Complex numbers and polynomials - University College Cork

2. For which n ∈ N is x 2 +x+1 a factor of x 2n +x n +1? [Hint: Say x 2n +x n +1 =(x 2 + x + 1)q(x), for some poly q.]3. (IMO 1973) Find the minimum value of a 2 + b 2 , where a, b are real numbersfor which the equationhas at least one real root.x 4 + ax 3 + bx 2 + ax + 1 = 04. USAMO75. Let p be a poly of degree n and suppose thatp(k) =k , k = 0, 1, 2, . . . , n.k + 1Determine p(n + 1). [Hint: Consider the poly q(x) = (x + 1)p(x) − x.]5. IRMO89. Let a be a positive real number, and letb = 3 √a + √ a 2 + 1 + 3 √a − √ a 2 + 1.Prove that b is a positive integer if, and only if, a is a positive integer of theform 1 2 n(n2 + 3), for some positive integer n.6. IRMO91. Find all polynomialsf(x) = a 0 + a 1 x + · · · + a n x nsatisfying the equationfor all real numbers x.f(x 2 ) = (f(x)) 27. IRMO91. Find all polynomialsf(x) = x n + a 1 x n−1 + · · · + a nwith the following properties:(a) all the coefficients a 1 , a 2 , . . . , a n belong to the set {−1, 1}(b) all the roots of the equationf(x) = 0are real.23