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Source: STEMS 2023 Maths CAT A Part A P1

January 8, 2023
combinatoricsSTEMS

Problem Statement

The following 100100 numbers are written on the board: 21−1,22−1,23−1,…,2100−1.2^1 - 1, 2^2 - 1, 2^3 - 1, \dots, 2^{100} - 1.
Alice chooses two numbers a,b,a,b, erases them and writes the number ab−1a+b+2\dfrac{ab - 1}{a+b+2} on the board. She keeps doing this until a single number remains on the board.
If the sum of all possible numbers she can end up with is pq\dfrac{p}{q} where p,qp, q are coprime, then what is the value of log⁡2(p+q)\log_{2}(p+q)?
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