Problem 3. In a certain company, there are 100 shareholders, and any 66 of them collectively own at least $50\%$ of the company's shares. What is the largest percentage of all shares that one shareholder can own? (The percentage of shares in the company owned by a shareholder can be non-integer.)
Answer: $25 \%$.

Solution. Consider any shareholder $A$. Divide the other 99 shareholders into three groups $B, C, D$ of 33 shareholders each. By the condition, $B$ and $C$ together have at least $50 \%$ of the company's shares, similarly for $C$ and $D$, and for $B$ and $D$. Adding all this up and dividing by two, we get that $B, C, D$ together have at least $((50+50+50) : 2) \% = 75 \%$ of the shares, so $A$ has no more than the remaining $25 \%$ of the shares.

Moreover, $A$ can have exactly $25 \%$ of the shares if all other shareholders have an equal number of shares, each with $\frac{75}{99} \% = \frac{25}{33} \%$. Clearly, in this case, any 66 shareholders of the company together have at least $66 \cdot \frac{25}{33} \% = 50 \%$ of the shares.

## Criteria

The largest applicable criterion is used:

The following criteria are summed:

5 p. It is proven that any shareholder has no more than $25 \%$ of the shares.

2 p. It is proven that some shareholder can have exactly $25 \%$ of the shares.