Difference between revisions of "G285 2021 Summer Problem Set"
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\end{cases}</cmath> If <math>y</math> is a positive integer, find the sum of all values of <math>x</math> such that <math>f(x,y) \neq k</math> for some constant <math>k</math>. | \end{cases}</cmath> If <math>y</math> is a positive integer, find the sum of all values of <math>x</math> such that <math>f(x,y) \neq k</math> for some constant <math>k</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 2|Solution]] | ||
==Problem 3== | ==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== | ||
− | + | <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)}\ 5760\qquad\textbf{(E)}\ 6172</math> | ||
− | + | [[G285 2021 Summer Problem Set Problem 4|Solution]] | |
==Problem 5== | ==Problem 5== | ||
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<math>\textbf{(A)}\ 152 \qquad\textbf{(B)}\ 162 \qquad\textbf{(C)}\ 164 \qquad\textbf{(D)}\ 214\qquad\textbf{(E)}\ 224</math> | <math>\textbf{(A)}\ 152 \qquad\textbf{(B)}\ 162 \qquad\textbf{(C)}\ 164 \qquad\textbf{(D)}\ 214\qquad\textbf{(E)}\ 224</math> | ||
+ | |||
+ | [[G285 2021 Summer Problem Set Problem 5|Solution]] | ||
==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== | ||
− | Geometry285 is playing the game "Guess And Choose". In this game, Geometry285 selects a subset of not necessarily distinct integers <math>P=\{a,b,c \cdots \}</math> from the set <math>S=\{1,2,3,4 \cdots k-1,k \}</math> such that the sum of all elements in <math>P</math> is <math>k</math>. Each distinct is selected chronologically and placed in <math>P</math>, such that <math>1 \le a \le k</math>, <math>1 \le b \le a</math>, <math>1 \le c \le b</math>, and so on. Then, the elements are randomly arranged. Suppose <math>S_{p,k}</math> represents the total number of outcomes that a subset <math>P</math> containing <math>p</math> integers sums to <math>k</math>. If distinct permutations of the same set <math>P</math> are considered unique, find the remainder when <cmath>\sum_{p=1}^{1000}\sum_{k=1}^{1000} S_{ | + | Geometry285 is playing the game "Guess And Choose". In this game, Geometry285 selects a subset of not necessarily distinct integers <math>P=\{a,b,c \cdots \}</math> from the set <math>S=\{1,2,3,4 \cdots k-1,k \}</math> such that the sum of all elements in <math>P</math> is <math>k</math>. Each distinct is selected chronologically and placed in <math>P</math>, such that <math>1 \le a \le k</math>, <math>1 \le b \le a</math>, <math>1 \le c \le b</math>, and so on. Then, the elements are randomly arranged. Suppose <math>S_{p,k}</math> represents the total number of outcomes that a subset <math>P</math> containing <math>p</math> integers sums to <math>k</math>. If distinct permutations of the same set <math>P</math> are considered unique, find the remainder when <cmath>\sum_{p=1}^{1000}\sum_{k=1}^{1000} S_{p,k}</cmath> is divided by <math>100</math>. |
<math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 51 \qquad\textbf{(E)}\ 124</math> | <math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 51 \qquad\textbf{(E)}\ 124</math> | ||
− | [[G285 Summer Problem Set Problem 7|Solution]] | + | [[G285 2021 Summer Problem Set Problem 7|Solution]] |
==Problem 8== | ==Problem 8== | ||
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==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)}\ 28 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 39 \qquad\textbf{(D)}\ 45 \qquad\textbf{(E)}\ 54</math> | ||
+ | |||
+ | [[G285 2021 Summer Problem Set Problem 9|Solution]] | ||
+ | |||
+ | ==Problem 10== | ||
