of degree 
having 
distinct real roots 
)
: 
.
defined on all real numbers and taking real values such that ![\[f(f(y)) + f(x - y) = f(xf(y) - x),\]](http://latex.artofproblemsolving.com/f/5/2/f528fc4047e21247663d4993913eadf346315be7.png)
for all real numbers 
.
such that 
.
![$P\in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/7/4/c/74cb2247069abb8a22666e87fcbd69a55a015004.png)
for which there exists 
such that for all 
we have: ![\[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]](http://latex.artofproblemsolving.com/9/0/9/90954cdeade9d13149e1aaa03df2c87dff157041.png)
Y by Adventure10, Mango247
players participated in a competition. Any two players have played exactly one game, and there was no tie game. For a set of 
players, if it is able to line the players up so that each player won every player at the back, we call the set ranked. For each player who participated in the competition, the set of players who lost to the player is ranked. Prove that the whole set of players can be split into three or less ranked sets.This post has been edited 1 time. Last edited by Starrysky, Jul 26, 2019, 7:33 AM
![https://i.imgur.com/1cMbOvy.jpg?1/img]](https://i.imgur.com/1cMbOvy.jpg?1/img])
