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N

Part of 2021 Taiwan TST Round 1

Problems(1)

Not eventually odd and not eventually even

Source: 2021 Taiwan TST Round 1 Independent Study 1-N

7/22/2021
For each positive integer nn, define Vn=⌊2n2020⌋+⌊2n2021⌋V_n=\lfloor 2^n\sqrt{2020}\rfloor+\lfloor 2^n\sqrt{2021}\rfloor. Prove that, in the sequence V1,V2,…,V_1,V_2,\ldots, there are infinitely many odd integers, as well as infinitely many even integers.
Remark. ⌊x⌋\lfloor x\rfloor is the largest integer that does not exceed the real number xx.
floor functionSequencenumber theoryTaiwan