IMO Shortlist 1967 problem 4
Dodao/la:
arhiva2. travnja 2012. Prove the following statement: If
![r_1](/media/m/9/0/1/901ecb943995b3585cd44466e1b750cb.png)
and
![r_2](/media/m/9/0/6/90608ee2be6d3b5c7f96a6ca45780ec4.png)
are real numbers whose quotient is irrational, then any real number
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
can be approximated arbitrarily well by the numbers of the form
![\ z_{k_1,k_2} = k_1r_1 + k_2r_2](/media/m/8/3/0/83043089460bad43dc14ae9c1d11cd5c.png)
integers, i.e. for every number
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
and every positive real number
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
two integers
![k_1](/media/m/3/5/6/35656cbf3adb55dddd30996fc068363b.png)
and
![k_2](/media/m/6/a/b/6abbe24dbf6713b55498fe55ab050d06.png)
can be found so that
![|x - (k_1r_1 + k_2r_2)| < p](/media/m/0/9/a/09a6ac61d61a65b3d792a1931b8d9c01.png)
holds.
%V0
Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.
Izvor: Međunarodna matematička olimpijada, shortlist 1967