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2011 Issue 405

  1. Which of the following two numbers is greater? $$A=\frac{326}{1955}+\frac{988}{1975}+\frac{662}{1985},\quad B=\dfrac{3951}{3950}+\dfrac{1}{5955}+\dfrac{1}{11730}.$$
  2. Let $A B C$ be an isosceles triangle with $\widehat{B A C}=96^{\circ} .$ A point $M$ is inside the triangle such that $\widehat{M B C}=12^{\circ}$, $\widehat{M C B}=24^{\circ} .$ Prove that $M A=M C$.
  3. Find the maximum value of the expression $P=\max \{a, b, c\}-\min \{a, b, c\}$ where $a, b, c$ are real numbers satisfying the condition $$a+b+c=a^{3}+b^{3}+c^{3}-3 a b c=2.$$
  4. Solve the equation $$21 x-25+2 \sqrt{x-2}=19 \sqrt{x^{2}-x+2}+\sqrt{x+1}$$
  5. $A B C$ is a triangle inscribed the circle $(O)$ with $\widehat{B A C}=60^{\circ}$, $A K$ is the angle-bisector of $\widehat{B A C}$ ($K$ is on the circle $(O)$). Let $F$ be the midpoint of $A K,$ the ray $O F$ meets the altitude $C E$ of triangle $A B C$ at $H .$ Prove that $B H$ is perpendicular to $A C$.
  6. Find the minimum value of the expression $$P=\left(5 a+\frac{2}{b+c}\right)^{3}+\left(5 b+\frac{2}{c+a}\right)^{3}+\left(5 c+\frac{2}{a+b}\right)^{3}$$ where $a, b, c$ are positive real numbers satisfying $a^{2}+b^{2}+c^{2}=3$
  7. $OABC$ is a trirectangular tetrahedron at vertex $O$. $O H$ is the altitude from $O$ of tetrahedron. Let $R$ be the circumradius of triangle $A B C .$ Prove that $O H \leq \dfrac{R \sqrt{2}}{2} .$ When does equality occur?
  8. a) Find all distinct permutations of the word $TOANHOCTUOITRE$.
    b) How many permutations are there that has three consecutive $T - TTT$?
    c) How many permutations are there without adjacent $T$s?
  9. Let $k$ be a positive integer, $\alpha$ is an arbitrary real number. Find the limit of sequence $\left(a_{n}\right)$ where $$a_{n}=\frac{\left[1^{k} \cdot \alpha\right]+\left[2^{k} \cdot \alpha\right]+\ldots+\left[n^{k} \cdot \alpha\right]}{n^{k+1}},\, n=1,2, \ldots$$ here the notation $[x]$ is the largest integer that does not exceed $x .$
  10. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfy the following conditions
    • $f$ is strictly increasing;
    • $f(f(n))=4 n+9$ for all $n \in \mathbb{N}^{*}$
    • $f(f(n)-n)=2 n+9$ for all $n \in \mathbb{N}^{*}$
  11. Does there exist a positive integer $n \geq 2$ so that $$f(x)=1+4 x+4 x^{2}+\ldots+4 x^{2 n}$$ is a perfect square polynomial?
  12. Let $A B C$ be a triangle inscribed the circle $(O)$ and $A^{\prime}$ is a fixed point on $(O)$. $P$ moves on $B C$, $K$ belongs to $A C$ so that $P K$ is always parallel to a fixed line $d$. The circumcircle of triangle $A P K$ cuts the circle $(O)$ at a second point $E$. $A E$ cuts $B C$ at $M$. $A^{\prime} P$ cuts the circle $(O)$ at a second point $N .$ Prove that the line $M N$ passes through a fixed point.

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Mathematics & Youth: 2011 Issue 405
2011 Issue 405
Mathematics & Youth
https://www.molympiad.org/2020/09/2011-issue-405.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2011-issue-405.html
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