MathDB

119 pairs of (x,y) such that (x+my) and (mx+y) are integers , 0<x,y<1,

Source: Dutch NMO 2010 p4

September 6, 2019

number theoryInteger

Problem Statement

(a) Determine all pairs

(x, y)

of (real) numbers with

0 < x < 1

and

0 <y < 1

for which

x + 3y

and

3x + y

are both integer. An example is

(x,y) =( \frac{8}{3}, \frac{7}{8})

, because

x+3y =\frac38 +\frac{21}{8} =\frac{24}{8} = 3

and

3x+y = \frac98 + \frac78 =\frac{16}{8} = 2

.
(b) Determine the integer

m > 2

for which there are exactly

119

pairs

(x,y)

with

0 < x < 1

and

0 < y < 1

such that

x + my

and

mx + y

are integers.
Remark: if

u \ne v,

the pairs

(u, v)

and

(v, u)

are different.

Back to Problems View on AoPS