Let k be a fixed positive integer the n-th derivative of

Question

Putnam TNG - Polynomials

1: Find polynomials f(x), g(x), and h(x), if they exist, such that for all x,

2: Let k be a fixed positive integer. The n-th derivative of 1/x^k-1 has the form P_n(x)/(x^k-1)ⁿ⁺¹ where P_n(x) is a polynomial. Find P_n(1).

3: Determine necessary and sufficient conditions for the equations:

x + y + z = A        x² + y² + z² = B      x³ + y³ + z³ = C

to have a solution (possibly complex) for (x, y, z) with at least one of x, y or z equal to zero.

4: Let f(x) be a polynomial with real coefficients. If f(x) ≥ 0 for all real x, prove there are polynomials p₁(x), p₂(x), . . . , p_k(x) for which

f(x) = _i=1∑^kp_i²(x).

5: Let f be a non-constant polynomial with positive integer coefficients. Prove that if n is a positive integer, then f(n) divides f(f(n) + 1) if and only if n = 1.

6: Let P(x) be a polynomial of degree n with real coefficients and only real zeroes. Prove that for every real x, (n - 1) [P'(x)]² ≥ nP(x)P''(x), and determine for which P(x) equality is obtained.

7: Let p(x) = x⁵ + x and q(x) = x⁵ + x². Find all pairs (w, z) of complex numbers with w ≠ z for which p(w) = p(z) and q(w) = q(z).

8: Let n be a positive integer. Find the number of pairs P, Q of polynomials with real coefficients such that (P(X))² + (Q(X))² = X²ⁿ + 1 and deg P > deg Q.