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[a+\frac{b}{a}]=[b+\frac{a}{b}]

Source: 1999 KJMO

June 30, 2024
algebrafloor function

Problem Statement

Recall that [x][x] denotes the largest integer not exceeding xx for real xx. For integers a,ba, b in the interval 1≤a<b≤1001 \leq a<b \leq 100, find the number of ordered pairs (a,b)(a, b) satisfying the following equation.
[a+ba]=[b+ab][a+\frac{b}{a}]=[b+\frac{a}{b}]
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