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2023 SSMO Relay Round 5 Problems

Problem 1

Let $S_n$ be the set of all rational numbers of the form $0.\overline{a_1a_2a_3\dots a_n},$ where $n$ is an integer satisfying $n\geq 1$ and $a_1,a_2,\dots,a_n$ are nonzero integers. If \[n = 5\left(\sum_{n=1}^{\infty}\left(\sum_{a\in S_n}\frac{a}{10^{n}}\right)\right),\] find $n.$

Solution

Problem 2

Let $T=$ TNYWR. Let $a_n = \text{lcm} \{1, 2, \dots n\}$ for positive integers $n$.Compute \[\sum_{i=1}^N a_i \pmod {720}.\]

Solution

Problem 3

Let $T=$ TNYWR. Suppose that $x^2+y^2 = N.$ Find the remainder when the expected value of the square of the maximum value $ax+by$ is divided by $100,$ where $a$ and $b$ distinct members from the set $\{1,2,\dots,N\}.$

Solution

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