Wednesday, April 27, 2011

Floor And Ceiling

Show that there are positive integers $a,b,c,d$ with $b,d < 100$ such that:

$$\lfloor \frac{a}{b} k \rfloor = \lfloor \frac{73}{100} k \rfloor$$

and

$$\lceil \frac{c}{d} k \rceil = \lceil \frac{73}{100} k \rceil$$

For all $k = 1,2,\dots,99$

Note: $\lfloor x \rfloor$ denotes the floor function and $\lceil x \rceil$ denotes the ceiling function