Difference between revisions of "G285 2021 Summer Problem Set"
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<math>\textbf{(A)}\ -1 \qquad\textbf{(B)}\ -\frac{1}{2} \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ \frac{3}{8} \qquad\textbf{(E)}\ 1</math> | <math>\textbf{(A)}\ -1 \qquad\textbf{(B)}\ -\frac{1}{2} \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ \frac{3}{8} \qquad\textbf{(E)}\ 1</math> | ||
− | [[G285 2021 Summer Problem Set Problem | + | [[G285 2021 Summer Problem Set Problem 2|Solution]] |
+ | ==Problem 3== | ||
+ | <math>60</math> groups of molecules are gathered in a lab. The scientists in the lab randomly assign the <math>60</math> molecules into <math>5</math> groups of <math>12</math>. Within these groups, there will be <math>5</math> 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. | ||
− | + | <math>\textbf{(A)}\ 5^{60} \qquad\textbf{(B)}\ \frac{60!\cdot 5^{60}}{(12!)^4} \qquad\textbf{(C)}\ \frac{60!\cdot 5^{30}}{(12!)^4} \qquad\textbf{(D)}\ \frac{40!\cdot 5^{60}}{11!(12!)^3} \qquad\textbf{(E)}\ 60!5^{60}</math> | |
− | |||
− | + | [[G285 2021 Summer Problem Set Problem 3|Solution]] | |
==Problem 4== | ==Problem 4== | ||
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<math>\textbf{(A)}\ 67 \qquad\textbf{(B)}\ 69 \qquad\textbf{(C)}\ 70 \qquad\textbf{(D)}\ 71 \qquad\textbf{(E)}\ 72</math> | <math>\textbf{(A)}\ 67 \qquad\textbf{(B)}\ 69 \qquad\textbf{(C)}\ 70 \qquad\textbf{(D)}\ 71 \qquad\textbf{(E)}\ 72</math> | ||
− | [[G285 Summer Problem Set Problem 8|Solution]] | + | [[G285 2021 Summer Problem Set Problem 8|Solution]] |
==Problem 9== | ==Problem 9== | ||
− | <math> | + | Let circles <math>\omega_1</math> and <math>\omega_2</math> with centers <math>Q</math> and <math>L</math> concur at points <math>A</math> and <math>B</math> such that <math>AQ=20</math>, <math>AL=28</math>. Suppose a point <math>P</math> on the extension of <math>AB</math> is formed such that <math>PQ=29</math> and lines <math>PQ</math> and <math>PL</math> intersect <math>\omega_1</math> and <math>\omega_2</math> at <math>C</math> and <math>D</math> respectively. If <math>DC=\frac{16\sqrt{37}}{\sqrt{145}}</math>, the value of <math>\sin^2(\angle LAQ)</math> can be represented as <math>\frac{m \sqrt{n}}{r}</math>, where <math>m</math> and <math>r</math> are relatively prime positive integers, and <math>n</math> is square free. Find <math>2m+3n+4r</math> |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 28 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 39 \qquad\textbf{(D)}\ 45 \qquad\textbf{(E)}\ 54</math> |
− | [[G285 Summer Problem Set|Solution]] | + | [[G285 2021 Summer Problem Set Problem 9|Solution]] |
==Problem 10== | ==Problem 10== |
Revision as of 00:13, 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
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
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