The polynomial P(x) has the property that P(1), P(2), P(3), P(4), and P(5) are equal to 1, 2, 3, 4, 5 in some order. How many possibilities are there for the polynomial P, given that the degree of P is strictly less than 4?

(Duke Math Meet 2013 Tiebreaker round)

Discussion:

Let

Suppose P(1), P(2), P(3), P(4) are p,q,r,s respectively. Then in matrix notation we may write:

Here p, q, r, s is a permutation of a selection from 1, 2, 3, 4, 5.

This implies:

Note that or

This implies

Next we work with cases:

P(5) = 2 or 4 (even) implies p is even (hence 4 or 2 respectively). Since otherwise LHS will odd.

Hence . or implies .

Therefore is divisible by 3. This is possible only when q+s = 1+ 5 or 5 +1 and r = 3.