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The positive integers a,b,c,d,p,q

Source: IberoAmerican 1988 Q2

December 13, 2010
number theory proposednumber theory

Problem Statement

Let a,b,c,d,pa,b,c,d,p and qq be positive integers satisfying ad−bc=1ad-bc=1 and ab>pq>cd\frac{a}{b}>\frac{p}{q}>\frac{c}{d}.
Prove that:
(a)(a) q≥b+dq\ge b+d
(b)(b) If q=b+dq=b+d, then p=a+cp=a+c.
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