What is the largest value of k such that the equation 6x-x^2=k has at least one real solution?

By the quadratic formula, the largest possible value of k is 12.

Rewrite as x^2 - 6x + k = 0

This has at least one real solution as long as the discriminant is greater than or equal to zero.

i.e. 6^2 - 4k >= 0

4k <= 36

k<= 9

So largest value of k is 9.