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Cartesian coordinates

Source: 1981 National High School Mathematics League, Problem 6

February 20, 2020
analytic geometrygeometry

Problem Statement

In Cartesian coordinates, two areas M,NM,N are defined below: M:y≥0,y≤x,y≤2−xM:y\geq0,y\leq x,y\leq 2-x; N:t≤x≤t+1N:t\leq x\leq t+1. tt is a real number that t∈[0,1]t\in[0,1]. Then the area of M∩NM\cap N is (A)−t2+t+12(B)−2t2+2t(C)1−2t2(D)12(t−2)2\text{(A)}-t^2+t+\frac{1}{2}\qquad\text{(B)}-2t^2+2t\qquad\text{(C)}1-2t^2\qquad\text{(D)}\frac{1}{2}(t-2)^2
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