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is not divisible by 
for any positive integer 
.
is called 
if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo for every 
.
be an acute-angled triangle with 
, orthocenter 
, circumcircle 
and circumcentre 
. Let 
be the midpoint of 
and let 
be a point such that 
is a parallellogram. Suppose that there exists a point 
on and on the opposite side of 
to 
such that 
. Let 
be the midpoint of 
. Prove that if 
, then 
.
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
and 
. It is clear that 
and 
are midlines of 
and 
respectively; whence, 
. Therefore the result follows by Jensen's inequality. In fact, we find the follow generalization
for all 
and the corollary result
for all 
and 
(where 
equals to the area of 
) Summing these 4 inequalities gives the desired result.
be a convex quadrilateral and 
be points on ![$[AB],[BC],[CD],[DA]$](http://latex.artofproblemsolving.com/e/7/a/e7a0e13d1bdc509f3d60c8daa3eebfda74541dac.png)
,![\[
\sqrt[3]{s_{1}}+\sqrt[3]{s_{2}}+\sqrt[3]{s_{3}}+\sqrt[3]{s_{4}}\leq 2\sqrt[3]{s}
\]](http://latex.artofproblemsolving.com/b/9/2/b924b280c0166a655512c9f2439f5196b03a537c.png)
,
,
,
and 
.
