AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
+3 w
be non-negative real numbers such that 
Prove that
+1 w
be an acute triangle with orthocenter 
and circumcenter 
. Let 
and 
be points on the line segments 
and 
respectively such that 
is a parallelogram. Prove that 
.
be the set of positive real numbers. Determine all functions 
such that ![\[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]](http://latex.artofproblemsolving.com/e/7/1/e71b6e0d0b858eb46b81149f1e6be8c41e13d301.png)
for every 
.
, 
and 
be pairwise distinct nonnegative real numbers. Prove that![\[
(a + b + c) \left( \frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2} \right) > 4.
\]](http://latex.artofproblemsolving.com/2/0/9/209f9807481e7a1e97f023bf65a603cb462c62cb.png)
(Karl Czakler)
that can be written in the form![\[
n = \frac{a^2 - b^2}{b},
\]](http://latex.artofproblemsolving.com/f/9/0/f90a3ce2e1b51d6e1cb53a15cad6f36c2b30492a.png)
where and are positive integers.. Initially, the numbers 
are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers 
and 
are on the board, then 
can be erased as 
, or 
as 
, or 
as 
.) is it possible to end with only one number remaining on the board?
st sees the other three, the 
nd sees the rd and 
th, and the rd sees only the th. The devil has 
hats, numbered from to , he puts one hat on each wise man, and hides the extra 
hats. After that, each wise man (in turn: first the first, then the second, etc.) loudly calls a number, trying to guess the number of his hat. The numbers mentioned should not be repeated. When all the wise men have spoken, they take off their hats and check which one of them has guessed. Can the sages to act in such a way that at least three of them knowingly guessed?
, and are such that 
. Prove that ![\[\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}.\]](http://latex.artofproblemsolving.com/3/e/4/3e4b336137b17e55bf172d45bd18fe6d5561a827.png)
be real numbers such that 
.![\[
3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c
\]](http://latex.artofproblemsolving.com/5/a/8/5a83eaa7470bcca6546599ff4e296e4efec69b16.png)
and determine all the cases when the equality occurs.
, let 
be the midpoint of side 
. Let the line perpendicular to at point 
intersect 
at 
. If 
is tangent to at 
, find 
.
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
