Problem Solving Set 1

04 July 2012

1. Let f ∈ C^{1}(a, b), lim_{x→a+} f(x) = +∞, lim_{x→b−} f(x) = −∞ and f′(x) + f^{2}(x) ≥ −1 for x ∈ (a, b). Prove that b − a ≥ π and give an example where b − a = π.

2. A collection of subsets of {1, 2, …, n} has the property that each pair of subsets has at least one element in common. Prove that there are at most 2^{n−1} subsets in the collection.