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G285 2021 Summer Problem Set

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 $k$.

$\textbf{(A)}\ -1 \qquad\textbf{(B)}\ -\frac{1}{2} \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ \frac{3}{8} \qquad\textbf{(E)}\ 1$

Solution

Problem 3

$60$ groups of molecules are gathered in a lab. The scientists in the lab randomly assign the $60$ molecules into $5$ groups of $12$. Within these groups, there will be $5$ 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.

$\textbf{(A)}\ 5^{60} \qquad\textbf{(B)}\ \frac{60!\cdot 5^{60}}{(12!)^4} \qquad\textbf{(C)}\ \frac{60!\cdot 5^{30}}{(12!)^4} \qquad\textbf{(D)}\ \frac{40!\cdot 5^{60}}{11!(12!)^3} \qquad\textbf{(E)}\ 60!5^{60}$

Solution

Problem 4

$16$ 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.

$\textbf{(A)}\ 720 \qquad\textbf{(B)}\ 1440 \qquad\textbf{(C)}\ 2520 \qquad\textbf{(D)}\ 5760\qquad\textbf{(E)}\ 6172$

Solution

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 $m+n+o+p$

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

Solution

Problem 6

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 $25 \cdot OD \cdot PC$

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

Solution

Problem 7

Geometry285 is playing the game "Guess And Choose". In this game, Geometry285 selects a subset of not necessarily distinct integers $P=\{a,b,c \cdots \}$ from the set $S=\{1,2,3,4 \cdots k-1,k \}$ such that the sum of all elements in $P$ is $k$. Each distinct is selected chronologically and placed in $P$, such that $1 \le a \le k$, $1 \le b \le a$, $1 \le c \le b$, and so on. Then, the elements are randomly arranged. Suppose $S_{p,k}$ represents the total number of outcomes that a subset $P$ containing $p$ integers sums to $k$. If distinct permutations of the same set $P$ are considered unique, find the remainder when \[\sum_{p=1}^{1000}\sum_{k=1}^{1000} S_{p,k}\] is divided by $100$.

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 51 \qquad\textbf{(E)}\ 124$

Solution

Problem 8

Let $p(x)=x^{12}-9x^{11}+16x^{10}+256x^5+1$, Let $r_1, r_2, r_3, r_4, r_5, r_6, ..., r_{12}$ be the twelve roots that satisfies $p(x)=0$, find the least possible value of \[\left \lfloor \sum_{n=1}^{12}\sum_{k=1}^{12} r_nr_k-\sum_{s=1}^{11} r_s \right \rfloor\]

$\textbf{(A)}\ 67 \qquad\textbf{(B)}\ 69 \qquad\textbf{(C)}\ 70 \qquad\textbf{(D)}\ 71 \qquad\textbf{(E)}\ 72$

Solution

Problem 9

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 $2m+3n+4r$

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

Solution

Problem 10

Let $k \in \mathbb{N}$ for $k>1$. Suppose $\lfloor \omega_k \rfloor$ makes $k=(p_1p_2p_3 \cdots p_e)^1$ for distinct prime factors $p$. If $\tau(p)$ for $p>1$ is \[\sum_{j=1}^{e} p_j\] where $p_j$ must satisfy that $\frac{\lfloor \omega_k \rfloor}{p_j}$ is an integer, and $p_j$ is divisible by the $p$th and $(p-1)$th triangular number. Find $\tau(3)+\tau(4)+\tau(5)+ \cdots +\tau(99)+\tau(100)$

$\textbf{(A)}\ 1024 \qquad\textbf{(B)}\ 1331 \qquad\textbf{(C)}\ 1539 \qquad\textbf{(D)}\ 2000 \qquad\textbf{(E)}\ 2719$

Solution

Problem 11

Let a recursive sequence $a_n$ be defined such that $a_1=20$, and $a_n=16a_{n-1}+4$. Find the last $3$ digits of $a_{100}$

Solution

Problem 12

Suppose the function \[P(a,b,c)=a^2b^4+b^2c^4+c^2a^4+8c+8b+8a+8a^3+8b^3+8c^3-3\sqrt[3]{abc}-21\]. If $P(a)+P(b)+P(c)=P(a,b,c)=P(k)$, and the polynomial $P(k)$ contains the points $(P(k),P(k)+1)$,$(P(k)+3,P(k)+5)$, and $(P(k)+8,11)$, find the smallest value of $P(23)$ for which $P(P(P(a,b,c))=abc(P(a)+P(b)+P(c))$

Solution

Problem 13

Let circles $O_1$,$O_2$, and $O_3$ concur at $E$, where $EP$ is the common chord shared by $\{O_1,O_3 \}$, $QE$ is the common chord shared by $\{O_1,O_2 \}$, and $E$ lies on the common internal tangent of $\{O_2,O_3 \}$. Let the extension of $PE$ and $QE$ intersect $O_2$ and $O_3$ again at $F$ and $G$ respectively. If $\overline{CF} \cap \overline{BG} \in D$, prove $ABDC$ is a parallelogram.

Solution

Problem 14

Bobby the frog is hopping around the unit circle. Suppose Bobby starts at $(1,0)$. After every $n$th minute, Bobby moves to $(a,bi)$ such that $a^2+b^2 \le 1$, and $(a,bi)$ is an $n$th root of unity for $n>1$. Suppose Bobby is unidirectional for every $3$ minutes, and randomly chooses to reverse his direction after each cycle. In how many ways can Bobby travel around the unit circle exactly $6$ times?

Solution

Problem 15

Find the average of all values $z$ such that \[\sum_{n=1}^{119} \prod_{j=1}^{7} (z^j)^{n} = \left(\sum_{p=1}^{60} z^{2p-1}-\sum_{n=1}^{59} z^{2n} \right)^{5040}+2\]

Solution

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