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Let

q<50

be a prime number. Call a sequence of polynomials

P_0(x), P_1(x), P_2(x), ..., P_{q^2}(x)

tasty if it satisfies the following conditions:
[*]

P_i

has degree

i

for each

i

(where we consider constant polynomials, including the

0

polynomial, to have degree

0

) [*] The coefficients of

P_i

are integers between

0

and

q-1

for each

i

. [*] For any

0\le i,j\le q^2

, the polynomial

P_i(P_j(x)) - P_j(P_i(x))

has all its coefficients divisible by

q

.
As

q

varies over all such prime numbers, determine the total number of tasty sequences of polynomials.
Proposed by Vincent Huang

Problem Statement