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Number theory hidden behind a geometrical statement

Source: Bulgaria NMO 2022 P5

April 17, 2022
geometrynumber theorygeometric sequenceratio

Problem Statement

Let ABCABC be an isosceles triangle with AB=4AB=4, BC=CA=6BC=CA=6. On the segment ABAB consecutively lie points X1,X2,X3,…X_{1},X_{2},X_{3},\ldots such that the lengths of the segments AX1,X1X2,X2X3,…AX_{1},X_{1}X_{2},X_{2}X_{3},\ldots form an infinite geometric progression with starting value 33 and common ratio 14\frac{1}{4}. On the segment CBCB consecutively lie points Y1,Y2,Y3,…Y_{1},Y_{2},Y_{3},\ldots such that the lengths of the segments CY1,Y1Y2,Y2Y3,…CY_{1},Y_{1}Y_{2},Y_{2}Y_{3},\ldots form an infinite geometric progression with starting value 33 and common ratio 12\frac{1}{2}. On the segment ACAC consecutively lie points Z1,Z2,Z3,…Z_{1},Z_{2},Z_{3},\ldots such that the lengths of the segments AZ1,Z1Z2,Z2Z3,…AZ_{1},Z_{1}Z_{2},Z_{2}Z_{3},\ldots form an infinite geometric progression with starting value 33 and common ratio 12\frac{1}{2}. Find all triplets of positive integers (a,b,c)(a,b,c) such that the segments AYaAY_{a}, BZbBZ_{b} and CXcCX_{c} are concurrent.
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