Found problems: 100
2016 Spain Mathematical Olympiad, 5
From all possible permutations from $(a_1,a_2,...,a_n)$ from the set $\{1,2,..,n\}$, $n\geq 1$, consider the sets that satisfies the $2(a_1+a_2+...+a_m)$ is divisible by $m$, for every $m=1,2,...,n$. Find the total number of permutations.
1969 IMO Shortlist, 31
$(GDR 3)$ Find the number of permutations $a_1, \cdots, a_n$ of the set $\{1, 2, . . ., n\}$ such that $|a_i - a_{i+1}| \neq 1$ for all $i = 1, 2, . . ., n - 1.$ Find a recurrence formula and evaluate the number of such permutations for $n \le 6.$
2006 Korea Junior Math Olympiad, 8
De ne the set $F$ as the following: $F = \{(a_1,a_2,... , a_{2006}) : \forall i = 1, 2,..., 2006, a_i \in \{-1,1\}\}$
Prove that there exists a subset of $F$, called $S$ which satis es the following:
$|S| = 2006$
and for all $(a_1,a_2,... , a_{2006})\in F$ there exists $(b_1,b_2,... , b_{2006}) \in S$, such that $\Sigma_{i=1} ^{2006}a_ib_i = 0$.
2006 Germany Team Selection Test, 3
Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$.
Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have
\[ n\mid a_i \minus{} b_i \minus{} c_i
\]
[i]Proposed by Ricky Liu & Zuming Feng, USA[/i]
1963 IMO, 6
Five students $ A, B, C, D, E$ took part in a contest. One prediction was that the contestants would finish in the order $ ABCDE$. This prediction was very poor. In fact, no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $ DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.
2025 Taiwan TST Round 1, N
Find all positive integers $n$ such that there exist two permutations $a_0,a_1,\ldots,a_{n-1}$ and $b_0,b_1,\ldots,b_{n-1}$ of the set $\lbrace0,1,\ldots,n-1\rbrace$, satisfying the condition
$$ia_i\equiv b_i\pmod{n}$$
for all $0\le i\le n-1$.
[i]Proposed by Fysty[/i]
2011 Ukraine Team Selection Test, 12
Let $ n $ be a natural number. Consider all permutations $ ({{a} _ {1}}, \ \ldots, \ {{a} _ {2n}}) $ of the first $ 2n $ natural numbers such that the numbers $ | {{a} _ {i +1}} - {{a} _ {i}} |, \ i = 1, \ \ldots, \ 2n-1, $ are pairwise different. Prove that $ {{a} _ {1}} - {{a} _ {2n}} = n $ if and only if $ 1 \le {{a} _ {2k}} \le n $ for all $ k = 1, \ \ldots, \ n $.
2022 Singapore MO Open, Q4
Let $n,k$, $1\le k\le n$ be fixed integers. Alice has $n$ cards in a row, where the card has position $i$ has the label $i+k$ (or $i+k-n$ if $i+k>n$). Alice starts by colouring each card either red or blue. Afterwards, she is allowed to make several moves, where each move consists of choosing two cards of different colours and swapping them. Find the minimum number of moves she has to make (given that she chooses the colouring optimally) to put the cards in order (i.e. card $i$ is at position $i$).
NOTE: edited from original phrasing, which was ambiguous.
2021 Poland - Second Round, 6
Let $p\ge 5$ be a prime number. Consider the function given by the formula $$f (x_1,..., x_p) = x_1 + 2x_2 +... + px_p.$$
Let $A_k$ denote the set of all these permutations $(a_1,..., a_p)$ of the set $\{1,..., p\}$, for integer number $f (a_1,..., a_p) - k$ is divisible by $p$ and $a_i \ne i$ for all $i \in \{1,..., p\}$. Prove that the sets $A_1$ and $A_4$ have the same number of elements.
2023 pOMA, 1
Let $n$ be a positive integer. Marc has $2n$ boxes, and in particular, he has one box filled with $k$ apples for each $k=1,2,3,\ldots,2n$. Every day, Marc opens a box and eats all the apples in it. However, if he eats strictly more than $2n+1$ apples in two consecutive days, he gets stomach ache. Prove that Marc has exactly $2^n$ distinct ways of choosing the boxes so that he eats all the apples but doesn't get stomach ache.
2016 Indonesia MO, 8
Determine with proof, the number of permutations $a_1,a_2,a_3,...,a_{2016}$ of $1,2,3,...,2016$ such that the value of $|a_i-i|$ is fixed for all $i=1,2,3,...,2016$, and its value is an integer multiple of $3$.
2021 Kyiv City MO Round 1, 8.3
The $1 \times 1$ cells located around the perimeter of a $3 \times 3$ square are filled with the numbers $1,
2, \ldots, 8$ so that the sums along each of the four sides are equal. In the upper left corner cell is the number $8$, and in the upper left is the number $6$ (see the figure below).
[img]https://i.ibb.co/bRmd12j/Kyiv-MO-2021-Round-1-8-2.png[/img]
How many different ways to fill the remaining cells are there under these conditions?
