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2024 SSMO Tiebreaker Round Problems/Problem 3

Revision as of 16:03, 2 May 2025 by Pinkpig (talk | contribs) (Created page with "==Problem== Let <math>A=\dots a_2a_1a_0.a_{-1}a_{-2}a_{-3}\dots</math> be a terminating decimal. The length of <math>A</math> is defined to be the length of the shortest sub-...")
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Problem

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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