MathDB

5

Part of 1991 Irish Math Olympiad

Problems(2)

find all polynomials 2 (IrMO 1991)

Source:

1/17/2014
Find all polynomials
f(x)=xn+a1xn−1+⋯+anf(x) = x^{n} + a_{1}x^{n-1} + \cdots + a_{n}
with the following properties
(a) all the coefficients a1,a2,...,ana_{1}, a_{2}, ..., a_{n} belong to the set {−1,1}\{ -1, 1 \}; and (b) all the roots of the equation f(x)=0f(x)=0 are real.
algebrapolynomialalgebra unsolved
Subset of the Rationals

Source: 1991 IrMO Paper 2 Problem 5

10/1/2017
Let Q\mathbb{Q} denote the set of rational numbers. A nonempty subset SS of Q\mathbb{Q} has the following properties:
(a) 00 is not in SS;
(b) for each s1,s2s_1,s_2 in SS, the rational number s1/s2s_1/s_2 is in SS;
(c) there exists a nonzero number q∈Q\Sq\in \mathbb{Q} \backslash S that has the property that every nonzero number in Q\S\mathbb{Q} \backslash S is of the form qsqs for some ss in SS.
Prove that if xx belongs to SS, then there exists elements y,zy,z in SS such that x=y+zx=y+z.