Let be a natural number and let be mutually distinct real numbers none of which equals , , , . Prove that
for all . Adding these fractions together we get that
where is a polynomial in of degree at most and for . Since does not equal , , , we see that for . Hence is a polynomial of degree at most with at least roots. Therefore and . We will show that this implies
Suppose , on the contrary , that . Then
and hence contradicting the fact that .
Euler Competition 2010