Consider sequences $a_0$,$a_1$,$a_2$,$\cdots$ of non-negative integers defined by selecting any $a_0$,$a_1$,$a_2$ (not all 0) and for each $n$ $\geq$ 3 letting $a_n$ = |$a_n-1$ - $a_n-3$| 1-In the particular case that $a_0$ = 1,$a_1$ = 3 and $a_2$ = 2, calculate the beginning of the sequence, listing $a_0$,$a_1$,$\cdots$,$a_{19}$,$a_{20}$. 2-Prove that for each sequence, there is a constant $c$ such that $a_i$ $\leq$ $c$ for all $i$ $\geq$ 0. Note that the constant $c$ my depend on the numbers $a_0$,$a_1$ and $a_2$ 3-Prove that, for each choice of $a_0$,$a_1$ and $a_2$, the resulting sequence is eventually periodic. 4-Prove that, the minimum length p of the period described in (3) is the same for all permitted starting values $a_0$,$a_1$,$a_2$ of the sequence
Problem
Source: GMO 2016
Tags: GMO-Gulf Mathmatical Olympiad, algebra
08.09.2017 17:29
m2121 wrote: Consider sequences $a_0$,$a_1$,$a_2$,$\cdots$ of non-negative integers defined by selecting any $a_0$,$a_1$,$a_2$ (not all 0) and for each $n$ $\geq$ 3 letting $a_n$ = |$a_{n-1}$ - $a_{n-3}$| 1-In the particular case that $a_0$ = 1,$a_1$ = 3 and $a_2$ = 2, calculate the beginning of the sequence, listing $a_0$,$a_1$,$\cdots$,$a_{19}$,$a_{20}$. 2-Prove that for each sequence, there is a constant $c$ such that $a_i$ $\leq$ $c$ for all $i$ $\geq$ 0. Note that the constant $c$ my depend on the numbers $a_0$,$a_1$ and $a_2$ 3-Prove that, for each choice of $a_0$,$a_1$ and $a_2$, the resulting sequence is eventually periodic. 4-Prove that, the minimum length p of the period described in (3) is the same for all permitted starting values $a_0$,$a_1$,$a_2$ of the sequence I think this is the problem
08.09.2017 18:14
2 and 3) are easy. 2)$ a_n \leq max( a_{n-1},a_{n-3})$ so $a_n \leq c=max (a_0,a_1,a_2)$ 3.Let $a_n \leq c$. Let triple $b_n=(a_n,a_{n+1},a_{n+2})$ As $a_n$ can take only $c+1$ different values, then every triple $b_n$ can take only $(c+1)^3$ different values, so there are such $0 \leq i<j \leq(c+1)^3+1$ that $b_i=b_j$ and so $a_i=a_j,a_{i+1}=a_{j+1},a_{i+2}=a_{j+2},...\to a_n$ periodic with period $j-i$
23.08.2019 06:57
3,4 also solved here as I am creating the Gulf Math Olympiad post collections, it is better to can find all solutions given from the post of the related collection