[/quote]
,
be the number of positive integers that divide the integer 
.
of 
,
?
is a square of an integer.
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
let 
be the circumcircle of 
and let 
be its radius. Point 
is chosen on 
such taht 
and point 
is the foot of the perpendicular from 
.
mets 
.
is chosen on line 
such that 
and 
lie on a line in that order. Finally, let 
be a point satisfying 
and 
.
has two real roots, so that the one is the square of the other.
such that: ![\[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]](http://latex.artofproblemsolving.com/e/c/f/ecf7fb3c2805290f2cb07b7fdd7c73e1c7a55b99.png)
be a convex pentagon such that 
.
is the circumcenter of triangle 
.
be the circumcenter of triangle 
.
passes through the midpoint of segment 
.
.
and 
meet at a point belonging to the diameter 
and 
are the incenter and the 
-
on the circumcircle, and 
.
intersects the circumcircle again at 
. For
on the circumcircle of triangle 
intersect 
. Prove
are concyclic.
.
are the midpoints of 
respectively. 
be the incenter and tangent of angles 
respectively. Prove that 
have a common point
Y by Adventure10, Rounak_iitr
Let 
be a right triangle and isosceles, with 
. Let 
be the midpoint of 
and 
the midpoint of 
. Let 
be such that 
is an equilateral triangle with inside the quadrilateral 
. Calculate the measure of 
be a right triangle and isosceles, with 
. Let 
be the midpoint of 
and 
the midpoint of 
. Let 
be such that 
is an equilateral triangle with inside the quadrilateral 
. Calculate the measure of 
This post has been edited 1 time. Last edited by parmenides51, Sep 27, 2021, 5:39 PM
Reason: latex
Reason: latex
.
.
and 
.
,
