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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 $$\frac{\cos B \cos C}{\cos \frac{B-C}{2}}+\frac{\cos C \cos A}{\cos \frac{C-A}{2}}+\frac{\cos A \cos B}{\cos \frac{A-B}{2}} \leq \frac{3}{4}$$

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Mathematics & Youth: 2012 Issue 425
2012 Issue 425
Mathematics & Youth
https://www.molympiad.org/2020/09/2012-issue-425.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2012-issue-425.html
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