Question

if x+y+z+w=15, then at least k of the numbers x,z,y,w must be positive.find k

Answer 1

The minimum value of k is 1, because if we take any of the variables, say, x, then x=15-(y+z+w)>0, so, as long as (y+z+w)<15 (e.g., y=1, z=2, w=3; 4, 5, 5; 1, 5, 8; etc.), x can be made to exceed their sum. k cannot be zero as a minimum because all the variables could be negative, so k must be at least 1.