be a convex quadrilateral and points 
on sides 
such that![\[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\]](http://latex.artofproblemsolving.com/b/4/2/b42cd6305fab848a60c79689f17bdb55e1ba0d31.png)
is the area of 
show that 
and can teleport to other Cartesian coordinates with only 
of 
moves: 
,
when 
,
,
when 
.
be any polynomial with positive integer coefficients that passes through 
.
,
denote the increasing arithmetic sequence of integers whose first term is 
is the sequence 
.
?
is the set of positive real solutions to the system![\[
x^3 + y^3 + z^3 = x + y + z
\]](http://latex.artofproblemsolving.com/7/7/a/77a748a8b55a6b44761bced9bb1031193b6127dc.png)
![\[
x^2 + y^2 + z^2 = xyz
\]](http://latex.artofproblemsolving.com/9/3/a/93a4dffc6fd17b9f19f738d3b3865f56aa0151a9.png)
then find the number of elements in 
satisfy![\[
a + \frac{1}{a + 2015} = b - 4030 + \frac{1}{b - 2015}
\]](http://latex.artofproblemsolving.com/8/c/0/8c0b0cfdd569ea250e97590a3364794920d86e04.png)
and 
.![\[
\frac{ab}{2015} - a + b.
\]](http://latex.artofproblemsolving.com/1/f/6/1f620dea7a0c769137bf13004916e8b283e00eee.png)
,
.
,
is a regular tetrahedron.
with respect to a circle 
,
,
numebr on his list? Write its last two digits as your answer.
grid of square tiles. Robert chooses four of these squares uniformly at random. The probability that the centers of these four squares form a square themselves is 
where 
are coprime naturals. Write the largest 
digit odd factor of 
.
.
which touches one face of the cube and passes through all 
are of the form 
,
,
,
be the greatest common divisor of 
and 
.
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 
