G285 2021 Summer Problem Set
Welcome to the Birthday Problem Set! In this set, there are multiple choice AND free-response questions. Feel free to look at the solutions if you are stuck:
Contents
Problem 1
Find
Problem 2
Let If
is a positive integer, find the sum of all values of
such that
for some constant
.
Problem 3
Let circles and
with centers
and
concur at points
and
such that
,
. Suppose a point
on the extension of
is formed such that
and lines
and
intersect
and
at
and
respectively. If
, the value of
can be represented as
, where
and
are relatively prime positive integers, and
is square free. Find
Problem 4
Let be a rectangle with
and
. Let points
and
lie on
such that
is the midpoint of
and
lies on
. Let point
be the center of the circumcircle of quadrilateral
such that
and
lie on the circumcircle of
and
respectively, along with
and
. If the shortest distance between
and
is
,
and
are degenerate, and
, find
Problem 5
Suppose is an equilateral triangle. Let points
and
lie on the extensions of
and
respectively such that
and
. If there exists a point
outside of
such that
, and there exists a point
outside outside of
such that
, the area
can be represented as
, where
and
are squarefree,. Find
Problem 6
people are attending a hotel conference,
of which are executives, and
of which are speakers. Each person is designated a seat at one of
round tables, each containing
seats. If executives must sit at least one speaker and executive, there are
ways the people can be seated. Find
. Assume seats, people, and table rotations are distinguishable.
Problem 7
Geometry285 is playing the game "Guess And Choose". In this game, Geometry285 selects a subset of not necessarily distinct integers from the set
such that the sum of all elements in
is
. Each distinct is selected chronologically and placed in
, such that
,
,
, and so on. Then, the elements are randomly arranged. Suppose
represents the total number of outcomes that a subset
containing
integers sums to
. If distinct permutations of the same set
are considered unique, find the remainder when
is divided by
.
Problem 8
Let , Let
be the twelve roots that satisfies
, find the least possible value of
Problem 9
groups of molecules are gathered in a lab. The scientists in the lab randomly assign the
molecules into
groups of
. Within these groups, there will be
distinguishable labels (Strong acid, weak acid, strong base, weak base, nonelectrolyte), and each molecule will randomly be assigned a label such that teams can be empty, and each label is unique in the group. Find the number of ways that the molecules can be arranged by the scientists.
Problem 10
Let for
. Suppose
makes
for distinct prime factors
. If
for
is
where
must satisfy that
is an integer, and
is divisible by the
th and
th triangular number. Find