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2022 CMWMC Guts Round 5/8 - Carnegie Mellon University Womens' Competition

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August 12, 2023
CMWMCalgebrageometrycombinatoricsnumber theory

Problem Statement

Set 5
p13. An equiangular 1212-gon has side lengths that alternate between 22 and 3\sqrt3. Find the area of the circumscribed circle of this 1212-gon.
p14. For positive integers nn, let σ(n)\sigma(n) denote the number of positive integer factors of nn. Then σ(17280)=σ(27⋅33⋅5)=64\sigma(17280) = \sigma (2^7 \cdot 3^3 \cdot 5)= 64. Let SS be the set of factors kk of 1728017280 such that σ(k)=32\sigma(k) = 32. If pp is the product of the elements of SS, find σ(p)\sigma(p).
p15. How many odd 33-digit numbers have exactly four 11’s in their binary (base 22) representation? For example, 22510=111000012225_{10} = 11100001_2 would be valid.
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