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there are exactly $k$ pairs $(x_1,x_2)$...

Source: 4-th Taiwanese Mathematical Olympiad 1995

January 16, 2007

vectorgeometryparallelograminequalitiesanalytic geometrycombinatorics unsolvedcombinatorics

Problem Statement

Let

a,b,c,d

are integers such that

(a,b)=(c,d)=1

and

ad-bc=k>0

. Prove that there are exactly

k

pairs

(x_{1},x_{2})

of rational numbers with

0\leq x_{1},x_{2}<1

for which both

ax_{1}+bx_{2},cx_{1}+dx_{2}

are integers.