2009 Issue 380

  1. Compare $\dfrac{1}{6}$ with $$A=\frac{1}{5}-\frac{1}{7}+\frac{1}{17}-\frac{1}{31}+\frac{1}{65}-\frac{1}{127}$$
  2. Let $ABC$ be an isosceles triangle (at vertex $C$) with $\widehat{A C B}=100^{\circ}$. $M$ is a point chosen on the ray $C A$ such that $C M=A B$. Find the measure of the angle $\widehat{C M B}$. 
  3. Find all triple $(x, y, z)$ of integers such that $$\begin{cases}y^{3} &=x^{3}+2 x^{2}+1 \\ x y &=z^{2}+2\end{cases}$$
  4. Find all pair of numbers $x$ and $y$ such that the following conditions hold $x>1$, $0<y<1$, $x+y \leq \sqrt{5}$, $\dfrac{1}{x}+\dfrac{1}{y} \leq \sqrt{5}$ and $\dfrac{x}{x+1}+\dfrac{y}{1-y} \leq \sqrt{5}$.
  5. Let $A B C D$ be an isosceles trapezoid inscribed inside a circle $\left(O_{1}: R\right)$ and circumscribes the circle $\left(O_{2} ; r\right)$. Let $d=O_{1} O_{2}$. Prove that $$\frac{1}{r^{2}} \geq \frac{2}{R^{2}+d^{2}} .$$ When does equality occur? 
  6. Prove that in any triangle $A B C$, the following inequality holds $$\frac{\cot A \cdot \cot B \cdot \cot C}{\sin A \cdot \sin B \cdot \sin C} \leq\left(\frac{2}{3}\right)^{3}$$ 
  7. There are $17$ ornament betel-nut trees around a circular pond. How many ways are there to chopped off $4$ trees with the condition that no two consecutive trees be removed? 
  8. Solve the following system of equations with parameter $a$ $$\begin{cases} 2 x\left(y^{2}+a^{2}\right) &=y\left(y^{2}+9 a^{2}\right) \\ 2 y\left(z^{2}+a^{2}\right) &=z\left(z^{2}+9 a^{2}\right) \\ 2 z\left(x^{2}+a^{2}\right) &=x\left(x^{2}+9 a^{2}\right)\end{cases}$$
  9. Let $\left(x_{n}\right)$, $n=0,1,2, \ldots$ be a sequence given by the following recursive formula $$x_{0}=a,\quad x_{n+1}=x_{n}+\sin x+2 \pi,\, n=0,1,2, \ldots$$ where $a, x \in \mathbb{R}$. Prove that the limit $\displaystyle\lim_{n \rightarrow+\infty} \frac{x_{1}+\ldots+x_{n}}{n^{2}}$ exists and find its exact value. 
  10. Prove that
    a) $\displaystyle\sum_{k=0}^{n}\left(\frac{k}{n}-x\right)^{2} C_{n}^{k} x^{k}(1-x)^{n-k}=\frac{x(1-x)}{n},\, \forall x \in \mathbb{R}$
    b) $\displaystyle\sum_{k=0}^{n}\left|\frac{k}{n}-x\right|C_{n}^{k} x^{k}(1-x)^{n-k} \leq \frac{1}{2 \sqrt{n}},\, \forall x \in[0 ; 1]$. 
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f\left(n^{2}\right)=f(m+n) f(n-m)+m^{2},\, \forall m, n \in \mathbb{R}$$
  12. On a given plane, choose a point $A$ outside a circle whose center is at a point $O$ and whose radius equals $R$. A chord $M N$ with constant length moves on the circle such that the two line segments $M N$ and $O A$ always intersect. Determine the positions of $M$ and $N$ such that the sum $A M+A N$ is
    a) greatest possible.
    b) smallest possible.




Mathematics & Youth: 2009 Issue 380
2009 Issue 380
Mathematics & Youth
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