Maths Olympiad Prep

Track / Stage 6 / 68 of 400 #1068 of 1964

Problem 1068

National olympiad, first round
Algebra Difficulty 6.1 Prove it

Example 1.16 (Han Jingjun) x,y∈Rx, y \in \mathbf{R}, and x2+y2=1x^{2}+y^{2}=1, prove that
1−x+1−12x−32y⩾22\sqrt{1-x}+\sqrt{1-\frac{1}{2} x-\frac{\sqrt{3}}{2} y} \geqslant \frac{\sqrt{2}}{2}

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Prove that after completing the square inside the square root, it is equivalent to
22[(1−x)2+y2+(x−12)2+(y−32)2]⩾22\frac{\sqrt{2}}{2}\left[\sqrt{(1-x)^{2}+y^{2}}+\sqrt{\left(x-\frac{1}{2}\right)^{2}+\left(y-\frac{\sqrt{3}}{2}\right)^{2}}\right] \geqslant \frac{\sqrt{2}}{2}

By Minkowski's inequality, we have
(1−x)2+y2+(x−12)2+(y−32)2⩾(1−12)2+(32)2=1\sqrt{(1-x)^{2}+y^{2}}+\sqrt{\left(x-\frac{1}{2}\right)^{2}+\left(y-\frac{\sqrt{3}}{2}\right)^{2}} \geqslant \sqrt{\left(1-\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}=1

Hence, the proof is complete.

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