Maths Olympiad Prep

Library / /33 of 841

Algebra Difficulty 4.6 AIME Find the answer

Two reals x x and y y are such that x−y=4 x-y=4 and x3−y3=28 x^{3}-y^{3}=28 . Compute xy x y .

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

Solution

We have 28=x3−y3=(x−y)(x2+xy+y2)=(x−y)((x−y)2+3xy)=4⋅(16+3xy) 28=x^{3}-y^{3}=(x-y)\left(x^{2}+x y+y^{2}\right)=(x-y)\left((x-y)^{2}+3 x y\right)=4 \cdot(16+3 x y) , from which xy=−3 x y=-3 .

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: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.