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be a triangle with 
, 
, 
. Let 
, 
, 
be the midpoints of arcs 
, 
, 
(not containing the opposite vertices) respectively on the circumcircle of . Let 
be the midpoint of arc containing point 
. Suppose lines 
and 
meet at 
, while lines 
and 
meet at 
. Find the square of the distance from to 
. be an acute triangle with 
, and circumradius 
. Let have circumcenter 
, symmedian point 
, and nine-point center . Consider all non-degenerate hyperbolas 
with perpendicular asymptotes passing through 
. Of these , exactly one has the property that there exists a point 
such that 
is tangent to and 
. Let 
be the reflection of over . If 
meets 
at 
, then the length of 
can be expressed in the form 
, where 
are positive integers such that 
is not divisible by the square of any prime. Compute 
. be a triangle with 
, 
, and 
. Denote the reflections of 
across 
by 
, respectively, and let be the circumcenter of triangle . Let be a point such that 
, and let and be the midpoints of the major and minor arcs 
of the circumcircle of triangle . Find 
., 
, 
, and 
. Let 
be the point on segment satisfying 
, and let 
be the unique point such that 
and line 
is tangent to the circumcircle of . Find the length of segment ., 
, and 
be distinct points on a line with 
. Square 
and equilateral triangle 
are drawn on the same side of line . What is the degree measure of the acute angle formed by lines 
and 
?
such that when 
is written in base , it has at least trailing zeroes.
cookies, each labeled with a unique nonempty subset of 
. Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie labeled with 
, he eats that cookie as well as the cookies with 
and 
). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form 
, where 
and 
are relatively prime positive integers. Find 
. has sides 
, 
, and 
. Let 
be the points on segments 
, respectively, such that 
and 
. Suppose lines 
and intersect at 
and the circumcircles of 
and 
intersect at a second point 
. If the length of segment 
can be expressed in the form 
for positive integers 
, where is not divisible by the square of any prime, find 
.
, 
, be the distinct roots of the polynomial 
.
is monic and has distinct roots 
, 
, 
.? be real nonzero numbers such that 
and ![\[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}=1.\]](http://latex.artofproblemsolving.com/6/d/a/6da709be271e908a29f5a90b71e5ff8f499c6fbb.png)
Compute the largest possible value of 
.
is a solution to both equations in the integers.
is a solution to both equations in the integers.
Then, 
.
We have,
By QM-AM, 
,
.
.
variable function ![$$f(a,b,c,x,y):abc-a-2b+3c+x(a+b+c-12)+y[a^{-1}+b^{-1}+c^{-1}+(abc)^{-1}-1]=0.$$](http://latex.artofproblemsolving.com/2/a/e/2aefbef3b054e9f31a9742ac85825530b9a51e5c.png)
.
.
.
.
,
.
,
.
,
,
.
.
.
to the left side, and 12 to the right.
and permutations for 
.
and permutations for 
.
.
.
.
.![$2(3)(7)-[3+2(2)-3(-7)] = 42-(3+4-21) = 42-(-14) = 42+14 = \boxed{56}$](http://latex.artofproblemsolving.com/a/2/2/a2295561adebef8223e6101cb5c2261b884dee4f.png)
.
.