Define polynomials fnx for n ge 0 by f0x 1 fn0 0 for n ge

Question

Putnam TNG - Polynomials and Binomials

1: Define polynomials f_n(x) for n ≥ 0 by f₀(x) = 1, f_n(0) = 0 for n ≥ 1, and d/dxf_n+1(x) = (n + 1)f_n(x + 1) for n ≥ 0. Find, with proof, the explicit factorization of f₁₀₀(1) into powers of distinct primes.

2: Prove that the number of odd binomial coefficients in any finite binomial expression is a power of 2.

3: Consider all lines which meet the graph y = 2x⁴ + 7x³ + 3x - 5 in four distinct points, say (x_i, y_i), i = 1, 2, 3, 4. Show that x₁ + x₂ + x₃ + x₄/4, is independent of the line, and find its value.

4: Let a, b, c be real numbers such that a + b + c = 0. Prove that

a⁵ + b⁵ + c⁵/5 = (a³ + b³ + c³/3)·(a² + b² + c²/2).

5: Let p be a prime number. Show that

6: Let x⁽ⁿ⁾ = x(x - 1)· · ·(x - n + 1) for n a positive integer, and let x⁽⁰⁾ = 1. Prove that for all real numbers x and y

7: Let p(x) be a polynomial that is nonnegative for all real x. Prove that for some k, there are polynomials f₁(x), ..., f_k(x) such that

p(x) = _j=1∑^k(f_j (x))².