AoPS Academy
satisfying
for all integers 
.
be a cyclic quadrilateral. A circle centered at 
passes through 
and 
and meets lines 
and 
again at points 
and 
(distinct from 
). Let 
denote the orthocenter of triangle 
Prove that if lines 
are concurrent, then triangle 
and 
are similar.
and 
be positive integers. For a connected simple graph 
on vertices and edges, we consider the number 
of orientations of (all of) its edges so that, in the resulting directed graph, every vertex has even outdegree. only depends on and , and determine its value.
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that Prove that
Equality holds iff 
or 
and any cyclic permutations.
be non-negative real numbers such that Prove that
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that![\[\sum a^2+8\sum ab\ge 3\sum a\sum\sqrt{ab}.\]](http://latex.artofproblemsolving.com/2/b/a/2bac168124825dbc152f3085a0d3723fa34caf47.png)
Changing 
it becomes a fourth degree inequality, so is linear in 
. Thus it remains to check only the cases 
and 
which is easy.
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
be positive real variables with 
Prove that
Equality holds iff 
be non-negative real variables with 
Prove thatEquality holds iff or where ![\[\frac{12a(a+2b)}{4ab+bc+ca} \geqslant \frac{32a^3+3(33b+56c)a^2+3(26b^2+102bc+13c^2)a-4(4b+c)(b-2c)^2}{11[ab(a+b)+bc(b+c)+ca(c+a)]+51abc}.\]](http://latex.artofproblemsolving.com/a/b/0/ab0f41dc6c9cf3e143885b3e7e7079048d3841c3.png)
be three non-negative real numbers with Prove that
When does equality hold?
be three non-negative real numbers with 
Prove that
When does equality hold?
and 
.
and 
.
,
and 
be non-negative numbers such that 
.
and set 
The inequality becomes
ưhere
which ends the proofs 