Competition problem — Algebra — Maths Olympiad Prep

Library / /42 of 520

Algebra Difficulty 4.7 AIME Find the answer

2. If $1=\frac{x y}{x+y}, 2=\frac{y z}{y+z}, 3=\frac{z x}{2+x}$ , then the value of $x$ is

Your answer

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

Solution

(Given: from the equations $\frac{1}{x}+\frac{1}{y}=1, \frac{1}{y}+$ $\frac{1}{z}$ $=\frac{1}{2} \cdot \frac{1}{z}+\frac{1}{x}=\frac{1}{3} \cdot x=\frac{12}{5}.$ )

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.