AoPS Academy
fair coins are flipped. If 
of them land heads, find the expected value of 
.
have angles 
and 
such that 
and 
. Moreover, suppose that the product of the side lengths of the triangle is equal to its area. Let 
denote the circumcircle of 
. Let 
intersect 
at 
, 
intersect 
at 
, and 
intersect 
at 
. If the area of 
can be written as 
for relatively prime integers 
and 
and squarefree 
, find the sum of all prime factors of , counting multiplicities (so the sum of prime factors of 
is 
), given that has 
divisors.
and 
, find the area of the shaded region.
be a rectangle. Points 
and are chosen on 
and 
respectively such that there exists a point G such that quadrilateral 
is a rectangle. If 
, 
, and 
, then 
can be expressed in the form 
, where 
and 
are relatively prime positive integers. Find 
.
, it is known that 
is acute and 
. Prove that the unit circles with centers 
cover if and only if 
.
