Difference between revisions of "2025 IMO Problems/Problem 1"
(Created page with "A line in the plane is called <math>sunny</math> if it is not parallel to any of the <math>x</math>–axis, the <math>y</math>–axis, or the line <math>x+y=0</math>. Let <ma...") |
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| − | A line in the plane is called < | + | A line in the plane is called <i>sunny</i> if it is not parallel to any of the <math>x</math>–axis, the <math>y</math>–axis, or the line <math>x+y=0</math>. |
Let <math>n\ge3</math> be a given integer. Determine all nonnegative integers <math>k</math> such that there exist <math>n</math> distinct lines in the plane satisfying both of the following: | Let <math>n\ge3</math> be a given integer. Determine all nonnegative integers <math>k</math> such that there exist <math>n</math> distinct lines in the plane satisfying both of the following: | ||
| − | + | * For all positive integers <math>a</math> and <math>b</math> with <math>a+b\le n+1</math>, the point <math>(a,b)</math> lies on at least one of the lines; and | |
| − | + | * Exactly <math>k</math> of the <math>n</math> lines are sunny. | |
| + | |||
== Video Solution == | == Video Solution == | ||
https://www.youtube.com/watch?v=kJVQqugw_JI [includes motivational discussion] | https://www.youtube.com/watch?v=kJVQqugw_JI [includes motivational discussion] | ||
Revision as of 01:03, 16 July 2025
A line in the plane is called sunny if it is not parallel to any of the 
–axis, the 
–axis, or the line 
.
Let 
be a given integer. Determine all nonnegative integers 
such that there exist 
distinct lines in the plane satisfying both of the following:
- For all positive integers
and 
with 
, the point 
lies on at least one of the lines; and - Exactly
of the 
lines are sunny.
and 
with 
, the point 
lies on at least one of the lines; and
Video Solution
https://www.youtube.com/watch?v=kJVQqugw_JI [includes motivational discussion]
