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In the country of Distant, a province is called large if more than $7 \%$ of the country's population lives in it. It is known that for each large province, there are two provinces with a smaller population such that their combined population is greater than that of this large province. What is the smallest number of provinces that can be in the country of Distant?
We will order the provinces by increasing population. The first and second provinces are not large, as for each of them, there will not be two provinces with a smaller population. In the third province, less than $14\%$ of the population lives, since in both provinces with a smaller population, the total is no more than $7\%+7\%$ $=14\%$. In the fourth province, less than $21\%$ of the population lives, as the population of any two smaller provinces is less than $7\%+14\%$ $=21\%$. For the same reasons, in the fifth province, less than $14\%+21\%$ $=35\%$ of the population lives. Therefore, in the first five provinces, less than $7\%+7\%+14\%+21\%+35\%$ $=84\%$ of the population lives. Consequently, there are more than 5 provinces.

Example of population distribution across six provinces: $7\%$, $7\%$, $11\%$, $16\%$, $25\%$, $34\%$ of the population.

## Answer

6 provinces.