Part 1’s link https://nttuan.org/2016/01/01/topic-733/
Problem 1. Let be a polynomial of degree at most satisfying Determine .
Problem 2. A polynomial has degree at most , where . Given that for an integer , the inequality implies , prove that for all real numbers , with , the following inequality holds
Problems 3. Prove that at least one of the numbers is greater than or equal to Where
Problem 4. Prove that any monic polynomial (a polynomial with leading coefficient ) of degree with real coefficients is the average of two monic polynomials of degree with real roots.
Problem 5. Let be a prime number and an integer polynomial of degree such that and is congruent to or modulo for every integer . Prove that .
Problem 6. Let be a polynomial of degree satisfying Prove that .
Problem 7. is a polynomial of degree such that
Problem 8. Let be a set of distinct complex numbers , exactly of which are real.
Prove that there are at most two quadratic polynomials with complex
coefficients such that (that is, permutes the elements of ).