Maths Olympiad Prep

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Algebra Difficulty 5.3 AIME, harder Prove it Philippines

Problem:

The nonzero numbers xx, yy, and zz satisfy the equations
xy=2(x+y),yz=4(y+z),andxz=8(x+z) x y = 2(x + y), \quad y z = 4(y + z), \quad \text{and} \quad x z = 8(x + z)
Solve for xx.

Solution

Solution:

163\dfrac{16}{3}

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.