7.5. On a line, one hundred points are marked: green, blue, and red. It is known that between any two red points there is a blue one, and between any two blue points there is a green one. In addition, there are no fewer red points than blue ones, and no fewer blue points than green ones. How many points are painted blue?
Answer: 33.

Solution: Let the number of red points be $n$, then the number of blue points is not less than $n-1$ (the number of intervals between "adjacent" red points), and since by condition the number of red points is not less than the number of blue points, the number of blue points is either $n$ or $n-1$. Similarly, if the number of blue points is $m$, then the number of green points is either $m$ or $m-1$. Thus, there are 4 possible cases.

| red | blue | green | total |
| :--: | :--: | :--: | :--: |
| $n$ | $n$ | $n$ | $3n$ |
| $n$ | $n$ | $n-1$ | $3n-1$ |
| $n$ | $n-1$ | $n-1$ | $3n-2$ |
| $n$ | $n-1$ | $n-2$ | $3n-3$ |

In each case, it is easy to find the total number of points (see table), which by condition is 100. Thus, we get 4 equations, only one of which has an integer solution $100=3n-2$. From this, the number of red points is 34, blue - 33, and green - 33.

Criteria: only answer, answer with verification - 0 points. Obtaining the estimate that the number of blue points is not less than the number of red points minus 1, and the number of green points is not less than the number of blue points minus 1 - 3 points (for both estimates together, not for each).

## Solutions to the tasks of the first stage of the All-Siberian Open Olympiad for schoolchildren in mathematics 2014-15, November 9, 2014

## 8th grade

Each task is worth 7 points.