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Log areas

Source: AMC 12 2006A, Problem 21

February 5, 2006
geometrylogarithmsratioinequalitiesfunctionAMC

Problem Statement

Let S_1 \equal{} \{ (x,y)\ | \ \log_{10} (1 \plus{} x^2 \plus{} y^2)\le 1 \plus{} \log_{10}(x \plus{} y)\} and S_2 \equal{} \{ (x,y)\ | \ \log_{10} (2 \plus{} x^2 \plus{} y^2)\le 2 \plus{} \log_{10}(x \plus{} y)\}. What is the ratio of the area of S2 S_2 to the area of S1 S_1? <spanclass=latexbold>(A)</span>98<spanclass=latexbold>(B)</span>99<spanclass=latexbold>(C)</span>100<spanclass=latexbold>(D)</span>101<spanclass=latexbold>(E)</span>102 <span class='latex-bold'>(A) </span> 98\qquad <span class='latex-bold'>(B) </span> 99\qquad <span class='latex-bold'>(C) </span> 100\qquad <span class='latex-bold'>(D) </span> 101\qquad <span class='latex-bold'>(E) </span> 102