$show=home

2010 Issue 399

  1. Find a four digits perfect square, given that all four digits are distinct, and if these digits are written in reverse order, the result is also a perfect square, and is divisible by the original number. 
  2. Determine all possible choices of three integers $x$, $y$ and $z$ such that $$x^{2}+y^{2}+z^{2}+3<x y+3 y+2 z.$$
  3. Let $A B C$ be a triangle where the length of the altitudes $A H$ is $6 \mathrm{cm}, B H$ is $3 \mathrm{cm}$ and the measure of angle $C A H$ is three times the measure of angle $B A H$. Find the area of this triangle. 
  4. Find the greatest value of the expression $$M=\frac{232 y^{3}-x^{3}}{2 x y+24 y^{2}}+\frac{783 z^{3}-8 y^{3}}{6 y z+54 z^{2}}+\frac{29 x^{3}-27 z^{3}}{3 x z+6 x^{2}}$$ where $x, y$ and $z$ are positive numbers satisfying the condition $x+2 y+3 z=\dfrac{1}{4}$ 
  5. Let $A B C$ be a right triangle with right angle at $A$ and $A B<A C$. Let $H$ be the projection of $A$ onto $B C$, let $M$ be the point reflection of $H$ across $A B$. $M C$ meets the circumcircle of triangle $A B H$ at $P$ $(P \neq M)$, $H P$ meets the circumcircle of triangle $A P C$ at $N$ $(N \neq P)$. Let $E$ and $K$ be respectively the intersections of $A B$ and $B C$ with the circumcircle of triangle $A P C$ $(E \neq A, K \neq C)$. Prove that
    a) $E N$ is parallel to $B C$.
    b) $H$ is the midpoint of $B K$. 
  6. Find the interger part of $A$, where $$A=\sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\ldots+2010 \sqrt{\frac{2010}{2009}}$$
  7. Find the least value of the following expression $$A=\frac{x}{1+y^{2}}+\frac{y}{1+x^{2}}+\frac{z}{1+t^{2}}+\frac{t}{1+z^{2}}$$ where $x, y, z, t$ are nonnegative real numbers satisfying $x+y+z+t=k$ ($k$ is a given positive number).
  8. Let $A B C$ be a given triangle and $M$ is a point which is not on its sides. Prove that $$P_{A /(M C B)}=P_{B /(M C A)}=P_{A /(M A B)}$$ if and only if $M$ is the centroid of triangle $A B C$. ($P_{T /(X Y Z))}$ is the power of the point $T$ with respect to the circle through $X$, $Y$, $Z$.)
  9. Let $A B C$ be a triangle. A circle intersects with the sides $B C$, $C A$ and $A B$ at pairs of two points $(M, N)$; $(P, Q)$ and $(S, T)$ respectively, where $M$ lies between $B$ and $N$; $P$ lies between $C$ and $Q,$ and $S$ lies between $A$ and $T$. Let $K$, $H$, $L$ be respectively the intersections of $S N$ and $Q M$; $Q M$ and $T P$; $T P$ and $S N$. Prove that the lines $A K$, $B H$, $C L$ are concurrent.
  10. Let $\left(a_{n}\right)$ be a sequence of numbers such that $$a_{0}=10,\quad \left(6-a_{n}\right)\left(16+a_{n-1}\right)=96,\, n=0,1,2, \ldots$$ Find the sum $$S=\frac{1}{a_{0}}+\frac{1}{a_{1}}+\ldots+\frac{1}{a_{2010}}$$
  11. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that the following equality holds $$f(x+y)+f(x y)=x+y+x y,\,\forall x, y \in \mathbb{R}^{+}.$$
  12. Let $A B C$ be a triangle. Prove that $$\cos A \cos B \cos C+8 \sqrt{3} \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} \geq 24 \cos ^{2} \frac{A}{2} \cos ^{2} \frac{B}{2} \cos ^{2} \frac{C}{2}-1$$

$type=three$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: 2010 Issue 399
2010 Issue 399
Mathematics & Youth
https://www.molympiad.org/2020/09/2010-issue-399.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2010-issue-399.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