[/quote]
,
,
and 
be the circumcenter and orthocenter, respectively. Let 
and 
be the midpoints of 
and 
,
and 
be the feet of the perpendiculars from 
and 
to the opposite sides, respectively. Show that if 
is the intersection of 
and 
,
is perpendicular to 
.
.
.![$(a+b+c)^4(ab+bc+ca) - 9\sum{\frac{a}{c}} \ge 54[(a+b)(b+c)(c+a)+abc-1]$](http://latex.artofproblemsolving.com/d/3/c/d3cf439c79cb84ca23083e51b6190ab5118f34e8.png)
denote the sum of all integers 
such that 
and exactly one of 
and 
is divisible by 
.
.
.
.
and 
,
of positive integers that satisfy
.
![\[\sqrt x - \frac 1y - 2w + 3z = 1,\]](http://latex.artofproblemsolving.com/d/0/f/d0ff4819db851e6c12f738dcdb6b81bbd2ed0ca7.png)
![\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]](http://latex.artofproblemsolving.com/6/5/3/653c1547620f85ff761817a5cc333f16bce9c4c6.png)
![\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]](http://latex.artofproblemsolving.com/1/b/9/1b9c2ed383acc931aa3607ba82031e182e18f3d1.png)
![\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]](http://latex.artofproblemsolving.com/e/0/3/e032ce109cd74bbae47dc6bca04dcab2d8f9ef29.png)
and 
.
.
and 
intersect at point 
is the point on segment 
such that 
.
is isosceles.
![\[\sqrt{a+h_B}+\sqrt{b+h_C}+\sqrt{c+h_A}=\sqrt{a+h_C}+\sqrt{b+h_A}+\sqrt{c+h_B}\]](http://latex.artofproblemsolving.com/5/d/6/5d6881be29327bcdc3e7bd76c221968578c7f50b.png)
is isosceles.
,
,
are the roots of 
,
.
such that the set of prime divisors of the numbers 
and 
coincides.
be a median, and let 
and 
be the altitudes. Let 
be the line perpendicular to 
.
be the intersection point of 
be the circumscribed circle of the triangle 
, and let 
and 
be any two circles that have two points 
and 
in common, and are tangent to the segment 
are collinear.
and 
. Prove that
,such that 
P
Y by
You are given a 
grid with each cell colored with an integer from 
to 
. Two players take turns. On a turn, a player may increase any one cell’s value by a power of 2 (i.e., add 1, 2, 4, or 8 mod 16). The first player wins if, after their move, the sum of each row and the sum of each column is congruent to 0 modulo 16. Prove whether or not Player 1 has a forced win strategy from any starting configuration.
grid with each cell colored with an integer from 
to 
. Two players take turns. On a turn, a player may increase any one cell’s value by a power of 2 (i.e., add 1, 2, 4, or 8 mod 16). The first player wins if, after their move, the sum of each row and the sum of each column is congruent to 0 modulo 16. Prove whether or not Player 1 has a forced win strategy from any starting configuration.
