MathDB

Let

p = 9001

be a prime number and let

\mathbb{Z}/p\mathbb{Z}

denote the additive group of integers modulo

p

. Furthermore, if

A, B \subset \mathbb{Z}/p\mathbb{Z}

, then denote

A+B = \{a+b \pmod{p} | a \in A, b \in B \}.

Let

s_1, s_2, \dots, s_8

are positive integers that are at least

2

. Yang the Sheep notices that no matter how he chooses sets

T_1, T_2, \dots, T_8\subset \mathbb{Z}/p\mathbb{Z}

such that

|T_i| = s_i

for

1 \le i \le 8,

T_1+T_2+\dots + T_7

is never equal to

\mathbb{Z}/p\mathbb{Z}

, but

T_1+T_2+\dots+T_8

must always be exactly

\mathbb{Z}/p\mathbb{Z}

. What is the minimum possible value of

s_8

?
Proposed by Yang Liu

22