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2010 Issue 396

  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 $$\max \left\{\frac{H B}{H C}+\frac{H C}{H B^{\prime}} + \frac{H C}{H A}+\frac{H A}{H C^{\prime}} + \frac{H A}{H B}+\frac{H B}{H A}\right\} \geq \frac{2 R}{r}-2$$

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Mathematics & Youth: 2010 Issue 396
2010 Issue 396
Mathematics & Youth
https://www.molympiad.org/2020/09/2010-issue-396.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2010-issue-396.html
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