Found problems: 7
2025 Nordic, 2
Let $p$ be a prime and suppose $2^{2p} \equiv 1 (\text{mod}$ $ 2p+1)$ is prime. Prove that $2p+1$ is prime$^{1}$
[size=75]$^{1}$This is a special case of Pocklington's theorem. A proof of this special case is required.[/size]
2025 Nordic, 4
Denote by $S_{n}$ the set of all permutations of the set $\{1,2,\dots, n\}$. Let $\sigma \in S_{n}$ be a permutation. We define the $\textit{displacement}$ of $\sigma$ to be the number $d(\sigma)=\sum_{i=1}^{n} \vert \sigma(i)-i \vert$. We saw that $\sigma$ is $\textit{maximally}$ $\textit{displacing}$ if $d(\sigma)$ is the largest possible, i.e. if $d(\sigma) \geq d({\pi})$, for all $\pi \in S_{n}$.
$\text{a)}$ Suppose $\sigma$ is a maximally displacing permutation of $\{1,2, \dots, 2024\}$. Prove that $\sigma(i)\neq i$, for all $i \in \{1,2, \dots, 2024.\}$
$\text{b)}$ Does the statement of part a) hold for permutations of $\{1,2, \dots, 2025\}$?
2024 Nordic, 3
Find all functions $f: \mathbb{R} \to \mathbb{R}$
$f(f(x)f(y)+y)=f(x)y+f(y-x+1)$
For all $x,y \in \mathbb{R}$
2025 Nordic, 1
Let $n$ be a positive integer greater than $2$. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying:
$(f(x+y))^{n} = f(x^{n})+f(y^{n}),$ for all integers $x,y$
2024 Nordic, 1
Let $T(a)$ be the sum of digits of $a$. For which positive integers $R$ does there exist a positive
integer $n$ such that $\frac{T(n^2)}{T(n)}=R$?
2024 Nordic, 4
Alice and Bob are playing a game. First, Alice chooses a partition $\mathcal{C}$ of the positive integers
into a (not necessarily finite) set of sets, such that each positive integer is in exactly one of the
sets in $\mathcal{C}$. Then Bob does the following operation a finite number of times.
Choose a set $S \in \mathcal{C}$ not previously chosen, and let $D$ be the set of all positive integers dividing at least one element in $S$. Then add the set $D \setminus S$ (possibly the empty set) to $\mathcal{C}$.
Bob wins if there are two equal sets in $\mathcal{C}$ after he has done all his moves, otherwise, Alice wins.
Determine which player has a winning strategy.
1999 Nordic, 3
The infinite integer plane $Z\times Z = Z^2$ consists of all number pairs $(x, y)$, where $x$ and $y$ are integers. Let $a$ and $b$ be non-negative integers. We call any move from a point $(x, y)$ to any of the points $(x\pm a, y \pm b)$ or $(x \pm b, y \pm a) $ a $(a, b)$-knight move. Determine all numbers $a$ and $b$, for which it is possible to reach all points of the integer plane from an arbitrary starting point using only $(a, b)$-knight moves.