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compare terms of 3 sequences

Source: Polish MO Finals 1970 p2

August 22, 2024
algebrainequalities

Problem Statement

Consider three sequences (an)n=1∞(a_n)_{n=1}^{^\infty}, (bn)n=1∞(b_n)_{n=1}^{^\infty} , (cn)n=1∞(c_n)_{n=1}^{^\infty}, each of which has pairwisedistinct terms. Prove that there exist two indices kk and ll for which k<lk < l, ak<al,bk<bl,   and   ck<cl.a_k < a_l , b_k < b_l , \,\,\, and \,\,\, c_k < c_l.
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