Question

It wil be greatful if u help me in solving these problems:

1.)The number of solutions of x + [x] = 2013, where [x] is greatest integer less than or equal
to x, is

Answer 1

1006 + [1006] = 2012

1007 + [1007] = 2014

so, for c>0,
1006+c + [1006+c] = 1006+c + 1006 = 2012 + c

for 2012+c = 2013, we need c=1, but we already said c<1.

So, there are no real solutions.