2023 SSMO Team Round Problems
Contents
Problem 1
Let be a permutation of
. Find the largest possible value of
Problem 2
A plane and a car start both move northward. The car moves northbound at 60 miles per hour. The plane moves northeast and increases in altitude at an angle of Let
the speed in feet per second that the plane must fly at to move north at the same speed as the car. Find
.
Problem 3
Let be a triangle such that
and
Let
be the circumcircle of
. Let
be on the circle such that
Let
be the point diametrically opposite of
. Let
be the point diametrically opposite
. Find the area of the quadrilateral
in terms of a mixed number
. Find
.
Problem 4
Find the sum of values for prime such that
Problem 5
Joshy is playing a game with a dartboard that has two sections. If Joshy hits the first section, he gets points, and if he hits the second section, he gets
points. Assume Joshy always hits one of the two sections. Let
be the maximum value that Joshy cannot achieve. Let
be the number of positive integer scores Joshy cannot achieve. Let
be the number of ways for Joshy to achieve
points. Find
.
Problem 6
Suppose that are positive reals satisfying
Find the sum of all possible values of
If you believe there are no solutions, put
as your answer. If you believe the sum is infinity, put
as your answer.
Problem 7
Let and let there be randomly chosen sets
where
. The probability that
can be expressed as
. Let
be the largest power of
such
. Find
.
Problem 8
Three rabbits run away from the origin at the same speed and constant velocity such that the angle between any two rabbits' directions is . After
seconds, a hunter with a speed
times that of the rabbits runs from the origin. Let the minimal time in seconds needed for her to meet (and subsequently) catch all three rabbits be
. Find
.
Problem 9
Let ,
, and
be the total number of possible moves for a bishop, knight, or rook from any position of a
by
grid.
Find
.
(A bishop moves along diagonals, a rook moves along rows, and a knight moves in the form
of a "L" shape)
Problem 10
There exists a lane of infinite cars. Each car has a chance of being high quality and a
chance of being low quality. John goes down the row of cars buying high-quality cars. However, after John sees 3 low-quality cars, he gives up on buying additional cars. Let the probability that he buys at least
cars before giving up as
. Find
.
Problem 11
Let be a cyclic quadrilateral such that
is the diameter. Let
be the orthocenter of
. Define
, and
. If
,
, and
, suppose
Find
.
Problem 12
Let be the set of rationals of the form
for nonnegative
and
. Define the function
such that, for
such
is minimal, we have that
Suppose
equals
. Find
.
Problem 13
Let denote the product of all divisors of
Let
denote the set of all integers that are both a multiple of
and a factor of
Let
\[
-F(a) = \sqrt{\left|\log_{10}\left(\frac{D(10^{a})}{\prod_{\omega\in P(10^2,10^{a+2})}\omega}\right)\right|}\text{ and }G(n) = \sqrt[n-1]{\prod_{i=2}^{n}10^{F(i)}}.
\]
\sum_{k=2}^{\infty}G(k)
\frac{a+b\sqrt{c}}{d}
a+b+c+d$.
Problem 14
Find the sum of all perfect squares of the form where
and
are positive integers such
is prime and
.
Problem 15
Consider a piece of paper in the shape of a regular pentagon with sidelength We fold it in half. We then fold it such that the vertices of the longest side become the same side. The area of the folded figure can be expressed as
where
are integers and
is squarefree. Find
(For convenience, note that
)