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A3

Part of 1962 Putnam

Problems(1)

Putnam 1962 A3

Source: Putnam 1962

5/15/2022
In a triangle ABCABC, let A′A' be a point on the segment BCBC, B′B' be a point on the segment CACA and C′C' a point on the segment ABAB such that AB′B′C=BC′C′A=CA′A′B=k, \frac{AB'}{B'C}= \frac{BC'}{C'A} =\frac{CA'}{A'B}=k, where kk is a positive constant. Let △\triangle be the triangle formed by the interesctions of AA′AA', BB′BB' and CC′CC'. Prove that the areas of △\triangle and ABCABC are in the ratio (k−1)2k2+k+1.\frac{(k-1)^{2}}{k^2 +k+1}.
PutnamratioTrianglearea of a triangle