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Part of 2022 Malaysia IMONST 2

Problems(3)

Prove multiple of 2^2022 that has 2022 digits can be written with 1 and 2 only

Source: IMONST2 2022 Senior #3

Prove that there is a multiple of

2^{2022}

2022

digits, and can be written using digits

1

2

[IMONST2 2022 Primary #3] sum of n(n+3)

Source:

1\cdot 4 + 2\cdot 5 + 3\cdot 6 + \cdots + n(n+3) = \frac{n(n+1)(n+5)}{3}

for all positive integer

n

arithmeticalgebrainduction

[IMONST2 2022 Junior #3] m+n=999999999

Source:

Given an integer

n

. We rearrange the digits of

n

to get another number

m

. Prove that it is impossible to get

m+n = 999999999