Olympiad problems

1979 Bundeswettbewerb Mathematik, 3

In base $10$ there exist two-digit natural numbers that can be factorized into two natural factors such that the two digits and the two factors form a sequence of four consecutive integers (for example, $12 = 3 \cdot 4$). Determine all such numbers in all bases.

2021 IMO, 6

Tags: combinatorics , linear algebra , number system , other bases , algebra

Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements.

2011 Gheorghe Vranceanu, 3

Tags: number theory , other bases , binary representation

Prova that any integer $ Z $ has a unique representation $$ a_0+a_12+a_22^2+a_32^3+\cdots +a_n2^n, $$ where $ n $ is natural, $ a_i\in\{ -1,0,+1\} $ for $ i=\overline{0,n} $ and $ a_ka_{k-1}=0 $ for $ k=\overline{1,n} . $

2021 IMO Shortlist, A6

Tags: combinatorics , linear algebra , number system , other bases , algebra

Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements.