MathDB

2

Part of 2020 Tuymaada Olympiad

Problems(2)

Coefficients counting

Source: 2020 Tuymaada Junior P2

10/6/2020
All non-zero coefficients of the polynomial f(x)f(x) equal 11, while the sum of the coefficients is 2020. Is it possible that thirteen coefficients of f2(x)f^2(x) equal 99?
(S. Ivanov, K. Kokhas)
algebrapolynomial
Inequality strikes again.

Source: Tuymaada 2020 Senior, Problem 2

10/6/2020
Given positive real numbers a1,a2,…,ana_1, a_2, \dots, a_n. Let m=min⁡(a1+1a2,a2+1a3,…,an−1+1an,an+1a1). m = \min \left( a_1 + \frac{1}{a_2}, a_2 + \frac{1}{a_3}, \dots, a_{n - 1} + \frac{1}{a_n} , a_n + \frac{1}{a_1} \right). Prove the inequality a1a2…ann+1a1a2…ann≥m. \sqrt[n]{a_1 a_2 \dots a_n} + \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} \ge m.
Inequalityinequalitiesn-variable inequality