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To find minimum, assume that everyone else got maximum.
Let the minimum score is x.
Other 4 scores are 100,100,100 and 100.

Given that mean is 89

 \frac{100+100+100+100+x}{5}=89\\ \\  \frac{400+x}{5}=89\\ \\ 400+x=89 \times 5\\ \\400+x=445\\ \\ x=445-400\\ \\x=45

So minimum score that can be achieved is 45.
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