$show=home

2009 Issue 381

  1. Rearrange the following rational numbers in increasing order $$\frac{1005}{2002}, \frac{1007}{2006}, \frac{1009}{2010}, \frac{1011}{2014}$$
  2. In an isosceles triangle $A B C$ (at vertex $A$ ), choose a point $M$ such that $$\widehat{M A C}=\widehat{M B A}=\widehat{M C B}.$$ Compare the areas of the triangles $A B M$ and $C B M$.
  3. Determine the sum of all rational numbers of the form $\dfrac{a}{b}$ where $a$, $b$ are natural divisors of $27000$ and $\gcd(a, b)=1$.
  4. Given $a$, $b$, $c$ such that $a>0$, $b>c$, $a^{2}=b c$, $a+b+c=a b c$. Prove the inequalities $$a \geq \sqrt{3},\quad b \geq \sqrt{3},\quad 0<c \leq \sqrt{3}.$$
  5. Let $ABCD$ be a quadrilateral where $\widehat{A B C}=\widehat{A D C}=90^{\circ}$ and $\widehat{B C D}<90^{\circ}$. Choose a point $E$ on the opposite ray of $A C$ such that $D A$ is the angle-bisector of $B D E$. Let $M$ be chosen arbitrarily between $D$ and $E$, choose another point $N$ on the opposite ray of $B E$ such that $\widehat{N C B}=\widehat{M C D}$. Prove that $M C$ is the angle bisector of $D M N$.
  6. Prove that the equation $x^{3}+3 y^{3}=5$ has infinitely many rational solutions.
  7. If $a$, $b$, $c$ are the length of the sides of a triangle, prove that $$\frac{1}{\sqrt{a b+a c}}+\frac{1}{\sqrt{b c+b a}}+\frac{1}{\sqrt{c a+c b}} \geq \frac{1}{\sqrt{a^{2}+b c}}+\frac{1}{\sqrt{b^{2}+a c}}+\frac{1}{\sqrt{c^{2}+a b}}.$$
  8. Let $A B C D . A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a parallelepiped and let $S_{1}$, $S_{2}$, $S_{3}$ denote the areas of the sides $A B C D$, $A B B^{\prime} A^{\prime}$ and $A D D^{\prime} A^{\prime}$ respectively. Given that the sum of squares of the areas of all sides of the tetrahedron $A B^{\prime} C D^{\prime}$ equals $3$, find the smallest possible value of the following expression $$T=2\left(\frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{3}}\right)+3\left(S_{1}+S_{2}+S_{3}\right)$$

$type=three$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: 2009 Issue 381
2009 Issue 381
Mathematics & Youth
https://www.molympiad.org/2020/09/2009-issue-381.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2009-issue-381.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