MathDB
Greater than weighted mean ==> greater than mean

Source: 4th QEDMO, created by myself

March 9, 2007
algebra proposedalgebra

Problem Statement

Let (a1, a2, a3, ...)\left(a_{1},\ a_{2},\ a_{3},\ ...\right) be a sequence of reals such that an≥(n−1)an−1+(n−2)an−2+...+2a2+1a1(n−1)+(n−2)+...+2+1a_{n}\geq\frac{\left(n-1\right)a_{n-1}+\left(n-2\right)a_{n-2}+...+2a_{2}+1a_{1}}{\left(n-1\right)+\left(n-2\right)+...+2+1} for every integer n≥2n\geq 2. Prove that an≥an−1+an−2+...+a2+a1n−1a_{n}\geq\frac{a_{n-1}+a_{n-2}+...+a_{2}+a_{1}}{n-1} for every integer n≥2n\geq 2. Generalization. Let (b1, b2, b3, ...)\left(b_{1},\ b_{2},\ b_{3},\ ...\right) be a monotonically increasing sequence of positive reals, and let (a1, a2, a3, ...)\left(a_{1},\ a_{2},\ a_{3},\ ...\right) be a sequence of reals such that an≥bn−1an−1+bn−2an−2+...+b2a2+b1a1bn−1+bn−2+...+b2+b1a_{n}\geq\frac{b_{n-1}a_{n-1}+b_{n-2}a_{n-2}+...+b_{2}a_{2}+b_{1}a_{1}}{b_{n-1}+b_{n-2}+...+b_{2}+b_{1}} for every integer n≥2n\geq 2. Prove that an≥an−1+an−2+...+a2+a1n−1a_{n}\geq\frac{a_{n-1}+a_{n-2}+...+a_{2}+a_{1}}{n-1} for every integer n≥2n\geq 2. darij
Back to ProblemsView on AoPS