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International Math Olympiad 2010 Problem 1 - A beautiful functional equation from the IMO
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This is a problem from the 2010 International Math Olympiad, Problem 1.
Theory for the this problem:
- you understand what a functional equation is and how they are generally solved in math problems
- you understand what the floor function is
This is really a problem with little theory needed and is quite approachable if you have a little familiarity in solving functional equations. If you believe you are generally good in algebra and you want to begin solving problems from the IMO, I think this is a pretty good problem to begin with.
Timestamps:
00:00 - 00:25 Suggestions for how long to try the problem on your own
00:28 - 00:40 Summary of problem
00:40 - 00:48 What to do in functional equations
00:48 - 01:48 Plugging stuff in and a general principle in functional equations
01:43 - 01:49 Invitation to pause for 5-10 mins and try to solve the case 1
01:53 - 03:10 Case 1 solution
03:10 - 03:20 Invitation to pause for 5-10 mins and try to solve the case 2
03:20 - 03:50 First idea i
03:50 - 04:50 Plugging stuff in and conclusion
04:50 - 05:35 Solving sub-case 1
05:40 - 05:55 Invitation to pause for 5 mins and try to solve sub-case 2
05:55 - 11:35 Solving sub-case 2
11:35 - 13:40 Summary of everything we did
13:40 - 13:46 Thanks for problem solving
Theory for the this problem:
- you understand what a functional equation is and how they are generally solved in math problems
- you understand what the floor function is
This is really a problem with little theory needed and is quite approachable if you have a little familiarity in solving functional equations. If you believe you are generally good in algebra and you want to begin solving problems from the IMO, I think this is a pretty good problem to begin with.
Timestamps:
00:00 - 00:25 Suggestions for how long to try the problem on your own
00:28 - 00:40 Summary of problem
00:40 - 00:48 What to do in functional equations
00:48 - 01:48 Plugging stuff in and a general principle in functional equations
01:43 - 01:49 Invitation to pause for 5-10 mins and try to solve the case 1
01:53 - 03:10 Case 1 solution
03:10 - 03:20 Invitation to pause for 5-10 mins and try to solve the case 2
03:20 - 03:50 First idea i
03:50 - 04:50 Plugging stuff in and conclusion
04:50 - 05:35 Solving sub-case 1
05:40 - 05:55 Invitation to pause for 5 mins and try to solve sub-case 2
05:55 - 11:35 Solving sub-case 2
11:35 - 13:40 Summary of everything we did
13:40 - 13:46 Thanks for problem solving
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