MathDB
2006 China Second Round Olympiad Test 1 #5

Source:

September 28, 2014
logarithms

Problem Statement

Suppose f(x)=x3+log⁡2(x+x2+1)f(x) = x^3 + \log_2(x + \sqrt{x^2+1}). For any a,b∈Ra,b \in \mathbb{R}, to satisfy f(a)+f(b)≥0f(a) + f(b) \ge 0, the condition a+b≥0a + b \ge 0 is
<spanclass=′latex−bold′>(A)</span> necessary and sufficient<spanclass=′latex−bold′>(B)</span> not necessary but sufficient<spanclass=′latex−bold′>(C)</span> necessary but not sufficient <span class='latex-bold'>(A)</span>\ \text{necessary and sufficient}\qquad<span class='latex-bold'>(B)</span>\ \text{not necessary but sufficient}\qquad<span class='latex-bold'>(C)</span>\ \text{necessary but not sufficient}\qquad <spanclass=′latex−bold′>(D)</span> neither necessary nor sufficient<span class='latex-bold'>(D)</span>\ \text{neither necessary nor sufficient}\qquad
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