Maths Olympiad Prep

Track / Stage 5 / 221 of 400 #821 of 1964

Problem 821

AIME late
Algebra Difficulty 5.5 Prove it

Lemma 2.4. If xx and yy are real numbers, then max⁡(x,y)+min⁡(x,y)\max (x, y)+\min (x, y) =x+y=x+y

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

Proof. If x⩾yx \geqslant y, then min⁡(x,y)=y\min (x, y)=y and max⁡(x,y)=x\max (x, y)=x, so that max⁡(x,y)+min⁡(x,y)=x+y\max (x, y)+\min (x, y)=x+y. If x<yx<y, then min⁡(x,y)=x\min (x, y)=x and max⁡(x,y)=y\max (x, y)=y, and again we find that max⁡(x,y)+min⁡(x,y)=x+y\max (x, y)+\min (x, y)=x+y

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