MathDB

3

Part of 2020 China Team Selection Test

Problems(1)

Involuted sum with NT structure

Source: China Additional TST for IMO 2020, P3

10/19/2020
For a non-empty finite set AA of positive integers, let lcm(A)\text{lcm}(A) denote the least common multiple of elements in AA, and let d(A)d(A) denote the number of prime factors of lcm(A)\text{lcm}(A) (counting multiplicity). Given a finite set SS of positive integers, and fS(x)=∑∅≠A⊂S(−1)∣A∣xd(A)lcm(A).f_S(x)=\sum_{\emptyset \neq A \subset S} \frac{(-1)^{|A|} x^{d(A)}}{\text{lcm}(A)}. Prove that, if 0≤x≤20 \le x \le 2, then −1≤fS(x)≤0-1 \le f_S(x) \le 0.
number theoryalgebraleast common multiple