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## IMC 2012 Problem 10

Let ${c \geq 1}$ be a real number. Let ${G}$ be an abelian group and let ${A \subset G}$ be a finite set satisfying ${|A+A|\leq c|A|}$ where ${X+Y=\{x+y : x \in X , \ y \in Y\}}$ and ${|Z|}$ denotes the cardinality of ${Z}$. Prove that

$\displaystyle |\underbrace{A+A+...+A}_{k \text{ times}}| \leq c^k |A|$

for every positive integer ${k}$.