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2020 Taiwan APMO Preliminary Problem 3

Source: 2020 Taiwan APMO Preliminary

July 23, 2020

number theory

Problem Statement

Let

M

is a four digit positive interger. Write

M

backwards and get a new number

N

.(e.g

M=1234

then

N=4321

) Let

C

is the sum of every digit of

M

. If

M,N,C

satisfies (i)

d=\gcd(M-C,N-C)

and

d<10

(ii)

\dfrac{M-C}{d}=\lfloor\dfrac{N}{2}+1\rfloor

(1)Find

d

. (2)If there are "m(s)"

M

satisfies (i) and (ii), and the largest

M

=

M_{max}

. Find

(m,M_{max})

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