Problem Solving Set 5

08 July 2012

1. Let f be a real-valued function with n + 1 derivatives at each point of ℝ. Show that for each pair of real numbers a, b, a < b, such that

ln (\frac{f(b) + f′(b) + ⋯ + f^{(n}(b)}{f(a) + f′(a) + ⋯ + f^{(n}(a)}) = b − a

there is a number c in the open interval (a, b) for which

f^{(n+1)}(c) = f(c).

2. Given one million non-zero digits a_{1}, a_{2}, …, a_{1000000} (ie each a_{i} ∈ {1, 2, …, 9}) show that at most 100 of the million numbers

a_{1}, a_{1}a_{2}, a_{1}a_{2}a_{3}, …, a_{1}a_{2}a_{3} … a_{1000000}

are perfect squares