Geometry :3c
by popop614, Apr 3, 2025, 12:19 AM
Quadrilateral 
has an incenter 
Suppose 
. Let 
be the midpoint of 
. Suppose that 
. 
meets 
again at point 
. Let points 
and 
be such that is the midpoint of 
and is the midpoint of 
. Point 
lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.
The angle bisectors of
and 
meet 
at 
and 
. Prove that 
.
has an incenter 
Suppose 
. Let 
be the midpoint of 
. Suppose that 
. 
meets 
again at point 
. Let points 
and 
be such that is the midpoint of 
and is the midpoint of 
. Point 
lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.The angle bisectors of
and 
meet 
at 
and 
. Prove that 
.This post has been edited 1 time. Last edited by popop614, 2 hours ago
Reason: asfdasdf
Reason: asfdasdf
be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:
Choose a number of the form 
,
a non-negative integer, and place it in an empty square.
Choose two (not necessarily consecutive) squares containing the same number, say 
and erase the number in the other square.
,
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![\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]](http://latex.artofproblemsolving.com/c/8/9/c895ae7fdf8d7f284ac9fc94cc077d6edad6cbf0.png)
be a triangle. Let 
and 
be the intersections of the 
and 
angle bisectors with the opposite sides. Let 
.
where 
is the major 
arc midpiont.
and 
again at 
or 
and 
concur on 
and 
such that 
is a prime number.
that satisfy the equation
,
,
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
,
is an integer. Prove that there is a positive integer 
for all 
.
,
for 
as
Determine all values of 
such that there exists a number 
such that 
for infinitely many values of 
and 
intersect at points 
is tangent to 
(on 
(on 
.
and 
through 
and 
.
lies on line 
,
.
April 2021
February 2021