$show=home

2011 Issue 410

  1. Find all pairs of coprime positive integers $x, y$ so that $$\frac{x+y}{x^{2}+y^{2}}=\frac{7}{25}$$
  2. Let $A B C$ be an equilateral triangle whose altitudes $A H$, $B K$ intersect at $G .$ The angle-bisector of angle $B K H$ meets $C G$, $A H$, $B C$ at $M$, $N$, $P$ respectively. Prove that $K M=N P$.
  3. Find the minimum value of the expression $S=2011 c a-a b-b c$ where $a, b, c$ satisfy $a^{2}+b^{2}+c^{2} \leq 2$.
  4. Let $A B C$ be an isosceles right triangle with right angle at $A .$ Let $M$, $N$, $O$ be respectively the midpoints of $A B$, $A C$, $B C$. The line perpendicular to $C M$ from $O$ cuts $M N$ at $G .$ Compare the lengths of the two segments $G M$ and $G N$.
  5. Solve the equation $$\sqrt{7 x^{2}+25 x+19}-\sqrt{x^{2}-2 x-35}=7 \sqrt{x+2}$$
  6. Let $A B C$ be a triangle. Let $A M$, $B N$, $C P$ be its internal angle-bisectors ($M \in B C$, $N \in C A$, $P \in A B$). Find the measure of angle $B A C$ so that $P M$ is perpendicular to $N M$
  7. Solve the equation $$(\sin x-2)\left(\sin ^{2} x-\sin x+1\right)=3 \sqrt[3]{3 \sin x-1}+1.$$
  8. Find all values of $a, b$ so that the equation $$x^{4}+a x^{3}+b x^{2}+a x+1=0$$ has at least one solution and the sum $a^{2}+b^{2}$ is smallest possible.
  9. Let $a, b, c$ be positive numbers. Prove that $$\left(a^{2012}-a^{2010}+3\right)\left(b^{2012}-b^{2010}+3\right)\left(c^{2012}-c^{2010}+3\right) \geq 9(a b+b c+c a).$$ When does equality occur?
  10. Let $A B C$ be a triangle. An arbitrary line cuts the lines $B C$, $C A$, $A B$ at $M$, $N$, $P$ respectively. Let $X$, $Y$, $Z$ be respectively the centroids of triangles $A N P$, $B P M$, $C M N$. Prove that $$S_{X Y Z}=\frac{2}{9} S_{A B C}$$
  11. Let $\left(a_{n}\right)$ $\left(n \in \mathbb{N}^{*}\right)$ be a sequence given by $$a_{1}=0 ,\, a_{2}=38,\, a_{3}=-90,\quad a_{n+1}=19 a_{n-1}-30 a_{n-2},\, \forall n \geq 3.$$ Prove that $a_{2011}$ is divisible by 2011 .
  12. For all positive integers $n$ greater than $2$. Find the number of functions $$f:\{1,2,3, \ldots, n\} \rightarrow\{1,2,3,4,5\}$$ satisfying $|f(k+1)-f(k)| \geq 3$ where $k \in\{1,2, \ldots, n-1\}$.

$type=three$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

2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
ltr
item
Mathematics & Youth: 2011 Issue 410
2011 Issue 410
Mathematics & Youth
https://www.molympiad.org/2020/09/2011-issue-410.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2011-issue-410.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