2011 Issue 404

  1. Given seven distinct prime numbers $a$, $b$, $c$, $a+b+c$, $a+b-c$, $a-b+c$, $-a+b+c$ in which the sum of two of three numbers $a$, $b$, $c$ equals $800$. Let $d$ be the difference between the largest and the smallest number among these seven integers. What is the maximum value of $d ?$
  2. A triangle $A B C$ has sides $A B=2cm$, $A C=4cm$ and median $A M=\sqrt{3}cm$. Find the measure of the angle $B A C$, the length of side $B C$ and the area of triangle $A B C$.
  3. Find the largest natural number $k$ so that $n^{5}-2011 n$ is divisible by $k$ for all natural number $n$.
  4. Solve the equation $$\left(x^{4}-625\right)^{2}-100 x^{2}-1=0$$
  5. Let $A B C$ be an acute triangle $(A B \neq A C)$ inscribed in the circle $(O),$ and $H$ is its orthocenter. Let $d$ be an arbitrary line passing through $H$. Draw the line $d^{\prime}$ symmetric to $d$ through $B C$. Find the position of the line $d$ so that $d^{\prime}$ touches the circumcircle $(O)$.
  6. Find the largest constant $k$ such that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a} \geq k\left(a^{2}+b^{2}+c^{2}\right)$$ for all positive real numbers $a$, $b$, and $c$ whose sum equals $1 .$
  7. Let $A B C$ be a triangle inscribed in the circle $(O)$, and let $G$ be its centroid; $D$, $E$, and $F$ are the circumcenters of triangles $G B C$, $G C A$, $G A B$ respectively. Prove that $O$ is the centroid of $D E F$.
  8. Solve the equation $$\sqrt[3]{\cos 5 x+2 \cos x}-\sqrt[3]{2 \cos 5 x+\cos x}=2 \sqrt[3]{\cos x}(\cos 4 x-\cos 2 x).$$
  9. Let $x$, $y$, $z$ be real numbers such that $x \geq 1$, $y \geq 2$, $z \geq 3$ and $$\frac{1}{x+\sqrt{x-1}}+\frac{2}{y+\sqrt{y-2}}+\frac{3}{z+\sqrt{z-3}}=12$$ Find the maximum and minimum value of the function $f(x, y, z)=x+y+z$.
  10. Let $\left(x_{n}\right)$ be a sequence given by $$x_{1}=\frac{5}{2},\quad x_{n+1}=\sqrt{x_{n}^{3}-12 x_{n}+\frac{20 n+21}{n+1}},\,\forall n \in \mathbb{N}^{*}.$$ Prove that the sequence $\left(x_{n}\right)$ converges and find its limit.
  11. Find all functions $f: \mathbb{R} \rightarrow(0 ; 2011]$ such that $$f(x) \leq 2011\left(2-\frac{2011}{f(y)}\right),\,\forall x>y.$$
  12. Given four points $A_{I}$ $(i=1,2,3,4),$ no three of them are colinear and a point $M$ so that $A_{i}$ $(i=1,2,3,4)$ and $M$ do not lie on the same circle. Let $T_{i}$ be a triangle having $A_{j}$ $(j=1,2,3,4 ; j \neq i)$ as its vertices, $C_{i}$ is the circle (or the line) passing through the feet of the projections through $M$ onto three sides (or extended sides) of triangle $T_{i}$. Prove that $C_{I}$ $(i=1,2,3,4)$ have a common point.




Mathematics & Youth: 2011 Issue 404
2011 Issue 404
Mathematics & Youth
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