Difference between revisions of "2020 IMO Problems/Problem 4"

 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
Problem 4. There is an integer n > 1. There are n
+
== Problem ==
2
+
There is an integer <math>n > 1</math>. There are <math>n^2</math> stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, <math>A</math> and <math>B</math>, operates <math>k</math> cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The <math>k</math> cable cars of <math>A</math> have <math>k</math> different starting points and <math>k</math> different finishing points, and a cable car that starts higher also finishes higher. The same conditions hold for <math>B</math>. 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).
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
+
Determine the smallest positive integer k for which one can guarantee that there are two stations that are linked by both companies.
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
 
  
 
== Video solution ==
 
== Video solution ==
 
https://www.youtube.com/watch?v=dTqwOoSfaAA [video covers all day 2 problems]
 
https://www.youtube.com/watch?v=dTqwOoSfaAA [video covers all day 2 problems]
 +
 +
==See Also==
 +
 +
{{IMO box|year=2020|num-b=3|num-a=5}}
 +
 +
[[Category:Olympiad Combinatorics Problems]]

Latest revision as of 11:31, 14 May 2021

Problem

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 that 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.

Video solution

https://www.youtube.com/watch?v=dTqwOoSfaAA [video covers all day 2 problems]

See Also

2020 IMO (Problems) • Resources
Preceded by
Problem 3
1 2 3 4 5 6 Followed by
Problem 5
All IMO Problems and Solutions
Invalid username
Login to AoPS