Olympiad problems

2020 CMIMC Team, 5

We say that a binary string $s$ [i]contains[/i] another binary string $t$ if there exist indices $i_1,i_2,\ldots,i_{|t|}$ with $i_1 < i_2 < \ldots < i_{|t|}$ such that $$s_{i_1}s_{i_2}\ldots s_{i_{|t|}} = t.$$ (In other words, $t$ is found as a not necessarily contiguous substring of $s$.) For example, $110010$ contains $111$. What is the length of the shortest string $s$ which contains the binary representations of all the positive integers less than or equal to $2048$?

team number theory

2021 MOAA, 16

Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle{XAC}=32^\circ$, and $\angle{XCA}=35^\circ$. Compute $\angle{BAC}$ in degrees. [i]Proposed by Raina Yang[/i]

team geometry

2022 CMIMC, 3

Let $ABCD$ be a rectangle with $AB=10$ and $AD=5.$ Suppose points $P$ and $Q$ are on segments $CD$ and $BC,$ respectively, such that the following conditions hold: [list] [*] $BD \parallel PQ$ [*] $\angle APQ=90^{\circ}.$ [/list] What is the area of $\triangle CPQ?$ [i]Proposed by Kyle Lee[/i]

team geometry

2018 CMIMC Team, 9-1/9-2

Andy rolls a fair 4-sided dice, numbered 1 to 4, until he rolls a number that is less than his last roll. If the expected number of times that Andy will roll the dice can be expressed as a reduced fraction $\frac{p}{q}$, find $p + q$. Let $T = TNYWR$. The solutions in $z$ to the equation \[\left(z + \frac Tz\right)^2 = 1\] form the vertices of a quadrilateral in the complex plane. Compute the area of this quadrilateral.

team algebra geometry probability