Pretty straightforward, actually.

Let the mass of the man be "m"

So the mass of the boy becomes m/2

If the speed of the runner be "v", his Kinetic energy = 0.5mv²

If the speed of the boy be v1, his KE = 0.25m(v1)²

Since the runner's KE is half the boy's ,

0.5mv² = 0.125  m(v1)²

i.e. (v1)² = 4v²

ie. v1 = 2v.....(1)

Now when the runner's speed increases by 1 unit, his KE becomes equal to that of the boy's,

i.e. 0.5m(v+1)² = 0.25m(v1)²

Since v1 =2v ,

0.5(v+1)² = 0.25(2v)²

i.e. 2(v+1)² = (2v)²

 so sqrt(2)(v+1) = 2v

so v(2-sqrt(2) = sqrt(2)

v = 1/(sqrt(2)-1) unit