[i]Proposed by Mariia Rozhkova[/i]
2024 SG Originals, Q1
Find all permutations $(a_1, a_2, \cdots, a_{2024})$ of $(1, 2, \cdots, 2024)$ such that there exists a polynomial $P$ with integer coefficients satisfying $P(i) = a_i$ for each $i = 1, 2, \cdots, 2024$.
1980 Tournament Of Towns, (003) 3
If permutations of the numbers $2, 3,4,..., 102$ are denoted by $a_i,a_2, a_3,...,a_{101}$, find all such permutations in which $a_k$ is divisible by $k$ for all $k$.
2011 IFYM, Sozopol, 8
Let $S$ be the set of all 9-digit natural numbers, which are written only with the digits 1, 2, and 3. Find all functions $f:S\rightarrow \{1,2,3\}$ which satisfy the following conditions:
(1) $f(111111111)=1$, $f(222222222)=2$, $f(333333333)=3$, $f(122222222)=1$;
(2) If $x,y\in S$ differ in each digit position, then $f(x)\neq f(y)$.
2016 IMAR Test, 1
Fix an integer $n \ge 3$ and let $a_0 = n$. Does there exist a permutation $a_1, a_2,..., a_{n-1}$ of the fi rst $n-1$ positive integers such that $\Sigma_{j=0}^{k-1} a_j$ is divisible by $a_k$ for all indices $k < n$?
2022 Tuymaada Olympiad, 5
Each row of a $24 \times 8$ table contains some permutation of the numbers $1, 2, \cdots , 8.$ In each column the numbers are multiplied. What is the minimum possible sum of all the products?
[i](C. Wu)[/i]
2022 IFYM, Sozopol, 8
Let $p$ and $q$ be mutually prime natural numbers greater than $1$. Starting with the permutation $(1, 2, . . . , n)$, in one move we can switch the places of two numbers if their difference is $p$ or $q$. Prove that with such moves we can get any another permutation if and only if $n \ge p + q - 1$.
2024 Mexican University Math Olympiad, 5
Consider two finite sequences of real numbers \( a_1, a_2, \dots, a_n \) and \( b_1, b_2, \dots, b_n \). Let \( \alpha(x) = \#\{i | a_i = x \} \) and \( \beta(x) = \#\{i | b_i = -x \} \). Prove that there exists a permutation \( \sigma \in S_n \) (the symmetric group of \( n \) elements) such that \( a_{\sigma(i)} + b_i \neq 0 \) for all \( i = 1, \dots, n \) if and only if \( \alpha(x) + \beta(x) \leq n \) for all \( x \in \mathbb{R} \).
2021 Canadian Mathematical Olympiad Qualification, 8
King Radford of Peiza is hosting a banquet in his palace. The King has an enormous circular table with $2021$ chairs around it. At The King's birthday celebration, he is sitting in his throne (one of the $2021$ chairs) and the other $2020$ chairs are filled with guests, with the shortest guest sitting to the King's left and the remaining guests seated in increasing order of height from there around the table. The King announces that everybody else must get up from their chairs, run around the table, and sit back down in some chair. After doing this, The King notices that the person seated to his left is different from the person who was previously seated to his left. Each other person at the table also notices that the person sitting to their left is different.
Find a closed form expression for the number of ways the people could be sitting around the table at the end. You may use the notation $D_{n},$ the number of derangements of a set of size $n$, as part of your expression.
2005 IMO Shortlist, 7
Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$.
Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have
\[ n\mid a_i \minus{} b_i \minus{} c_i
\]
[i]Proposed by Ricky Liu & Zuming Feng, USA[/i]
2006 Korea Junior Math Olympiad, 1
$a_1, a_2,...,a_{2006}$ is a permutation of $1,2,...,2006$.
Prove that $\prod_{i = 1}^{2006} (a_{i}^2-i) $ is a multiple of $3$. ($0$ is counted as a multiple of $3$)
2012 IMAC Arhimede, 1
Let $a_1,a_2,..., a_n$ be different integers and let $(b_1,b_2,..., b_n),(c_1,c_2,..., c_n)$ be two of their permutations, different from the identity. Prove that
$$(|a_1-b_1|+|a_2-b_2|+...+|a_n-b_n| , |a_1-c_1|+|a_2-c_2|+...+|a_n-c_n| ) \ge 2$$
where $(x,y)$ denotes the greatest common divisor of the numbers $x,y$
2020 OMMock - Mexico National Olympiad Mock Exam, 2
We say that a permutation $(a_1, \dots, a_n)$ of $(1, 2, \dots, n)$ is good if the sums $a_1 + a_2 + \dots + a_i$ are all distinct modulo $n$. Prove that there exists a positive integer $n$ such that there are at least $2020$ good permutations of $(1, 2, \dots, n)$.
[i]Proposed by Ariel García[/i]
2009 Danube Mathematical Competition, 5
Let $\sigma, \tau$ be two permutations of the quantity $\{1, 2,. . . , n\}$.
Prove that there is a function $f: \{1, 2,. . . , n\} \to \{-1, 1\}$ such that for any $1 \le i \le j \le n$,
we have $\left|\sum_{k=i}^{j} f(\sigma (k)) \right| \le 2$ and $\left|\sum_{k=i}^{j} f(\tau (k))\right| \le 2$