are positive real numbers and 
.
.
is given in the plane. The incircle of triangle 
has center 
and is tangent to diagonal 
at 
.
be the center of parallelogram 
and 
are parallel.
is valid while 
is not)
of positive integers such that 
is an integer.
for which 
is a perfect square.
,
,
away. What is the least number of moves that Carina can make in order for triangle 
in the coordinate plane where 
and 
are both integers, not necessarily positive.)
and incenter 
meets 
at 
and meets 
cuts 
.
and 
meet at 
,
is the midpoint of 
.
and 
intersect at points 
and 
.
or 
.
dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: 
be the number of distinct maximal grid-aligned configurations obtainable from 
that satisfy the equation
Y by GeoKing, Rounak_iitr
Two circles with centers 
and 
intersect at 
and 
. Let 
be the circumcircle of the triangle 
; the circle intersect the circles centered at and at points 
and 
, respectively. The point is inside triangle 
and 
intersects at 
. The point 
on is such that 
. Let 
and 
meet at 
, prove that 
.
Proposed by Amir Parsa Hoseini Nayeri, Iran
and 
intersect at 
and 
. Let 
be the circumcircle of the triangle 
; the circle intersect the circles centered at and at points 
and 
, respectively. The point is inside triangle 
and 
intersects at 
. The point 
on is such that 
. Let 
and 
meet at 
, prove that 
.Proposed by Amir Parsa Hoseini Nayeri, Iran
This post has been edited 1 time. Last edited by NJAX, May 31, 2024, 12:34 PM
,
,
and 
is tangent to 
.
.
be the circumcenter of 
.
Claim: 
is concyclic
Claim: 
is tangent to the circumcircle of 
as 
.
.
be the symmetric point of 
,
be the symmetric point of 
