MathDB

2

Part of 1991 Romania Team Selection Test

Problems(2)

concurrent sets of planes

Source: Romania IMO TST 1991 p2

2/19/2020
Let A1A2A3A4A_1A_2A_3A_4 be a tetrahedron. For any permutation (i,j,k,h)(i, j,k,h) of 1,2,3,41,2,3,4 denote: - PiP_i – the orthogonal projection of AiA_i on AjAkAhA_jA_kA_h; - BijB_{ij} – the midpoint of the edge AiAjA_iAj, - CijC_{ij} – the midpoint of segment PiPjP_iP_j - βij\beta_{ij}– the plane BijPhPkB_{ij}P_hP_k - δij\delta_{ij} – the plane BijPiPjB_{ij}P_iP_j - αij\alpha_{ij} – the plane through CijC_{ij} orthogonal to AkAhA_kA_h - γij\gamma_{ij} – the plane through CijC_{ij} orthogonal to AiAjA_iA_j. Prove that if the points PiP_i are not in a plane, then the following sets of planes are concurrent: (a) αij\alpha_{ij}, (b) βij\beta_{ij}, (c) γij\gamma_{ij}, (d) δij\delta_{ij}.
concurrent planesconcurrentplanestetrahedron
a_{n+2 }= a_{n+1} +a_n +k, least k for which a1991 and 1991 not coprime

Source: Romania IMO TST 1991 p2

2/19/2020
The sequence (ana_n) is defined by a1=a2=1a_1 = a_2 = 1 and an+2=an+1+an+ka_{n+2 }= a_{n+1} +a_n +k, where kk is a positive integer. Find the least kk for which a1991a_{1991} and 19911991 are not coprime.
recurrence relationSequencecoprimenumber theory