2022 SSMO Speed Round Problems
Contents
[hide]Problem 1
Bobby is bored one day and flips a fair coin until it lands on tails. Bobby wins if the coin lands on heads a positive even number of times in the sequence of tosses. Then the probability that Bobby wins can be expressed in the form , where
and
are relatively prime positive integers. Find
.
Problem 2
A bag is big enough to hold exactly 6 large pencils, 12 medium pencils, or 30 small pencils, with no space left over. Given that there is 1 large pencil and 3 medium pencils currently in the bag, what is the maximum number of small pencils that may be added to the bag? Note that there may still be space left over in the bag.
Problem 3
Let be a parallelogram such that
is a point on
such that
Suppose that
and
intersect at
If the area of triangle
is
find the area of
.
Problem 4
Consider a quadrilateral with area
and satisfying
. There exists a point
in 3D space such that the distances from
to
,
,
, and
are all equal to
. Find the volume of
.
Problem 5
Let be a square such that
is on
and
is on
If
and
then the value of
can be expressed as
, where
and
are relatively prime positive integers. Find
.
Problem 6
At the beginning of day , there is a single bacterium in a petri dish. During each day, each bacterium in the petri dish divides into
new bacteria, and
bacteria are added to the petri dish (these bacteria do not divide on the day they were added). For example, at the end of day
, there are
bacteria in the petri dish. If, at the end of day
, the number of bacteria in the petri dish is a multiple of
, find the minimum possible value of
.
Problem 7
Let . Define
as the image of
under a rotation of either
,
, or
clockwise about the origin, with each choice having a
chance of being selected. Find the expected value of the smallest positive integer
such that
coincides with
.
Problem 8
How many positive integers cannot be written as , where
,
, and
are positive integers (not necessarily distinct)?
Problem 9
Consider a triangle such that
,
,
and a square
such that
and
lie on
,
lies on
, and
lies on
. Suppose further that
,
,
, and
are distinct from
,
, and
. Let
be the center of
. If
intersects
at
, then the sum of all values of
can be expressed as
, where
and
are relatively prime positive integers. Find
.
Problem 10
Let be the set of all numbers for which the
element in
is the sum of the
triangular number and the
positive perfect square. Let
be the set which contains all unique remainders when the elements in
are divided by
. Find the number of elements in
.