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6

Part of 1995 Taiwan National Olympiad

Problems(1)

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

Source: 4-th Taiwanese Mathematical Olympiad 1995

1/16/2007
Let a,b,c,da,b,c,d are integers such that (a,b)=(c,d)=1(a,b)=(c,d)=1 and ad−bc=k>0ad-bc=k>0. Prove that there are exactly kk pairs (x1,x2)(x_{1},x_{2}) of rational numbers with 0≤x1,x2<10\leq x_{1},x_{2}<1 for which both ax1+bx2,cx1+dx2ax_{1}+bx_{2},cx_{1}+dx_{2} are integers.
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