AoPS Academy
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, 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.
, 
, 
, 
be non-negative real numbers such that![\[2a+2b+2c+2d+ab+bc+cd+da+3=abcd.\]](http://latex.artofproblemsolving.com/d/e/8/de8ccd816cfe032a97d940982192b1abea21dc18.png)
prove that : ![\[\sqrt[4]{abc}+\sqrt[4]{bcd}+\sqrt[4]{cda}+\sqrt[4]{dab}\le\sqrt[4]{27(1+a)(1+b)(1+c)(1+d)}.\]](http://latex.artofproblemsolving.com/a/f/c/afc556cd8f5c7f4dedf153ffb92fb1f978c56823.png)
be a given triangle with circumcenter 
and orthocenter 
. Let 
and 
be the feet of the perpendiculars from 
and 
to the opposite sides, respectively. Let 
be the reflection of 
with respect to 
. Prove that 
is a cyclic quadrilateral.
and 
be positive integers, and let 
be a 
grid of unit squares.
or 
rectangle. A subset 
of grid squares in is domino-tileable if dominoes can be placed to cover every square of exactly once with no domino extending outside of . Note: The empty set is domino tileable. to the upper-right corner of formed by exactly 
edges of the grid squares. and , the number of up-right paths that divide into two domino-tileable subsets.
such that the conditions
for all 
and 
for all 
where 
is the Euler's totient function. and be positive integers. Find all pairs of non-negative integers and that always satisfy the following condition: white dots and black dots on a circle, there always exist a line cutting the circle into two arcs, one of which consists of exactly white dots and black dots.
times as many students as Lara’s high school. The two high schools have a total of
students. How many students does Portia’s high school have?
divides 
for positive integers 
then
and 
Prove that
Where 
such that 
is also prime.
)
