Difference between revisions of "G285 2021 Summer Problem Set"
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==Problem 4== | ==Problem 4== | ||
− | + | <math>16</math> people are attending a hotel conference, <math>8</math> of which are executives, and <math>8</math> of which are speakers. Each person is designated a seat at one of <math>4</math> round tables, each containing <math>4</math> seats. If executives must sit at least one speaker and executive, there are <math>N</math> ways the people can be seated. Find <math>\left \lfloor \sqrt{N} \right \rfloor</math>. Assume seats, people, and table rotations are distinguishable. | |
+ | |||
+ | <math>\textbf{(A)}\ 720 \qquad\textbf{(B)}\ 1440 \qquad\textbf{(C)}\ 2520 \qquad\textbf{(D)}\ 3456\qquad\textbf{(E)}\ 5760</math> | ||
− | + | [[G285 2021 Summer Problem Set Problem 4|Solution]] | |
==Problem 5== | ==Problem 5== | ||
Line 33: | Line 35: | ||
==Problem 6== | ==Problem 6== | ||
− | <math> | + | Let <math>ABCD</math> be a rectangle with <math>BC=6</math> and <math>AB=8</math>. Let points <math>M</math> and <math>N</math> lie on <math>ABCD</math> such that <math>M</math> is the midpoint of <math>BC</math> and <math>N</math> lies on <math>AD</math>. Let point <math>Q</math> be the center of the circumcircle of quadrilateral <math>MNOP</math> such that <math>O</math> and <math>P</math> lie on the circumcircle of <math>\triangle MNP</math> and <math>\triangle MNO</math> respectively, along with <math>OD \perp QO</math> and <math>MP \perp BP</math>. If the shortest distance between <math>Q</math> and <math>AB</math> is <math>3</math>, <math>\triangle AOQ</math> and <math>\triangle QBP</math> are degenerate, and <math>BP=AO</math>, find <math>25 \cdot OD \cdot PC</math> |
+ | |||
+ | <math>\textbf{(A)}\ 209 \qquad\textbf{(B)}\ 228 \qquad\textbf{(C)}\ 54\sqrt{57} \qquad\textbf{(D)}\ 90\sqrt{19} \qquad\textbf{(E)}\ 72\sqrt{57}</math> | ||
− | + | [[G285 2021 Summer Problem Set Problem 6|Solution]] | |
==Problem 7== | ==Problem 7== |
Revision as of 00:18, 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