+ | Let <math>k \in \mathbb{N}</math> for <math>k>1</math>. Suppose <math>\lfloor \omega_k \rfloor</math> makes <math>k=(p_1p_2p_3 \cdots p_e)^1</math> for distinct prime factors <math>p</math>. If <math>\tau(p)</math> for <math>p>1</math> is <cmath>\sum_{j=1}^{e} p_j</cmath> where <math>p_j</math> must satisfy that <math>\frac{\lfloor \omega_k \rfloor}{p_j}</math> is an integer, and <math>p_j</math> is divisible by the <math>p</math>th and <math>(p-1)</math>th triangular number. Find <math>\tau(3)+\tau(4)+\tau(5)+ \cdots +\tau(99)+\tau(100)</math> | ||
+ | |||
+ | <math>\textbf{(A)}\ 1024 \qquad\textbf{(B)}\ 1331 \qquad\textbf{(C)}\ 1539 \qquad\textbf{(D)}\ 2000 \qquad\textbf{(E)}\ 2719</math> | ||
+ | |||
+ | [[G285 2021 Summer Problem Set Problem 10|Solution]] | ||
+ | |||
+ | ==Problem 11== | ||
+ | Let a recursive sequence <math>a_n</math> be defined such that <math>a_1=20</math>, and <math>a_n=16a_{n-1}+4</math>. Find the last <math>3</math> digits of <math>a_{100}</math> | ||
+ | |||
+ | [[G285 2021 MC-IME I Problem 1|Solution]] | ||
+ | |||
+ | ==Problem 12== | ||
+ | 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 the polynomial <math>P(k)</math> contains the points <math>(P(k),P(k)+1)</math>,<math>(P(k)+3,P(k)+5)</math>, and <math>(P(k)+8,11)</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]] | ||
+ | |||
+ | ==Problem 13== | ||
+ | Let circles <math>O_1</math>,<math>O_2</math>, and <math>O_3</math> concur at <math>E</math>, where <math>EP</math> is the common chord shared by <math>\{O_1,O_3 \}</math>, <math>QE</math> is the common chord shared by <math>\{O_1,O_2 \}</math>, and <math>E</math> lies on the common internal tangent of <math>\{O_2,O_3 \}</math>. Let the extension of <math>PE</math> and <math>QE</math> intersect <math>O_2</math> and <math>O_3</math> again at <math>F</math> and <math>G</math> respectively. If <math>\overline{CF} \cap \overline{BG} \in D</math>, prove <math>ABDC</math> is a parallelogram. | ||
+ | |||
+ | [[G285 2021 Summer Problem Set Problem 13|Solution]] | ||
+ | |||
+ | ==Problem 14== | ||
+ | Bobby the frog is hopping around the unit circle. Suppose Bobby starts at <math>(1,0)</math>. After every <math>n</math>th minute, Bobby moves to <math>(a,bi)</math> such that <math>a^2+b^2 \le 1</math>, and <math>(a,bi)</math> is an <math>n</math>th root of unity for <math>n>1</math>. Suppose Bobby is unidirectional for every <math>3</math> minutes, and randomly chooses to reverse his direction after each cycle. In how many ways can Bobby travel around the unit circle exactly <math>6</math> times? | ||
+ | |||
+ | [[G285 2021 Summer Problem Set Problem 14|Solution]] | ||
− | <math>\ | + | ==Problem 15== |
+ | Find the average of all values <math>z</math> such that <cmath>\sum_{n=1}^{119} \prod_{j=1}^{7} (z^j)^{n} = \left(\sum_{p=1}^{60} z^{2p-1}-\sum_{n=1}^{59} z^{2n} \right)^{5040}+2</cmath> | ||
− | [[G285 Summer Problem Set|Solution]] | + | [[G285 2021 Summer Problem Set Problem 15|Solution]] |
Latest revision as of 23:28, 28 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 the polynomial
contains the points
,
, and
, find the smallest value of
for which
Problem 13
Let circles ,
, and
concur at
, where
is the common chord shared by
,
is the common chord shared by
, and
lies on the common internal tangent of
. Let the extension of
and
intersect
and
again at
and
respectively. If
, prove
is a parallelogram.
Problem 14
Bobby the frog is hopping around the unit circle. Suppose Bobby starts at . After every
th minute, Bobby moves to
such that
, and
is an
th root of unity for
. Suppose Bobby is unidirectional for every
minutes, and randomly chooses to reverse his direction after each cycle. In how many ways can Bobby travel around the unit circle exactly
times?
Problem 15
Find the average of all values such that