P4

Part of 2024 Bangladesh Mathematical Olympiad

Problems(1)

Non-trivial sequence, do you exist?

Source: BdMO 2024 Secondary National P4 Higher Secondary National P4

a_1, a_2, \ldots, a_{11}

be integers. Prove that there exist numbers

b_1, b_2, \ldots, b_{11}

b_i

-1,0

1

i \in \{1, 2,\dots, 11\}

. [*] all numbers can't be zero at a time. [*] the number

N=a_1b_1+a_2b_2+\ldots+a_{11}b_{11}

is divisible by

2024

combinatoricsSequencemodular arithmeticpigeonhole principle