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0441 inequality 4th edition Round 4 p1

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May 7, 2021
algebrainequalities4th edition

Problem Statement

Let N0N_0 be the set of all non-negative integers and let f:N0×N0→[0,+∞)f : N_0 \times N_0 \to [0, +\infty) be a function such that f(a,b)=f(b,a)f(a, b) = f(b, a) and f(a,b)=f(a+1,b)+f(a,b+1),f(a, b) = f(a + 1, b) + f(a, b + 1), for all a,b∈N0a, b \in N_0. Denote by xn=f(n,0)x_n = f(n, 0) for all n∈N0n \in N_0. Prove that for all n∈N0n \in N_0 the following inequality takes place 2nxn≥x0.2^n x_n \ge x_0.
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