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Vietnam TST 2017 problem 2

Source: Vietnam TST 2017

March 26, 2017
binomial coefficientsInteger sequencePeriodic sequence

Problem Statement

For each positive integer nn, set xn=(2nn)x_n=\binom{2n}{n}. a. Prove that if 2017k2<n<2017k\frac{2017^k}{2}<n<2017^k for some positive integer kk then 20172017 divides xnx_n. b. Find all positive integer h>1h>1 such that there exists positive integers N,TN,T such that (xn)n>N(x_n)_{n>N} is periodic mod hh with period TT.
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