2023 SSMO Relay Round 4 Problems
Problem 1
Martha starts out with the number on her calculator, which has three buttons that multiply the current number by
,
, and
respectively. She randomly presses one of the buttons four times. After these
presses, she cubes the number. The expected value of the final number is
for relatively prime positive integers
and
Find
Problem 2
Let TNYWR. Let
Suppose that
points are chosen on the sides of a triangle with area 1 such that there is at least one point on each side. Let
be the area of the polygon formed by connecting the
points in counterclockwise order. Find the expected value of
(Note that
)
Problem 3
Let TNYWR.
numbers are chosen from the set
with replacement. If the probability that the median of these
numbers is greater than
is
such that the decimal representation of
has
's before the first nonzero digit of it, find
rounded to nearest multiple of