2

Part of 1983 Polish MO Finals

Problems(1)

a irrational in (0,1), p/q < a <r/s, r/s -p/q< 1/N, rq- ps = 1

Source: 1983 Polish MO Finals p2

Let be given an irrational number

a

in the interval

(0,1)

and a positive integer

N

. Prove that there exist positive integers

p,q,r,s

\frac{p}{q} < a <\frac{r}{s}, \frac{r}{s} -\frac{p}{q}<\frac{1}{N}

rq- ps = 1

inequalitiesalgebrairrational