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Romania Junior TST 2021 Day 3 P3

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June 7, 2021
number theorySetsmaximum valueromaniaRomanian TST

Problem Statement

Let p,qp,q be positive integers. For any a,b∈Ra,b\in\mathbb{R} define the sets P(a)={an=a + n ⋅ 1p:n∈N} and Q(b)={bn=b + n ⋅ 1q:n∈N}.P(a)=\bigg\{a_n=a \ + \ n \ \cdot \ \frac{1}{p} : n\in\mathbb{N}\bigg\}\text{ and }Q(b)=\bigg\{b_n=b \ + \ n \ \cdot \ \frac{1}{q} : n\in\mathbb{N}\bigg\}. The distance between P(a)P(a) and Q(b)Q(b) is the minimum value of ∣x−y∣|x-y| as x∈P(a),y∈Q(b)x\in P(a), y\in Q(b). Find the maximum value of the distance between P(a)P(a) and Q(b)Q(b) as a,b∈Ra,b\in\mathbb{R}.
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