MathDB
Putnam 1973 A1

Source: Putnam 1973

May 25, 2022
Putnamgeometryareatriangle inequalityarea of a triangle

Problem Statement

(a) Let ABCABC be any triangle. Let X,Y,ZX, Y, Z be points on the sides BC,CA,ABBC, CA, AB respectively. Suppose that BX≤XC,CY≤YA,AZ≤ZBBX \leq XC, CY \leq YA, AZ \leq ZB. Show that the area of the triangle XYZXYZ \geq 1\slash 4 times the area of ABC.ABC. (b) Let ABCABC be any triangle, and let X,Y,ZX, Y, Z be points on the sides BC,CA,ABBC, CA, AB respectively. Using (a) or by any other method, show: One of the three corner triangles AZY,BXZ,CYXAZY, BXZ, CYX has an area ≤\leq area of the triangle XYZ.XYZ.
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