2010 Issue 393

  1. Let $a_{1}, a_{2}, \ldots, a_{2010}$ be natural numbers such that $$\frac{1}{a_{1}^{11}}+\frac{1}{a_{2}^{11}}+\frac{1}{a_{3}^{11}}+\ldots+\frac{1}{a_{2010}^{11}}=\frac{1005}{1024}.$$ Determine the value of the following expression $$A=\frac{a_{2010}^{6}}{a_{1}^{5}}+\frac{a_{2009}^{6}}{a_{2}^{5}}+\frac{a_{2008}^{6}}{a_{3}^{5}}+\ldots+\frac{a_{1}^{6}}{a_{2010}^{5}}$$
  2. In a triangle $A B C$ where $B$ and $C$ are acute angles, let $B D$ and $A H$ be respectively the angle bisector and the altitude. Given that $\widehat{A D B}=\widehat{A H D}=\alpha,$ find the measure of $\alpha$.
  3. Find all integer solutions of the equation $$y^{3}=x^{6}+2 x^{4}-1000$$
  4. Find the minimum value of the expression $$P=\frac{a}{b+c+d-a}+\frac{b}{c+d+a-b}+\frac{c}{d+a+b-c}+\frac{d}{a+b+c-d}$$ where $a, b, c, d$ are the length of 4 sides of a convex quadrilateral.
  5. Let $A B C$ be an isosceles triangle with vertex at $C$. Let $O$, $I$ be its circumcenter and incenter respctively. $D$ is a point chosen on the side $B C$ so that $D O$ is perpendicular to $B I$. Prove that $D I$ is parallel to $A C$.
  6. Let $a, b, c$ be positive real numbers satisfying the condition $$\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1} \geq 1.$$ Prove that $$a+b+c=a b+b c+c a$$
  7. Let $A B C D E$ be a cyclic pentagon with $A C \parallel D E$ and $\widehat{A M B}=\widehat{B M C}$ where $M$ is the midpoint of $B D$. Prove that the line $B E$ passes through the midpoint of $A C$.
  8. Let $O A B C$ be a tetrahedron with right trihedral angle at vertex $O$. Prove that $$\cot A B \cdot \cot B C+\cot B C \cdot \cot C A+\cot C A \cdot \cot A B \leq \frac{3}{2}$$ where cot $A B$ is the cotangent of the dihedral angle of side $A B$.
  9. Let $A B C$ be an acute triangle. The altitudes $B K$ and $C L$ meet at $H$. The line passing through $H$ meets $A B$, $A C$ at $P$, $Q$ respectively. Prove that $H P=H Q$ if and only if $M P=M Q$ where $M$ is the midpoint of $B C$
  10. Find all real numbers $k$ and $m$ such that $$k\left(x^{3}+y^{3}+z^{3}\right)+m x y z \geq(x+y+z)^{3}$$ for any non-negative numbers $x, y, z$
  11. Let $\left(x_{n}\right)$ be a sequence of real numbers, $n=1,2, \ldots$ satisfying $$\ln \left(1+x_{n}^{2}\right)+n x_{n}=1$$ for any positive integers $n$. Find $$\lim _{n \rightarrow+\infty} \frac{n\left(1+n x_{n}\right)}{x_{n}}$$
  12. Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x+f(y))=2 y+f(x),\, \forall x, y \in \mathbb{R}$$




Mathematics & Youth: 2010 Issue 393
2010 Issue 393
Mathematics & Youth
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