MathDB

Problem 1

Part of 2013 VJIMC

Problems(2)

f(x)f'(x)≥cos, f(∞)=undef. if f is bounded

Source: VJIMC 2013 1.1

5/30/2021
Let f:[0,∞)→Rf:[0,\infty)\to\mathbb R be a differentiable function with ∣f(x)∣≤M|f(x)|\le M and f(x)f′(x)≥cos⁡xf(x)f'(x)\ge\cos x for x∈[0,∞)x\in[0,\infty), where M>0M>0. Prove that f(x)f(x) does not have a limit as x→∞x\to\infty.
functioncalculuslimits
Σ first n primes, gaps

Source: VJIMC 2013 2.1

5/31/2021
Let SnS_n denote the sum of the first nn prime numbers. Prove that for any nn there exists the square of an integer between SnS_n and Sn+1S_{n+1}.
number theoryprimesprime