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Find all functions $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ that satisfied the functional equation

$\displaystyle f\left(\frac{x+y+z}{3},\frac{a+b+c}{3}\right)=f(x,a)f(y,b)f(z,c)$

for every real number $x,y,z,a,b,c$ that $az+bx+cy\neq ay+bz+cx$

I don't know how to solve it. This question was awarded as the best problem in the competition.

Jiras K.
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  • Do you have an online link? Any similar questions to this you've solved before? (Similar functional equations etc. are welcome). The condition on $x,y,z,a,b,c$ looks weird, never seen anything like that before. Also, if you have any guesses for $f$ or any verified properties from your working, you can mention that as well. – Sarvesh Ravichandran Iyer Jun 26 '22 at 08:14
  • https://www.facebook.com/sense.in.soul/posts/5502813639738538 – Jiras K. Jun 26 '22 at 12:54
  • Thank you for adding the source. Please consider guessing some solutions. For example, merely using constants, $f = 0,1,-1$ are solutions. Also, working with the easier $f\left(\frac{a+b}{2},\frac{x+y}{2}\right) = f(a,x)f(b,y)$ might yield some useful information. – Sarvesh Ravichandran Iyer Jun 26 '22 at 14:16

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