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Two positive integers with a fixed sum: probability their product is at least a fraction of the maximum

This type of question, asked 2 times: 25 Jan 2023, 8 Apr 2024. All 2 questions below, oldest first.

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  1. Question 1

    25 Jan 2023

    73
    Let MM be the maximum value of the product of two positive integers when their sum is 66 . Let the sample space S={x∈Z:x(66−x)≥59M}S =\left\{ x\in Z:x(66 - x) \geq\frac{5}{9}M \right\} and the event A={x∈S:x\mathbf{A = \{ x \in S:x} is a multiple of 3}\}. Then P(A)P(A) is equal to

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  2. Question 2

    8 Apr 2024

    70
    Let the sum of two positive integers be 24. If the probability, that their product is not less than 34\frac{3}{4} times their greatest positive product, is mn\frac{m}{n}, where gcd(m,n)=1gcd(m,n) = 1, then n−mn-m equals:

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