MathDB
Three point sets

Source: 1988 National High School Mathematics League, Exam One, Problem 3

February 25, 2020

Problem Statement

M,N,PM,N,P are three point sets on a plane. M={(x,y)∣∣x∣+∣y∣<1}M=\{(x,y)||x|+|y|<1\}, N={(x,y)∣(x−12)2+(y+12)2+(x+12)2+(y−12)2<22}N=\{(x,y)|\sqrt{(x-\frac{1}{2})^2+(y+\frac{1}{2})^2}+\sqrt{(x+\frac{1}{2})^2+(y-\frac{1}{2})^2}<2 \sqrt2 \}, P={(x,y)∣∣x+y∣<1,∣x∣<1,∣y∣<1}P=\{(x,y)||x+y|<1,|x|<1,|y|<1\}.Then (A)M⊂P⊂N(B)M⊂N⊂P(C)P⊂N⊂M(D)\text{(A)}M\subset P\subset N\qquad\text{(B)}M\subset N\subset P\qquad\text{(C)}P\subset N\subset M\qquad\text{(D)} None of(A)(B)(C)\text{(A)(B)(C)}
Back to ProblemsView on AoPS