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x^6 - 2x^4 + x^2 = A

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April 23, 2009
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Problem Statement

Let n(A) n\left(A\right) be the number of distinct real solutions of the equation x^{6} \minus{}2x^{4} \plus{}x^{2} \equal{}A. When A A takes every value on real numbers, the set of values of n(A) n\left(A\right) is
<spanclass=latexbold>(A)</span> {0,1,2,3,4,5,6}<spanclass=latexbold>(B)</span> {0,2,4,6}<spanclass=latexbold>(C)</span> {0,3,4,6}<spanclass=latexbold>(D)</span> {0,2,3,4,6}<spanclass=latexbold>(E)</span> {0,2,3,4}<span class='latex-bold'>(A)</span>\ \left\{0,1,2,3,4,5,6\right\} \\ <span class='latex-bold'>(B)</span>\ \left\{0,2,4,6\right\} \\ <span class='latex-bold'>(C)</span>\ \left\{0,3,4,6\right\} \\ <span class='latex-bold'>(D)</span>\ \left\{0,2,3,4,6\right\} \\ <span class='latex-bold'>(E)</span>\ \left\{0,2,3,4\right\}