[ Counting in two ways ] $[\quad$ Trees $]$

A travel agency ran a promotion: "Buy a trip to Egypt, bring four friends who also buy a trip, and get the cost of your trip back." During the promotion, 13 buyers came on their own, and the rest were brought by friends. Some of them brought exactly four new customers, while the other 100 did not bring anyone. How many tourists went to the Land of the Pyramids for free?
Each of the $x$ "lucky ones" brought 4 friends. Therefore, the number of "brought" customers is $4x$, and 13 more came on their own, so the total number of tourists was $13 + 4x$. On the other hand, $x$ people brought new customers, while 100 did not, meaning the total number of tourists was $x + 100$. Thus, $13 + 4x = x + 100$, from which $x = 29$.

## Answer

29 tourists.