[/quote]
,
be positive real numbers.![\[\frac{ab}{a^2+3b^2}+\frac{bc}{b^2+3c^2}+\frac{ca}{c^2+3a^2}\leq \frac 34.\]](http://latex.artofproblemsolving.com/0/3/9/0396d1d0fe304dc632c7c209282cac784da85017.png)
be an acute triangle with 
and let 
and 
be its orthocenter and circumcenter, respectively. Let 
be the circle 
.
and the circle of radius 
cross 
and 
,
and circle 
are concurrent.
.
problems in a meeting. For each 
,
-
so that the chair can ensure that the longest idle time for any leader does not exceed 
and 
.
Where 
table is filled with numbers from 
to 
as shown in the figure. Is it possible to rearrange some numbers so that there is still one number in each cell, and so that the sum of the numbers does not change in all rectangles of three cells?
![\[ a^2+b^2+c^2=a^2b^2.\]](http://latex.artofproblemsolving.com/9/7/3/973246fd8d710bccf27fefe9ee8a81952122e085.png)
and 
are positive integers such that ![\[ 1 = \underbrace{\varphi( \varphi( \dots \varphi(}_{k\ \text{times}} n) \dots )). \]](http://latex.artofproblemsolving.com/b/6/4/b64fa5fbf6613768c78beb3ab2b712599bd7ea80.png)
Prove that 
.
denotes Euler's totient function, i.e. 
which are relatively prime to 
.
be the set of nonzero real numbers. Find all functions 
such that, for all 
,![\[(x-y)f(y^2)+f\left(xy\,f\left(\frac{x^2}{y}\right)\right)=f(y^2f(y)).\]](http://latex.artofproblemsolving.com/b/1/b/b1b1b156221baf8e9936027f352c50abcd2d05ff.png)
![$\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq \sqrt[n]{\frac{a^n + b^n}{2}} + \sqrt[n]{\frac{b^n + c^n}{2}} + \sqrt[n]{\frac{c^n + a^n}{2}}$](http://latex.artofproblemsolving.com/6/0/6/606cb6c4a384c641c90618f271d93b9a923ec8b8.png)
Y by
Let 
be a convex quadrilateral and 
an arbitrary point in its interior. Let also 
be the symmetric point of with respect to the midpoint 
of the side 
and let 
be the symmetric point of with respect to the midpoint 
of the side 
. Prove that 
.
be a convex quadrilateral and 
an arbitrary point in its interior. Let also 
be the symmetric point of with respect to the midpoint 
of the side 
and let 
be the symmetric point of with respect to the midpoint 
of the side 
. Prove that 
.
