4. In a joint-stock company, there are 2017 shareholders, and any 1500 of them hold a controlling stake (not less than $50 \%$ of the shares). What is the largest share of shares that one shareholder can have

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# Solution

Let's order the shareholders by the ascending share of their stocks. The 1500th shareholder has no less than $\frac{50}{1500} \%=\frac{1}{30} \%$ of the shares. If the 1500th has less, then the first 1500 shareholders together have less than $50 \%$ of the shares. Therefore, the first 2016 shareholders together have no less than $50 \%+(2016-1500) \frac{1}{30} \%=67.2 \%$ of the shares. The largest share of the 2017th shareholder is $32.8 \%$ of the shares.

## Criteria

A solution that arrives at the correct answer but assumes that each shareholder has no less than $\frac{1}{30} \%$ should be scored 4 points.

If the correct approach is used but one computational error leads to an incorrect answer, score 5 points.

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