[/quote]
,
,
onto 
be a point on the extension of 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
intersect 
.
.
and 
.
.
be the point on 
by 
,
for every positive integer 
.
leave distinct remainders when divided by 
.
of distinct positive integers such that there exists a positive integer 
for which the number of divisors of 
and of 
are equal.
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
and 
be two sequences of integers such that 
and 
for all 
.
divides 
for an infinity of integers 
such that 
for all 
are coprime. Consider the sequence 
given by 
.
such that 
is even and 
is an integer. Prove that the numerator in its simplest form of 
is an odd integer greater than 
.
be an integer and let 
be 
.
such that the n subsets 
are also disjoint.
s
satisfies 
.
,
.
such that for all irrational numbers 
and 
,
where 
satisfy that 
.![$$\sum_{cyc} \sqrt[4]{\frac{a^3}{b+c}}$$](http://latex.artofproblemsolving.com/e/f/0/ef089c4f2915bbf75b0f60b044bbe9273b301a81.png)
Y by Mango247
Let 
be an acute-angled triangle and let 
be a point located inside the triangle . The lines 
and 
intersect the circumscribed circle of triangle again at points 
and 
, respectively. The circumscribed circle of the triangle 
again intersects the line 
at point 
. The circumscribed circle of the triangle 
intersects again the line 
at point 
. Prove that the quadrilateral 
is a parallelogram.
be an acute-angled triangle and let 
be a point located inside the triangle . The lines 
and 
intersect the circumscribed circle of triangle again at points 
and 
, respectively. The circumscribed circle of the triangle 
again intersects the line 
at point 
. The circumscribed circle of the triangle 
intersects again the line 
at point 
. Prove that the quadrilateral 
is a parallelogram.
