Arcticturn Prep
Problem 5
Suppose that ,
, and
are complex numbers such that
,
, and
, where
. Then there are real numbers
and
such that
. Find
.
Problem 6
A real number is chosen randomly and uniformly from the interval
. The probability that the roots of the polynomial
are all real can be written in the form
, where
and
are relatively prime positive integers. Find
.
Problem 9
Octagon with side lengths
and
is formed by removing 6-8-10 triangles from the corners of a
rectangle with side
on a short side of the rectangle, as shown. Let
be the midpoint of
, and partition the octagon into 7 triangles by drawing segments
,
,
,
,
, and
. Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.
Problem 13
Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is where
and
are relatively prime positive integers. Find
.
Problem 6
Let be the number of complex numbers
with the properties that
and
is a real number. Find the remainder when
is divided by
.