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




Mathematics & Youth: 2010 Issue 396
2010 Issue 396
Mathematics & Youth
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy