Problem 5. An archipelago consists of several small islands and one large island. It was decided to build bridges between the islands so that the large island would be connected to each small island by two bridges, and any two small islands would be connected by one bridge.

By November 1, all the bridges between the small islands and several (at least one) bridges leading to the large island - a total of 28 bridges - had been built. How many islands are there in the archipelago?

Answer: 8 islands
Solution. Let's number the small islands of the archipelago. If a bridge connects islands with numbers $a$ and $b$, we write the smaller of these two numbers on this bridge.

Suppose the number of small islands in the archipelago is no more than six. Then there are no more than 5 bridges with the number 1, no more than 4 bridges with the number 2, and so on. No more than 12 bridges lead to the large island. Therefore, the total number of bridges built is no more than

$$
5+4+3+2+1+12=2728
$$

which also contradicts the condition.

Thus, there are 7 small islands in the archipelago, and a total of 8 islands.

## Criteria

4 points. A complete and justified solution is provided.

06 . The problem is not solved or solved incorrectly.

In other cases, sum the following criteria (but do not exceed 4 points):

2 points. It is proven that the number of islands is no more than eight (if there is an arithmetic error in the proof, then 16 .)

2 6. It is proven that the number of islands is no less than eight (if there is an arithmetic error in the proof, then 1 point).

16 . The correct answer is provided.

![](https://cdn.mathpix.com/cropped/2024_05_06_50551174acfc6b9a623bg-2.jpg?height=320&width=305&top_left_y=90&top_left_x=1552)