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Bundeswettbewerb Mathematik 1982 Problem 1.2

Source: Bundeswettbewerb Mathematik 1982 Round 1

September 22, 2022
equal lengthsgeometryquadrilateralSides

Problem Statement

In a convex quadrilateral ABCDABCD sides ABAB and DCDC are both divided into mm equal parts by points A,S1,S2,…,Sm−1,BA, S_1 , S_2 , \ldots , S_{m-1} ,B and D,T1,T2,…,Tm−1,C,D,T_1, T_2, \ldots , T_{m-1},C, respectively (in this order). Similarly, sides BCBC and ADAD are divided into nn equal parts by points B,U1,U2,…,Un−1,CB,U_1,U_2, \ldots, U_{n-1},C and A,V1,V2,…,Vn−1,DA,V_1,V_2, \ldots,V_{n-1}, D. Prove that for 1≤i≤m−11 \leq i \leq m-1 each of the segments SiTiS_i T_i is divided by the segments UjVjU_j V_j (1≤j≤n−11\leq j \leq n-1) into nn equal parts
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