MathDB

P2

Part of 2020 Bulgaria National Olympiad

Problems(1)

Inequality for real number and integers

Source: Bulgaria National Olympiad 2020

6/30/2020
Let b1b_1, …\dots , bnb_n be nonnegative integers with sum 22 and a0a_0, a1a_1, …\dots , ana_n be real numbers such that a0=an=0a_0=a_n=0 and ∣ai−ai−1∣≤bi|a_i-a_{i-1}|\leq b_i for each i=1i=1, …\dots , nn. Prove that ∑i=1n(ai+ai−1)bi≤2\sum_{i=1}^n(a_i+a_{i-1})b_i\leq 2 I believe that the original problem was for nonnegative real numbers and it was a typo on the version of the exam paper we had but I'm not sure the inequality would hold
inequalitiesalgebran-variable inequalitySequence