[/quote]
,
be an acute triangle with 
,
be the projection from 
onto 
.
past 
.
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 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
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 
Y by
In the triangle the 
, consider altitude 
. The points 
are the centers of the circles inscribed in the triangles 
respectively . The line 
intersects the segments 
and 
at the points 
and 
. Prove that the triangle 
is isosceles if and only if the triangle is isosceles or right.
, consider altitude 
. The points 
are the centers of the circles inscribed in the triangles 
respectively . The line 
intersects the segments 
and 
at the points 
and 
. Prove that the triangle 
is isosceles if and only if the triangle is isosceles or right.This post has been edited 1 time. Last edited by parmenides51, Aug 15, 2020, 11:23 AM
