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ratio which is not less than 1 / sqrt{3}

Source: Vietnam TST 2003 for the 44th IMO, problem 4

June 26, 2005
ratiogeometryperimetergeometry solved

Problem Statement

On the sides of triangle ABCABC take the points M1,N1,P1M_1, N_1, P_1 such that each line MM1,NN1,PP1MM_1, NN_1, PP_1 divides the perimeter of ABCABC in two equal parts (M,N,PM, N, P are respectively the midpoints of the sides BC,CA,ABBC, CA, AB). I. Prove that the lines MM1,NN1,PP1MM_1, NN_1, PP_1 are concurrent at a point KK. II. Prove that among the ratios KABC,KBCA,KCAB\frac{KA}{BC}, \frac{KB}{CA}, \frac{KC}{AB} there exist at least a ratio which is not less than 13\frac{1}{\sqrt{3}}.
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