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6

Part of 1976 Putnam

Problems(2)

Putnam 1976 A6

Source:

4/18/2022
Suppose f(x)f(x) is a twice continuously differentiable real valued function defined for all real numbers xx and satisfying ∣f(x)∣≤1|f(x)| \leq 1 for all x and (f(0))2+(f′(0))2=4.(f(0))^2+(f'(0))^2=4. Prove that there exists a real number x0x_0 such that f(x0)+f′′(x0)=0.f(x_0)+f''(x_0)=0.
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Putnam 1976 B6

Source:

4/20/2022
As usual, let σ(N)\sigma (N) denote the sum of all the (positive integral) divisors of N.N. (Included among these divisors are 11 and NN itself.) For example, if pp is a prime, then σ(p)=p+1.\sigma (p)=p+1. Motivated by the notion of a "perfect" number, a positive integer NN is called "quasiperfect" if σ(N)=2N+1.\sigma (N) =2N+1. Prove that every quasiperfect number is the square of an odd integer.
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