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2023 SSMO Relay Round 4 Problems

Problem 1

Martha starts out with the number $7$ on her calculator, which has three buttons that multiply the current number by $\frac{1}{2}$, $\frac{2}{3}$, and $\frac{5}{6}$ respectively. She randomly presses one of the buttons four times. After these $4$ presses, she cubes the number. The expected value of the final number is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution

Problem 2

Let $T=$ TNYWR. Let $n = \left\lfloor\sqrt{N}\right\rfloor.$ Suppose that $N$ points are chosen on the sides of a triangle with area 1 such that there is at least one point on each side. Let $m$ be the area of the polygon formed by connecting the $N$ points in counterclockwise order. Find the expected value of $\frac{30}{1-m}$ (Note that $\left\{x\right\} = x - \lfloor x \rfloor$)

Solution

Problem 3

Let $T=$ TNYWR. $N+1$ numbers are chosen from the set $\{1,2,3,\dots,N+1\}$ with replacement. If the probability that the median of these $N+1$ numbers is greater than $\frac{N+2}{2}$ is $M,$ such that the decimal representation of $\frac{1}{M}$ has $a$ $0$'s before the first nonzero digit of it, find $n$ rounded to nearest multiple of $5.$

Solution

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