Difference between revisions of "2017 IMO Problems/Problem 2"
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+ | ==Problem== | ||
+ | |||
Let <math>\mathbb{R}</math> be the set of real numbers , determine all functions | Let <math>\mathbb{R}</math> be the set of real numbers , determine all functions | ||
− | <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> such that for any real numbers <math>x</math> and <math>y</math> <math> | + | <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> such that for any real numbers <math>x</math> and <math>y</math> |
+ | |||
+ | <cmath>f(f(x)f(y)) + f(x+y)=f(xy)</cmath> | ||
+ | |||
+ | ==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, <math>f</math> is in the form <math>f(x)=mx+b</math> | ||
+ | |||
+ | Therefore, | ||
+ | |||
+ | <math>f((mx+b)(my+b))+m(x+y)+b=mxy+b</math> | ||
+ | |||
+ | <math>f(m^2xy+mb(x+y)+b^2)+m(x+y)+b=mxy+b</math> | ||
+ | |||
+ | <math>m(m^2xy+mb(x+y)+b^2)+b+m(x+y)+b=mxy+b</math> | ||
+ | |||
+ | <math>m^3xy+m^2b(x+y)+mb^2+m(x+y)+b=mxy</math> | ||
+ | |||
+ | <math>m(m^2-1)xy+m(bm+1)(x+y)+b(bm+1)=0</math> | ||
+ | |||
+ | Therefore, | ||
+ | |||
+ | <math>m(m^2-1)=0</math> [Equation 1] | ||
+ | |||
+ | <math>m(bm+1)=0</math> [Equation 2] | ||
+ | |||
+ | <math>b(bm+1)</math> [Equation 3] | ||
+ | |||
+ | From [Equation 1] we have, <math>m=0,\pm 1</math> | ||
+ | |||
+ | From [Equation 2] we have, <math>m=0, bm=-1\pm 1</math> | ||
+ | |||
+ | From [Equation 3] we have, <math>b=0, bm=-1\pm 1</math> | ||
+ | |||
+ | When <math>m=0</math>, <math>b=0</math>, then <math>f(x)=0</math> | ||
+ | |||
+ | When <math>bm=-1</math>, <math>b=\frac{-1}{m}</math>, then since <math>m=\pm 1</math>, then <math>b=\mp 1</math> | ||
+ | |||
+ | When <math>bm=-1</math>, <math>b=\frac{-1}{m}</math>, then since <math>m=\pm 1</math>, then <math>b=\mp 1</math> then <math>f(x)=\pm x \mp1</math> which gives these two functions: | ||
+ | |||
+ | <math>f(x)=x-1</math> and <math>f(x)=1-x</math>, which with <math>f(x)=0</math> provide all the three functions for this problem. | ||
+ | |||
+ | ~Tomas Diaz. orders@tomasdiaz.com | ||
+ | |||
+ | {{alternate solutions}} | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{IMO box|year=2017|num-b=1|num-a=3}} |
Latest revision as of 10:15, 20 November 2023
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 |