Difference between revisions of "G285 2021 Summer Problem Set"
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==Problem 12== | ==Problem 12== | ||
− | Suppose the function < | + | Suppose the function <cmath>P(a,b,c)=a^2b^4+b^2c^4+c^2a^4+8c+8b+8a+8a^3+8b^3+8c^3-3\sqrt[3]{abc}-21</cmath>. If <math>P(a)+P(b)+P(c)=P(a,b,c)=P(k)</math>, and <math>P(k)</math> contains the points <math>(4,6)</math>,<math>(8,11)</math>, and <math>(15,29)</math>, find the smallest value of <math>P(23)</math> for which <math>P(P(P(a,b,c))=abc(P(a)+P(b)+P(c))</math> |
[[G285 2021 Summer Problem Set Problem 12|Solution]] | [[G285 2021 Summer Problem Set Problem 12|Solution]] |
Revision as of 00:56, 26 June 2021
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
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 4
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 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
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 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
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 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
Problem 11
Let a recursive sequence be defined such that
, and
. Find the last
digits of
Problem 12
Suppose the function . If
, and
contains the points
,
, and
, find the smallest value of
for which