Two easy ones this week:
1.Let be a positive integer. Prove that the numbers and cannot both be perfect cubes.
2. Which regular n-gons can be inscribed in a non-circular ellipse?
If both n+2 and n^2+n+1 are both cubes then so is their product, but:
but this is imposible as no two cubes of integers differ by 1.
A regular n-gon can be inscribed in a circle, but the vetices also lie on2. Which regular n-gons can be inscribed in a non-circular ellipse?
the non-circular ellipse. But a pair of distinct conics intersect at no more
than four points, so n<=4.