# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 1. Compare$\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. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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