Maths Olympiad Prep

Track / Stage 4 / 311 of 340 #571 of 1964

Problem 571

AMC 12 late, AIME early
Algebra Difficulty 5.0 Find the answer

1. Solve the system:

x2+y2=z2xz=y2xy=10 \begin{aligned} & x^{2}+y^{2}=z^{2} \\ & x z=y^{2} \\ & x y=10 \end{aligned}

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

1. 2+18=(2+18)2=2+18+236=\sqrt{2}+\sqrt{18}=\sqrt{(\sqrt{2}+\sqrt{18})^{2}}=\sqrt{2+18+2 \sqrt{36}}=

=20+12=32=42 =\sqrt{20+12}=\sqrt{32}=4 \sqrt{2}

Formula

a+b=(a+b)2=a+b+2ab \sqrt{a}+\sqrt{b}=\sqrt{(\sqrt{a}+\sqrt{b})^{2}}=\sqrt{a+b+2 \sqrt{a b}}

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