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Algebra Difficulty 5.1 AIME, harder Find the answer

4・40 Let a>b>c>d>0a>b>c>d>0, and X=ab+cd,Y=ac+X=\sqrt{a b}+\sqrt{c d}, Y=\sqrt{a c}+ bd,Z=ad+bc\sqrt{b d}, Z=\sqrt{a d}+\sqrt{b c}. Then the size relationship of X,Y,ZX, Y, Z is

Pick one

Solution

［Solution］From the given, we have
X−Y=a(b−c)+d(c−b)=(a−d)(b−c)>0, Also, Y−Z=a(c−d)+b(d−c)=(a−b)(c−d)>0. \begin{aligned} X-Y & =\sqrt{a}(\sqrt{b}-\sqrt{c})+\sqrt{d}(\sqrt{c}-\sqrt{b}) \\ & =(\sqrt{a}-\sqrt{d})(\sqrt{b}-\sqrt{c})>0, \\ \text { Also, } \quad Y-Z & =\sqrt{a}(\sqrt{c}-\sqrt{d})+\sqrt{b}(\sqrt{d}-\sqrt{c}) \\ & =(\sqrt{a}-\sqrt{b})(\sqrt{c}-\sqrt{d})>0 . \end{aligned}

Thus, X>Y>ZX>Y>Z.
Therefore, the answer is (D)(D).

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.