MathDB

280

Part of 1979 All Soviet Union Mathematical Olympiad

Problems(1)

ASU 280 All Soviet Union MO 1979 1979 lines in space

Source:

7/14/2019
Given the point OO in the space and 19791979 straight lines l1,l2,...,l1979l_1, l_2, ... , l_{1979} containing it. Not a pair of lines is orthogonal. Given a point A1A_1 on l1l_1 that doesn't coincide with OO. Prove that it is possible to choose the points AiA_i on lil_i (i=2,3,...,1979i = 2, 3, ... , 1979) in so that 19791979 pairs will be orthogonal: A1A3A_1A_3 and l2l_2, A2A4A_2A_4 and l3l_3,... ... , Ai−1Ai+1A_{i-1}A_{i+1} and lil_i,... ... , A1977A1979A_{1977}A_{1979} and l1978l_{1978}, A1978A1A_{1978}A_1 and l1979l_{1979}, A1979A2A_{1979}A_2 and l1l_1
combinatorial geometrylinesorthogonal