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

## $show=home 1. Given the sum of$2012$terms $$S=\frac{1}{5}+\frac{2}{5^{2}}+\frac{3}{5^{3}}+\frac{4}{5^{4}}+\ldots+\frac{2012}{5^{2012}}$$ Compare$S$with$\dfrac{1}{3}$. 2. Let$A B C$be a triangle with$\widehat{A B C}=40^{\circ}, \widehat{A C B}=30^{\circ} .$Outside this triangle, construct triangle$A D C$with$\widehat{A C D}=\widehat{C A D}=50^{\circ} .$Prove that the triangle$B A D$is isosceles. 3. Find all natural numbers$a, b, c$such that$c < 20$and$a^{2}+a b+b^{2}=70 c$. 4. Find the largest possible value of the expression $$P=\sqrt{1-\frac{x}{y+z}}+\sqrt{1-\frac{y}{z+x}}+\sqrt{1-\frac{z}{x+y}}$$ where$x, y, z$are side lengths of a triangle. 5. Given a circle$(O),$with a fixed chord$B C$.$A$is a point moving on the line$B C$, outside the circle$(O)$.$AM$and$AN$are the tangent lines to circle$(O)(M, N \in (O))$. The line through$B$and parallel to$A M$meets$M N$at$E .$Prove that the circumcircle of triangle$B E N$always passes through two fixed points when point$A$moves on the line$B C$. 6. Given that$\dfrac{1}{3} < x \leq \dfrac{1}{2}$and$y \geq 1$. Find the minimum value of $$P=x^{2}+y^{2}+\frac{x^{2} y^{2}}{((4 x-1) y-x)^{2}}.$$ 7. Let$\left(a_{n}\right)$be a sequence of positive real numbers, given by •$a_{0}=1$, •$a_{m}<a_{n}$, for all$m, n \in \mathbb{N}$,$m<n$. •$a_{n}=\sqrt{a_{n+1} \cdot a_{n-1}}+1$and$4 \sqrt{a_{n}}=a_{n+1}-a_{n-1}$for all$n \in \mathbb{N}^{*}$. Determine the sum$T=a_{0}+a_{1}+a_{2}+\ldots+a_{2012}$. 1. The base of a triangular prism$A B C \cdot A^{\prime} B^{\prime} C^{\prime}$is an equilateral triangle with side lengths$a$and the lengths of its adjacent sides also equal$a$. Let$I$be the midpoint of$A B$and$B^{\prime} I \perp(A B C)$. Find the distance from$B^{\prime}$to the plane$\left(A C C^{\prime} A^{\prime}\right)$in term of$a$. 2. Find all polynomials$P(x)$with real coefficients satisfying $$P^{2}(x)-1=4 P\left(x^{2}-4 x+1\right)$$ 3. Find$\alpha, \beta$so that the largest value of $$y=|\cos x+\alpha \cos 2 x+\beta \cos 3 x|$$ is smallest possible. 4. Let$A B C$be a triangle with side lengths$a$,$b$and$c$. Let$S$and$p$be respectively the area and the semiperimeter of this triangle. Prove the inequality $$\frac{1}{a^{2}(p-a)^{2}}+\frac{1}{b^{2}(p-b)^{2}}+\frac{1}{c^{2}(p-c)^{2}} \geq \frac{9}{4 S^{2}}$$ 5. Given an acute triangle$A B C$inscribed the circle$(O)$with$B C>C A>A B$. On the circle$(O)$, select six distinct points$M$,$N$,$P$,$Q$,$R$and$S$(which are also distinct from the vertices of triangle$A B C$) so that$Q B=B C=C R$,$S C=C A=A M$and$N A=A B=B P$. Let$I_{A}$,$I_{B}$and$I_{C}$be the incenters of triangles$A P S$,$B N R$and$C M Q$respectively. Prove that$\Delta I_{A} I_{B} I_{C} \sim \Delta A B C$. ##$type=three$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: 2012 Issue 421
2012 Issue 421
Mathematics & Youth