Let be a natural number and let be mutually distinct real numbers none of which equals , , , . Prove that

**Solution**

We need to show that the rows of the above matrix are linearly idependent. Consider a linear combination with coefficients . Let

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