G285 2021 Fall Problem Set

Welcome to the Fall Problem Set! There are $15$ problems, $10$ multiple-choice, and $5$ free-response.

Problem 1

Larry is playing a logic game. In this game, Larry counts $1,2,3,6, \cdots$, and removes the number $r+p$ for every $r$th move, skipping $r+jp$ for $j \neq 0 \mod 3$, and then increments $p$ by one. If $(r,p)$ starts at $(1,3)$, what is $r+p$ when Larry counts his $100$th integer? Assume $\{r,p,j \} \in \mathbb{N}$

Problem 2

Let $\triangle ABC$ be a right triangle with right angle at $B$, and $AC=12$. Let $D$ denote the intersection of the cevian dropped from $B$ onto $AC$ such that $DA=DC$. If the reflection of point $B$ across $D$ lies on the circumcircle of $\triangle ABC$ as $E$, $\sin(BAC)<\frac{5}{8}$, and the circumradius of $\triangle ABC$ is an integer, find the smallest possible value of $AB^2+AE^2$.

Problem 8

Find \[\sum_{a=0}^{\infty} \sum_{b=0}^{\infty} \sum_{c=0}^{\infty} \sum_{d=0}^{\infty} \frac{a+2b+3c}{4^{(a+b+c+d)}}\]

\[\textbf{(A)}\ \frac{16}{27} \qquad \textbf{(B)}\ \frac{32}{27} \qquad \textbf{(C)}\ \frac{64}{27} \qquad \textbf{(D)}\ \frac{128}{27} \qquad \textbf{(E)}\ \frac{256}{27}\]

Solution

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