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HMMT November, 2014

Algebra Difficulty 4.9 AIME Prove it United States

Problem:

Consider the parabola consisting of the points (x,y)(x, y) in the real plane satisfying
(y+x)=(y−x)2+3(y−x)+3. (y+x) = (y-x)^2 + 3(y-x) + 3.
Find the minimum possible value of yy.

Solution

Solution:

Let w=y−xw = y - x. Adding ww to both sides and dividing by two gives
y=w2+4w+32=(w+2)2−12 y = \frac{w^2 + 4w + 3}{2} = \frac{(w+2)^2 - 1}{2}
which is minimized when w=−2w = -2. This yields y=−12y = -\frac{1}{2}.

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