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2010 Issue 391

  1. Does there exist a positive integer $k$ such that $2^{k}+3^{k}$ is a perfect square?
  2. Let $A B C$ be a triangle and let $M$ be the midpoint of $B C$ such that $A B=5cm$,$ A M=6cm$ and $A C=13cm$. The line through $B$ and perpendicular to $B C$ meets $A M$ at $D,$ the line through $C$ and perpendicular to $B C$ meets $A B$ at $E$. Prove that $C D$ is perpendicular to $M E$
  3. Find all pair of real numbers $a$ and $b$ so that $a+b=\dfrac{\sqrt[4]{8}}{2}$ and $A=a^{4}-6 a^{2} b^{2}+b^{4}$ is a positive integer.
  4. Let $O$ be the midpoint of a line segment $A B=2 a$. In the half-plane with edge $A B$, draw two rays $A x$, $B y$, both perpendicular to $A B$. Choose $M$ and $N$ on $A x$ and $B y$ respectively such that $M N=A M+B N$. Let $H$ be the foot of the altitude from $O$ onto $M N$. Find the positions of $M$ and $N$ such that the area of the triangle $H A B$ is greatest possible.
  5. Without using trigonometry formula, prove the following equalities
    a) $\cos 36^{\circ} \cdot \cos 72^{\circ}=\dfrac{1}{4}$.
    b) $\tan 36^{\circ} \cdot \tan 72^{\circ}=\sqrt{5}$.
  6. Solve the equation $$3 x^{4}-4 x^{3}=1-\sqrt{\left(1+x^{2}\right)^{3}}$$
  7. Does there exist a polynomial $P(x)$ of degree $2010$ such that $P\left(x^{2}-2010\right)$ is divisible to $P(x)$?.
  8. Let $Oxyz$ be a right trihedral with right angle at $O$ and $A$, $B$, $C$ move on the sides $O x$, $O y$ and $O z$ respectively so that the area of the triangle $A B C$ is a constant $S$. Let $S_{1}$, $S_{2}$, $S_{3}$ be the areas of the triangles $O B C$, $OCA$, $OAB$ respectively. Find the greatest value of the expression $$P=\frac{S_{1}}{S+2 S_{1}}+\frac{S_{2}}{S+2 S_{2}}+\frac{S_{3}}{S+2 S_{3}}$$
  9. Let $a, b, c$ be positive real numbers. Prove that $$\min \left\{\frac{a b}{c^{2}}+\frac{b c}{a^{2}}+\frac{c a}{b^{2}} ; \frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}\right\} \geq \max \left\{\frac{a}{b}+\frac{b}{c}+\frac{c}{a} ; \frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right\}$$
  10. For a given positive integer $n$, how many $n$ -digit natural numbers can be formed from five possible digits $1,2,3,4$ and $5$ so that an odd numbers of $1$ and even numbers of $2$ are used?
  11.  sequence $\left(x_{n}\right)$, $n=0,1, \ldots$ is given by $$x_{0}=\alpha,\quad x_{n}=\sqrt{1+\frac{1}{x_{n}+1}},\, n=0,1, \ldots$$ where $\alpha$ is greater than $1$. Determine $\displaystyle\lim_{n\to\infty} x_{n}$.
  12. Let $A B C$ be a triangle with the altitudes $A A^{\prime}$, $B B^{\prime}$, $C C^{\prime}$ meet at $H$. Prove that $$\frac{H A}{H A^{\prime}}+\frac{H B}{H B^{\prime}}+\frac{H C}{H C^{\prime}}+6 \sqrt{3} \geq 6+\frac{a}{H A^{\prime}}+\frac{b}{H B^{\prime}}+\frac{c}{H C^{\prime}}$$

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Mathematics & Youth: 2010 Issue 391
2010 Issue 391
Mathematics & Youth
https://www.molympiad.org/2020/09/2010-issue-391.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2010-issue-391.html
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