2017 IMO Problems/Problem 2
Problem
Let be the set of real numbers , determine all functions such that for any real numbers and
Solution
Looking at the equation one can deduce that the functions that will work will be linear. That is, a polynomial of at most a degree of 1.
Thus, is in the form
Therefore,
Therefore,
[Equation 1]
[Equation 2]
[Equation 3]
From [Equation 1] we have,
From [Equation 2] we have,
From [Equation 3] we have,
When , , then
When , , then since , then
When , , then since , then then which gives these two functions:
and , which with provide all the three functions for this problem.
~Tomas Diaz. orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
2017 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |