Count the Good Cyclic Shifts - Math Problems | Repovive

MATH3. Count the Good Cyclic Shifts Beta

Let $n$ be a positive integer, and let $a_1,\ldots,a_n$ be integers satisfying $a_i\le1$ for every $i$ . Suppose

$a_1+\cdots+a_n=k>0.$

For each $t\in\{0,\ldots,n-1\}$ , consider the cyclic shift

$a_{t+1},a_{t+2},\ldots,a_n,a_1,\ldots,a_t.$

Prove that exactly $k$ choices of $t$ make every nonempty prefix of this shift have strictly positive sum. Different starting indices count separately, even when they produce identical sequences.

Proof

You can write formulas in LaTeX or in plain text, and write your solution in any language.

Let $n$ be a positive integer, and let $a_1,\ldots,a_n$ be integers satisfying $a_i\le1$ for every $i$ . Suppose

$a_1+\cdots+a_n=k>0.$

For each $t\in\{0,\ldots,n-1\}$ , consider the cyclic shift

$a_{t+1},a_{t+2},\ldots,a_n,a_1,\ldots,a_t.$

Prove that exactly $k$ choices of $t$ make every nonempty prefix of this shift have strictly positive sum. Different starting indices count separately, even when they produce identical sequences.