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2008 Issue 375

  1. Find a natural number $x$ and two decimal digits $y$, $z$ such that $$\left(5.10^{n}-2\right) x=3 . \overline{y \ldots y z}$$ for any natural number $n>1,$ where $\overline{y \ldots y z}$ (in the decimal system) contains $n-1$ digits $y$.
  2. Prove that for any $x$ and $y$  $$\frac{|x|}{2008+|x|}+\frac{|y|}{2008+|y|} \geq \frac{|x-y|}{2008+|x-y|}$$
  3. Consider an integer $n>2008$ such that both $2 n-4015$ and $3 n-6023$ are perfect squares. Find the remainder of the division of $n$ by 40
  4. Solve the quation $$\sqrt{x-\frac{1}{x}}-\sqrt{1-\frac{1}{x}}=\frac{x-1}{x}.$$
  5. Let $A B C$ be an isosceles triangle with the apex angle at $A$ and $\widehat{B A C}=150^{\circ}$. Construct the triangles $A M B$ and $A N C$ such that the rays $A M$ and $A N$ lie in the angle $B A C$ and $\widehat{A B M}=\widehat{A C N}=90^{\circ}$, $\widehat{M A B}=30^{\circ}$, $\widehat{N A C}=60^{\circ} .$ Let $D$ be a point on $M N$ such that $N D=3 M D$. $B D$ intersects with $A M$ and $A N$ at $K$ and $E,$ respectively. $B C$ and $A N$ meets at $F$. Prove that
    a) $NCE$ is an isosceles triangle;
    b) $K F$ and $C D$ are parallel.
  6. Find all pairs of integers $m$ and $n$, both greater than $1$ such that the following equality $$a^{m+n}+b^{m+n}+c^{m+n}=\frac{a^{m}+b^{m}+c^{m}}{m} \cdot \frac{a^{n}+b^{n}+c^{n}}{n}$$ is true for all real numbers $a$, $b$, $c$ satisfying $a+b+c=0$.
  7. Let $k$ be a positive integer and let $a$, $b$, $c$ be positive real numbers such that $a b c \leq 1$. Prove the equality $$\frac{a}{b^{k}}+\frac{b}{c^{k}}+\frac{c}{a^{k}} \geq a+b+c.$$
  8. Let $C$ be a point on a fixed circle whose diameter is $A B=2 R$ ($C$ is different from $A$ and $B$). The incircle of $A B C$ touches $A B$ and $A C$ at $M$ and $N$, respectively. Find the maximum value of the length of $M N$ when $C$ moves on the given fixed circle.
  9. Let $\left(x_{n}\right)$ $(n=0.1,2, \ldots)$ be a sequence such that $$x_{0}=2,\quad x_{n+1}=\frac{2 x_{n}+1}{x_{n}+2},\,\forall n=0,1,2, \ldots.$$ Determine $\displaystyle \left[\sum_{k=1}^{n} x_{k}\right]$ where $[x]$ denote the largest integer not exceeding $x$.
  10. Prove that if $a$, $b$, $c$ are positive numbers whose product $a b c=1,$ then $$\frac{a}{\sqrt{8 c^{3}+1}}+\frac{b}{\sqrt{8 a^{3}+1}}+\frac{c}{\sqrt{8 b^{3}+1}} \geq 1.$$
  11. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(0)=0$ and $\dfrac{f(t)}{t}$ is a monotonic function on $\mathbb{R} \backslash\{0\}$. Prove that $$x \cdot f\left(y^{2}-z x\right)+y \cdot f\left(z^{2}-x y\right)+z \cdot f\left(x^{2}-y z\right) \geq 0$$ for all positive numbers $x$, $y$ and $z$.
  12. Let $A_{1} A_{2} A_{3} A_{4}$ be a tetrahedron. Denote by $B_{i}$ $(i=1,2,3,4)$ the feet of the altitude from a given point $M$ onto $A_{i} A_{i+1}$ (where we consider $A_{5}$ as identical to $A_{1}$). Find the smallest value of $\displaystyle\sum_{1 \leq i \leq 4} A_{i} A_{i+1} . A_{i} B_{i}$

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Mathematics & Youth: 2008 Issue 375
2008 Issue 375
Mathematics & Youth
https://www.molympiad.org/2020/09/2008-issue-375.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2008-issue-375.html
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