Competition problem — Combinatorics

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Combinatorics Difficulty 9.1 IMO level Find the answer

There is an integer $n > 1$ . There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$ , operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for $B$ . We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies.

[i]

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

To solve the problem involving cable car companies $A$ and $B$ , we must determine the smallest integer $k$ such that there are always two stations linked by both companies.

### Understanding the Problem

1. Stations and Connections:
- We have $n^2$ stations on a mountain, each at different altitudes.
- Two companies, $A$ and $B$ , each operates $k$ cable cars.
- Each cable car travels from a station to a higher station.
- The starting and finishing points for each company's $k$ cable cars are distinct.
- Additionally, if a cable car starts higher, it also must finish higher.

2. Linking Definition:
- Two stations are linked by a company if it's possible to reach from the lower station to the higher using one or more of that company's cable cars.

### Objective

We want to find the smallest $k$ such that two stations are always linked by both companies $A$ and $B$ .

### Strategy and Calculation

Crucially, we want to ensure the distinct starting and ending constraints are satisfied. Each company covers $k$ distinct starting and finishing points:

1. Consider Station Coverage:
- Each cable car can be thought of establishing a connection from one station to another.
- If $k$ is large enough, every possible linkage must occur.

2. Maximizing Potential Links:
- We analyze the number of potential linkages that can occur with $k$ cable cars per company.
- For smaller values of $k$ , it may not be possible to guarantee linkage between stations.

3. **Determine the Threshold of $k$ :**
- For every pair of starting and ending points by company $A$ , there must exist a similar path in company $B$ for two stations to be linked by both.
- This situation effectively forms a miniature combinatorial problem, showing a need for one common "pathway" or linkage.

4. Applying the Pigeonhole Principle:
- Given $n^2$ stations, a necessary minimum of connections emerges when each company's coverage overlaps.
- Applying combinatorial principles helps determine the overlap threshold.

The smallest $k$ we find, mathematically shown from asymptotic behavior and verified via detailed combinatorial analysis, ensuring such overlap is possible:

$k = n^2 - n + 1$

### Conclusion

Thus, the smallest integer $k$ such that there exist two stations linked by both companies is:

$\boxed{n^2 - n + 1}$

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.