MathDB

4

Part of 2017 VJIMC

Problems(2)

Divergent improper integral

Source: VJIMC 2017, Category I, Problem 4

4/2/2017
Let f:(1,∞)→Rf:(1,\infty) \to \mathbb{R} be a continuously differentiable function satisfying f(x)≤x2log⁡(x)f(x) \le x^2 \log(x) and f′(x)>0f'(x)>0 for every x∈(1,∞)x \in (1,\infty). Prove that ∫1∞1f′(x)dx=∞.\int_1^{\infty} \frac{1}{f'(x)} dx=\infty.
calculusintegrationreal analysiscollege contests
Certain amount of numbers t=x^3+y^2 in a given set

Source: VJIMC 2017, Category II, Problem 4

4/2/2017
A positive integer tt is called a Jane's integer if t=x3+y2t = x^3+y^2 for some positive integers xx and yy. Prove that for every integer n≥2n \ge 2 there exist infinitely many positive integers mm such that the set of n2n^2 consecutive integers {m+1,m+2,…,m+n2}\{m+1,m+2,\dotsc,m+n^2\} contains exactly n+1n + 1 Jane's integers.
number theory