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Number Theory Level 3

Let $S_1$ be a sequence with exactly five items. If we replace each item with a number, which shows that the how many times does the item occurs in $S_1$ , then we get the $S_2$ sequence.

For example if $S_1=(1,2,3,2,1)$ , then $S_2=(2,2,1,2,2)$ .

Which of the following sequences could be $S_2$ ?

(1,1,2,2,3) (1,1,1,2,2) (1,3,3,3,3) (1,1,2,2,2) (2,2,2,3,3)

1 solution

Marco Brezzi
Aug 14, 2017

When a number appears $n$ times in $S_1$ , this generates $n$ copies of $n$ in $S_2$ . So in $S_2$ there are $kn$ $(k\in\mathbb{N})$ copies of $n\phantom{c} \forall n$ .

Thus the only possible $S_2$ in the choices is

$(1,1,1,2,2)$

For example $(1,3,3,3,3)$ can be discarded because there are $4$ $3$ s, and $4\neq 3k$

A possible $S_1$ could be

$(\mathbin{\color{#D61F06}1},\mathbin{\color{#3D99F6}2},\mathbin{\color{#20A900}3},\mathbin{\color{#CEBB00}4},\mathbin{\color{#CEBB00}4}) \Longrightarrow (\mathbin{\color{#D61F06}1},\mathbin{\color{#3D99F6}1},\mathbin{\color{#20A900}1},\mathbin{\color{#CEBB00}2},\mathbin{\color{#CEBB00}2})$

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