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Difference between revisions of "G285 2021 Summer Problem Set"

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==Problem 2==
 
==Problem 2==
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Let <cmath>f(x,y) = \begin{cases}x^y & \text{ if } x^2>y \text{ and } |x|<y\\f(f(\sqrt{|x|},y),y) & \text{ otherwise}
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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>.
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==Problem 3==
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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>
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<math>\textbf{(A)}\ 28 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 39 \qquad\textbf{(D)}\ 45 \qquad\textbf{(E)}\ 54</math>
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==Problem 4==
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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>
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<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>
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==Problem 5==
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Suppose <math>\triangle ABC</math> is an equilateral triangle. Let points <math>D</math> and <math>E</math> lie on the extensions of <math>AB</math> and <math>AC</math> respectively such that <math>\angle AED=60^o</math> and <math>DE=14</math>. If there exists a point <math>P</math> outside of <math>\triangle ADE</math> such that <math>AP=PD=28</math>, and there exists a point <math>O</math> outside outside of <math>CBDE</math> such that <math>OE=OA</math>, the area <math>2APEO</math> can be represented as <math>m\sqrt{n}+o\sqrt{p}</math>, where <math>n</math> and <math>p</math> are squarefree,. Find <math>m+n+o+p</math>
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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>
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==Problem 6==
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<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.
 +
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<math>\textbf{(A)}\ 720 \qquad\textbf{(B)}\ 1440 \qquad\textbf{(C)}\ 2520 \qquad\textbf{(D)}\ 3456\qquad\textbf{(E)}\ 5760</math>
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==Problem 7==

Revision as of 12:24, 24 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:

Problem 1

Find $\left \lceil {\frac{3!+4!+5!+6!}{2+3+4+5+6}} \right \rceil$

$\textbf{(A)}\ 42\qquad\textbf{(B)}\ 43\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 45\qquad\textbf{(E)}\ 46$

Solution

Problem 2

Let \[f(x,y) = \begin{cases}x^y & \text{ if } x^2>y \text{ and } |x|<y\\f(f(\sqrt{|x|},y),y) & \text{ otherwise} \end{cases}\] If $y$ is a positive integer, find the sum of all values of $x$ such that $f(x,y) \neq k$ for some constant $k$.

Problem 3

Let circles $\omega_1$ and $\omega_2$ with centers $Q$ and $L$ concur at points $A$ and $B$ such that $AQ=20$, $AL=28$. Suppose a point $P$ on the extension of $AB$ is formed such that $PQ=29$ and lines $PQ$ and $PL$ intersect $\omega_1$ and $\omega_2$ at $C$ and $D$ respectively. If $DC=\frac{16\sqrt{37}}{\sqrt{145}}$, the value of $\sin^2(\angle LAQ)$ can be represented as $\frac{m \sqrt{n}}{r}$, where $m$ and $r$ are relatively prime positive integers, and $n$ is square free. Find $2m+3n+4r$

$\textbf{(A)}\ 28 \qquad\textbf{(B)}\ 31 \qquad\textbf{(C)}\ 39 \qquad\textbf{(D)}\ 45 \qquad\textbf{(E)}\ 54$

Problem 4

Let $ABCD$ be a rectangle with $BC=6$ and $AB=8$. Let points $M$ and $N$ lie on $ABCD$ such that $M$ is the midpoint of $BC$ and $N$ lies on $AD$. Let point $Q$ be the center of the circumcircle of quadrilateral $MNOP$ such that $O$ and $P$ lie on the circumcircle of $\triangle MNP$ and $\triangle MNO$ respectively, along with $OD \perp QO$ and $MP \perp BP$. If the shortest distance between $Q$ and $AB$ is $3$, $\triangle AOQ$ and $\triangle QBP$ are degenerate, and $BP=AO$, find $25 \cdot OD \cdot PC$

$\textbf{(A)}\ 209 \qquad\textbf{(B)}\ 228 \qquad\textbf{(C)}\ 54\sqrt{57} \qquad\textbf{(D)}\ 90\sqrt{19} \qquad\textbf{(E)}\ 72\sqrt{57}$

Problem 5

Suppose $\triangle ABC$ is an equilateral triangle. Let points $D$ and $E$ lie on the extensions of $AB$ and $AC$ respectively such that $\angle AED=60^o$ and $DE=14$. If there exists a point $P$ outside of $\triangle ADE$ such that $AP=PD=28$, and there exists a point $O$ outside outside of $CBDE$ such that $OE=OA$, the area $2APEO$ can be represented as $m\sqrt{n}+o\sqrt{p}$, where $n$ and $p$ are squarefree,. Find $m+n+o+p$

$\textbf{(A)}\ 152 \qquad\textbf{(B)}\ 162 \qquad\textbf{(C)}\ 164 \qquad\textbf{(D)}\ 214\qquad\textbf{(E)}\ 224$

Problem 6

$16$ people are attending a hotel conference, $8$ of which are executives, and $8$ of which are speakers. Each person is designated a seat at one of $4$ round tables, each containing $4$ seats. If executives must sit at least one speaker and executive, there are $N$ ways the people can be seated. Find $\left \lfloor \sqrt{N} \right \rfloor$. Assume seats, people, and table rotations are distinguishable.

$\textbf{(A)}\ 720 \qquad\textbf{(B)}\ 1440 \qquad\textbf{(C)}\ 2520 \qquad\textbf{(D)}\ 3456\qquad\textbf{(E)}\ 5760$

Problem 7

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