{9}{2(1+9 a b c-a-b-c)}$$ 7. Let$\alpha \in\left(0 ; \frac{\pi}{2}\right)$. Find the minimum value of the expression $$P=(\cos \alpha+1)\left(1+\frac{1}{\sin \alpha}\right)+(\sin \alpha+1)\left(1+\frac{1}{\cos \alpha}\right)$$ 8. Let$S . A B C$be a pyramid with$S A=a$,$S B=b$,$S C=c$and $$\widehat{A S B}=\widehat{B S C}=\widehat{C S A}=\alpha.$$ Compute its volume in term of$a, b, c$and$\alpha$. 9. Let$n$be a positive integer. Let$p(n)$be the product of its nonzero digits. (If$n$has a single digit then$p(n)=n$). Consider the expression$S=p(1)+p(2)+\ldots+p(999)$. What is the greatest prime divisor of$S ?$10. Let$(u_n)(n = 0, 1, 2, ...)$be the sequence given by $$u_{0}=0,\quad u_{n+1}=\frac{u_{n}+2008}{-u_{n}+2010}.$$ a) Prove that the sequence$\left(u_{n}\right)(n=0,1,2, \ldots)$converges and find limit of$u_{n}$. b) Put$\displaystyle T_{n}=\sum_{k=0}^{n} \frac{1}{u_{k}-2008} \cdot$Find$\displaystyle \lim_{n\to\infty} \frac{T_{n}}{n+2009}$. 11. Solve the equation $$4 \sqrt{x+2}+\sqrt{22-3 x}=x^{2}+8$$ 12. Let$A B C$be an acute triangle with orthocenter$H$. Let$R$,$r$be its the circumradius and inradius respectively. Prove that $$\max \left\{\frac{H B}{H C}+\frac{H C}{H B^{\prime}} + \frac{H C}{H A}+\frac{H A}{H C^{\prime}} + \frac{H A}{H B}+\frac{H B}{H A}\right\} \geq \frac{2 R}{r}-2$$ ##$type=three$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 396
2010 Issue 396
Mathematics & Youth