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← Opštinsko takmičenje (Municipal round) 2016City

Geometry · Vectors · Centroid · Cevians · Ratios on a segment

Problem 3, 2016

CityProof

Let \(ABC\) be a triangle. On side \(AB\) choose points \(C_1\) and \(C_2\) with

\[ AC_1 = \tfrac{2015}{3015}\,AB, \qquad AC_2 = \tfrac{2015}{3014}\,AB; \]

on side \(BC\) choose points \(A_1\) and \(A_2\) with

\[ BA_1 = \tfrac{1007}{2015}\,BC, \qquad BA_2 = \tfrac{1008}{2015}\,BC; \]

and on side \(AC\) choose points \(B_1\) and \(B_2\) with

\[ AB_1 = \tfrac{2015}{3031}\,AC, \qquad AB_2 = \tfrac{2015}{3030}\,AC. \]

The line \(AA_1\) meets the lines \(B_1C_1\) and \(B_2C_2\) at \(M\) and \(N\) respectively, and the line \(AA_2\) meets \(B_1C_1\) and \(B_2C_2\) at \(Q\) and \(P\) respectively. Prove that the centroid of triangle \(ABC\) lies inside the quadrilateral \(MNPQ\).

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Serbian Municipal Competition 2016, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source

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