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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

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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.