4

Part of the 15th XMO

Problems(1)

Classical NT in XMO

Source: XMO2024

p

be a prime number and

k

is a integer with

p|2^k-1

a\in \{ 1,2,\ldots,p-1\}

m_{a}

be the only element that satisfies

p|am_{a}-1

T_{a}= \{ x \in \{1,2,\ldots p-1\} | \{ \frac {m_{a}x}{p} - \frac {x}{ap} \} < \frac{1}{2}

and there exists integer y satisfying p | x-y^

k+1

\}

Try to proof that there exists an integer

m

1 \le a_1 <a_2< \ldots < a_{m} \le p-1

|T_{a_1}| = |T_{a_2}| = \ldots = |T_{a_{m}}| = m