2012 Issue 421

  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 $\Delta I_{A} I_{B} I_{C} \sim \Delta A B C$.




    Mathematics & Youth: 2012 Issue 421
    2012 Issue 421
    Mathematics & Youth
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