[/quote]
,
be an acute triangle with orthocenter 
and circumcenter 
.
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)
are
that can be written in the form![\[
n = \frac{a^2 - b^2}{b},
\]](http://latex.artofproblemsolving.com/f/9/0/f90a3ce2e1b51d6e1cb53a15cad6f36c2b30492a.png)
where 
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 
,
as 
,
as 
.
s
n
t
hats, numbered from 
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?
.![\[\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}.\]](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.
be the midpoint of side 
.
intersect 
at 
.
is tangent to 
,
.
divides 
Y by PikaPika999, Sedro
Hi people!
I have been and aops user for only a few months and not an active one either, just a user checking other posts and competitions. This is actually my first post. I am in 7th grade and only this year I have actually started locking in on competition math, before that, I didn’t really care about Mathcounts and AMCs. So, I hone my problem-solving skills by lightly doing some math problems. I have signed up for a lot of math competitions this month and am struggling to prepare properly with the resources I have. Do you guys have any resources you use to prepare for math competitions and that are actually helpful to use to prepare for competitions in a short amount of time. I am looking for resources that first teach you stuff and give you problems, like guided notes and handouts. I have few aops books like “Competition Math for Middle School” and “Intermediate Algebra”, but they’re not enough. I have learnt Algebra 1 and started doing high school geometry and a bit of Algebra 2 at home. Do let me know if you have any suggestions!
I have been and aops user for only a few months and not an active one either, just a user checking other posts and competitions. This is actually my first post. I am in 7th grade and only this year I have actually started locking in on competition math, before that, I didn’t really care about Mathcounts and AMCs. So, I hone my problem-solving skills by lightly doing some math problems. I have signed up for a lot of math competitions this month and am struggling to prepare properly with the resources I have. Do you guys have any resources you use to prepare for math competitions and that are actually helpful to use to prepare for competitions in a short amount of time. I am looking for resources that first teach you stuff and give you problems, like guided notes and handouts. I have few aops books like “Competition Math for Middle School” and “Intermediate Algebra”, but they’re not enough. I have learnt Algebra 1 and started doing high school geometry and a bit of Algebra 2 at home. Do let me know if you have any suggestions!
