Olympiad problems

2021 ASDAN Math Tournament, 1

Benny the Bear has $100$ rabbits in his rabbit farm. He observes that $53$ rabbits are spotted, and $73$ rabbits are blue-eyed. Compute the minimum number of rabbits that are both spotted and blue-eyed.

2021 ASDAN Math Tournament, 2

Tags: 2021 , Individual Tiebreaker

For a real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x,$ and let $\{x\} = x -\lfloor x\rfloor$ denote the fractional part of $x.$ The sum of all real numbers $\alpha$ that satisfy the equation $$\alpha^2+\{\alpha\}=21$$ can be expressed in the form $$\frac{\sqrt{a}-\sqrt{b}}{c}-d$$ where $a, b, c,$ and $d$ are positive integers, and $a$ and $b$ are not divisible by the square of any prime. Compute $a + b + c + d.$

2021 ASDAN Math Tournament, 3

Tags: 2021 , Individual Tiebreaker

Let $V$ be a set of eight points in $3\text{D}$ space that are the vertices of a cube with side length $1$. Compute the number of ways we can color the vertices in $V$ yellow or blue such that [list] [*] each vertex receives exactly one color, and [/*] [*] there exists a point in $3\text{D}$ space whose distance to each yellow vertex is less than $1$ and whose distance to each blue vertex is greater than $1$. [/*] [/list]