$show=home

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

    $type=three$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

    Name

    2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,4,Anniversary,4,
    ltr
    item
    Mathematics & Youth: 2012 Issue 421
    2012 Issue 421
    Mathematics & Youth
    https://www.molympiad.org/2020/09/2012-issue-421.html
    https://www.molympiad.org/
    https://www.molympiad.org/
    https://www.molympiad.org/2020/09/2012-issue-421.html
    true
    8958236740350800740
    UTF-8
    Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy