MathDB

2

Part of 1982 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 1982 Problem 1.2

Source: Bundeswettbewerb Mathematik 1982 Round 1

9/22/2022
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
equal lengthsgeometryquadrilateralSides
Bundeswettbewerb Mathematik 1981 Problem 2.2

Source: Bundeswettbewerb Mathematik 1981 Round 2

9/22/2022
Decide whether every triangle ABCABC in space can be orthogonally projected onto a plane such that the projection is an equilateral triangle A′B′C′A'B'C'.
TriangleEquilateralprojectionspacegeometry