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Algebra Difficulty 5.4 AIME, harder Find the answer

Let A=(1+22+33+66)(2+62+3+36)(3+2+63+26)(6+32+23+6)B=(1+32+23+66)(2+2+63+36)(3+62+3+26)(6+22+33+6)\begin{aligned} & A=(1+2 \sqrt{2}+3 \sqrt{3}+6 \sqrt{6})(2+6 \sqrt{2}+\sqrt{3}+3 \sqrt{6})(3+\sqrt{2}+6 \sqrt{3}+2 \sqrt{6})(6+3 \sqrt{2}+2 \sqrt{3}+\sqrt{6}) \\ & B=(1+3 \sqrt{2}+2 \sqrt{3}+6 \sqrt{6})(2+\sqrt{2}+6 \sqrt{3}+3 \sqrt{6})(3+6 \sqrt{2}+\sqrt{3}+2 \sqrt{6})(6+2 \sqrt{2}+3 \sqrt{3}+\sqrt{6}) \end{aligned} Compute the value of A/BA / B.

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Solution

Note that A=((1+22)(1+33))((2+3)(1+32))((3+2)(1+23))((3+3)(2+2))B=((1+32)(1+23))((2+2)(1+33))((3+3)(1+22))((2+3)(3+2))\begin{aligned} & A=((1+2 \sqrt{2})(1+3 \sqrt{3}))((2+\sqrt{3})(1+3 \sqrt{2}))((3+\sqrt{2})(1+2 \sqrt{3}))((3+\sqrt{3})(2+\sqrt{2})) \\ & B=((1+3 \sqrt{2})(1+2 \sqrt{3}))((2+\sqrt{2})(1+3 \sqrt{3}))((3+\sqrt{3})(1+2 \sqrt{2}))((2+\sqrt{3})(3+\sqrt{2})) \end{aligned} It is not difficult to check that they have the exact same set of factors, so A=BA=B and thus the ratio is 1.

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.