5

Part of 2012 IMC

Problems(2)

IMC 2012 Day 1, Problem 5

Source:

a

be a rational number and let

n

be a positive integer. Prove that the polynomial

X^{2^n}(X+a)^{2^n}+1

is irreducible in the ring

\mathbb{Q}[X]

of polynomials with rational coefficients.
Proposed by Vincent Jugé, École Polytechnique, Paris.

algebrapolynomialnumber theoryrelatively primeIMCcollege contests

Plunecke's Inequality

Source: IMC 2012, Day 2, Problem 5

c \ge 1

be a real number. Let

G

be an Abelian group and let

A \subset G

be a finite set satisfying

|A+A| \le c|A|

X+Y:= \{x+y| x \in X, y \in Y\}

|Z|

denotes the cardinality of

Z

|\underbrace{A+A+\dots+A}_k| \le c^k |A|

for every positive integer

k

.
Proposed by Przemyslaw Mazur, Jagiellonian University.

inequalitiesabstract algebrainductiongroup theoryIMCcollege contests