Olympiad problems

2023 Germany Team Selection Test, 3

Let $A$ be a non-empty set of integers with the following property: For each $a \in A$, there exist not necessarily distinct integers $b,c \in A$ so that $a=b+c$. (a) Proof that there are examples of sets $A$ fulfilling above property that do not contain $0$ as element. (b) Proof that there exist $a_1,\ldots,a_r \in A$ with $r \ge 1$ and $a_1+\cdots+a_r=0$. (c) Proof that there exist pairwise distinct $a_1,\ldots,a_r$ with $r \ge 1$ and $a_1+\cdots+a_r=0$.

2018 OMMock - Mexico National Olympiad Mock Exam, 6

Tags: combinatorics , elements of set

Let $A$ be a finite set of positive integers, and for each positive integer $n$ we define \[S_n = \{x_1 + x_2 + \cdots + x_n \;\vert\; x_i \in A \text{ for } i = 1, 2, \dots, n\}\] That is, $S_n$ is the set of all positive integers which can be expressed as sum of exactly $n$ elements of $A$, not necessarily different. Prove that there exist positive integers $N$ and $k$ such that $$\left\vert S_{n + 1} \right\vert = \left\vert S_n \right\vert + k \text{ for all } n\geq N.$$ [i]Proposed by Ariel García[/i]

1989 Nordic, 3

Tags: algebra , elements of set , sequence

Let $S$ be the set of all points $t$ in the closed interval $[-1, 1]$ such that for the sequence $x_0, x_1, x_2, ...$ defined by the equations $x_0 = t, x_{n+1} = 2x_n^2-1$, there exists a positive integer $N$ such that $x_n = 1$ for all $n \ge N$. Show that the set $S$ has infinitely many elements.