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2024 SSMO Tiebreaker Round Problems

Problem 1

Compute the exact value of $2^2+0^2+2^2+4^2+20^2+22^2+24^2+40^2+42^2+202^2.$

Solution

Problem 2

Bob is attempting to shoot a 3-point throw. Bob attempts the basket 97 times. Each time, Bob has a $35\%$ chance of making the shot. If $S_1$ denotes the expected number of points Bob will make and $S_2$ the number of points Bob is most likely to make, then $|S_1-S_2| = \frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution

Problem 3

Let $A=\dots a_2a_1a_0.a_{-1}a_{-2}a_{-3}\dots$ be a terminating decimal. The length of $A$ is defined to be the length of the shortest sub-sequence of consecutive digits that include all nonzero digits and at least one of $a_0,a_{-1}.$ So, the length of $12.03$ is $4$ and the length of $0.123$ is $3.$ Let $f(n)$ be the average of all numbers with a terminating decimal of length $n.$ Find the value of $\left\lfloor\sum_{n=0}^{10}(n+1)f(n)\right\rfloor.$

Solution

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