2012 Issue 425

  1. Find all natural numbers $N$ such that $N$ decreases by a factor of $1997$ after truncating the last several digits.
  2. Let $A B C$ be a right triangle with right angle at $A$ and $\widehat{A C B}=15^{\circ}$, Point $D$ on edge $A C$ such that the line passing through $D$ and perpendicular to $B D$ cuts $B C$ at $E$ and $D E=2 D A$. Find the measure of angle $A D B$.
  3. Find all positive integers $n$ such that $[A]=4951$ where $A$ is the sum of $n$ terms $$A=\left(1+\frac{1}{2}\right)+\left(2+\frac{2}{2^{2}}\right)+\left(3+\frac{3}{2^{3}}\right)+\ldots+\left(n+\frac{n}{2^{n}}\right).$$ Here $[x]$ denotes the largest integer not exceeding $x$
  4. Find the minimum value of the expression $$P=\frac{1+\sqrt[3]{x}+\sqrt[3]{y}+\sqrt[3]{z}}{x y+y z+z x},$$ where $x, y, z$ are positive numbers satisfying $x+y+z=3$
  5. Solve the equation $$x^{2}-2 x+7+\sqrt{x+3}=2 \sqrt{1+8 x}+\sqrt{1+\sqrt{1+8 x}}.$$
  6. Let $A B C$ be a non-isosceles triangle with medians $A A^{\prime}$, $B B^{\prime}$ and $C C^{\prime}$; and altitudes $A H$, $B F$ and CK. Given that $C K=B B^{\prime}$, $B F=A A^{\prime}$. Determine the ratio $\dfrac{C C^{\prime}}{A H}$.
  7. $a_{1}, a_{2}, \ldots, a_{n}$ $(n \geq 3)$ are positive numbers that $$\left(a_{1}+a_{2}+\ldots+a_{n}\right)^{2}>\frac{3 n-1}{3}\left(a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}\right).$$ Prove that for any triple $a_{i}, a_{j}, a_{k}$ are three edge lengths of some triangle, where natural numbers $i, j,$ $k$ satisfying $0<i<j<k \leq n$.
  8. The volume of a given parallelogrambased pyramid $S.ABCD$ is $V$. Assume that plane $(P)$ cuts$S A$, $S B$, $S C$, $S D$ at $A^{\prime}$, $B^{\prime}$, $C^{\prime}$, $D^{\prime}$ respectively such that $$\frac{S A}{S A^{\prime}}+\frac{S B}{S B^{\prime}}+\frac{S C}{S C^{\prime}}+\frac{S D}{S D^{\prime}}=8.$$ Denote the volume of the pyramid $S . A^{\prime} B^{\prime} C^{\prime}$ by $V_{1}$ and that of $S . A^{\prime} C^{\prime} D^{\prime}$ by $V_{2}$. Prove the inequality $$\frac{1}{\sqrt[3]{V_{1}}}+\frac{1}{\sqrt[3]{V_{2}}} \leq \frac{4 \sqrt[3]{2}}{\sqrt[3]{V}}.$$
  9. Write $2012^{2013}$ as a sum of $2013$ positive interger $a_{1}, a_{2}, a_{3}, \ldots, a_{2013} ;$ and let $$T=a_{1}^{13}+a_{2}^{13}+a_{3}^{13}+\ldots+a_{2013}^{13}.$$ Prove that $T+2012^{2013}$ is not a perfect square.
  10. The incircle $(I)$ of a triangle $A B C$ touches the edges $B C$, $C A$, $A B$ at $D$, $E$, $F$, respectively. $M$ is the intersection of $B C$ and the internal angle bisector of angle $B I C$, $N$ is the intersection of $E F$ and the internal angle bisector of angle $E D F$. Prove that $A$, $M$, $N$ are collinear.
  11. If $p(x)$ and $q(x)$ are polynomials with integer coefficients, write $p(x) \equiv q(x) \pmod 2$ if the coefficients of $p(x)-q(x)$ are all even. A sequence of polynomials $p_{n}(x)$ is such that $p_{1}(x)=p_{2}(x)=1$ and $$p_{n+2}(x)=p_{n+1}(x)+x p_{n}(x),\,\forall n \geq 1.$$ Prove that $p_{2^{n}}(x) \equiv 1\pmod 2, \forall n \in \mathbb{N}$.
  12. Let $A B C$ be an acute triangle. Prove the inequality $$\frac{\cos B \cos C}{\cos \frac{B-C}{2}}+\frac{\cos C \cos A}{\cos \frac{C-A}{2}}+\frac{\cos A \cos B}{\cos \frac{A-B}{2}} \leq \frac{3}{4}$$




Mathematics & Youth: 2012 Issue 425
2012 Issue 425
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