Competition problem — Algebra — Maths Olympiad Prep

Library / /73 of 520

Algebra Difficulty 6.0 AIME, harder Prove it

Question 1-1 Given that $x, y, z, r$ are all positive numbers, and $x^{2}+y^{2}$ $=z^{2}, z \sqrt{x^{2}-r^{2}}=x^{2}$ , prove: $x y=r z$ .

Solution

None

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.

Note: The provided instruction is a meta-instruction and not part of the text to be translated. Since the text to be translated is "None", the translation is also "None". Here is the formatted output as requested:

None

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

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