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1-1024, deleting 4k+3 numbers, successively for 5 times

Source: JBMO 2008 Shortlist A9

October 14, 2017
JBMOalgebra

Problem Statement

Consider an integer n≥4n \ge 4 and a sequence of real numbers x1,x2,x3,...,xnx_1, x_2, x_3,..., x_n. An operation consists in eliminating all numbers not having the rank of the form 4k+34k + 3, thus leaving only the numbers x3.x7.x11,...x_3. x_7. x_{11}, ...(for example, the sequence 4,5,9,3,6,6,1,84,5,9,3,6, 6,1, 8 produces the sequence 9,19,1). Upon the sequence 1,2,3,...,10241, 2, 3, ..., 1024 the operation is performed successively for 55 times. Show that at the end only one number remains and fi nd this number.
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