MathDB

2

Part of 2017 Vietnam Team Selection Test

Problems(2)

Vietnam TST 2017 problem 2

Source: Vietnam TST 2017

3/26/2017
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.
binomial coefficientsInteger sequencePeriodic sequence
Vietnam TST 2017 problem 5

Source:

3/26/2017
Given 20172017 positive real numbers a1,a2,…,a2017a_1,a_2,\dots ,a_{2017}. For each n>2017n>2017, set an=max⁡{ai1ai2ai3∣i1+i2+i3=n,1≤i1≤i2≤i3≤n−1}.a_n=\max\{ a_{i_1}a_{i_2}a_{i_3}|i_1+i_2+i_3=n, 1\leq i_1\leq i_2\leq i_3\leq n-1\}. Prove that there exists a positive integer m≤2017m\leq 2017 and a positive integer N>4mN>4m such that anan−4m=an−2m2a_na_{n-4m}=a_{n-2m}^2 for every n>Nn>N.
algebra