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exactly one hundred copies of 1 on the board, replacing numbers with1,10,25

Source: Singapore Junior Math Olympiad 2011 2nd Round p5 SMO

March 26, 2020

combinatoricsgameSum

Problem Statement

Initially, the number

10

is written on the board. In each subsequent moves, you can either (i) erase the number

1

and replace it with a

10

, or (ii) erase the number

10

and replace it with a

1

and a

25

or (iii) erase a

25

and replace it with two

10

. After sometime, you notice that there are exactly one hundred copies of

1

on the board. What is the least possible sum of all the numbers on the board at that moment?