Olympiad problems

2017 IFYM, Sozopol, 5

Tags: arrangement , combinatorics

In a group of $n$ people $A_1,A_2… A_n$ each one has a different height. On each turn we can choose any three of them and figure out which one of them is the highest and which one is the shortest. What’s the least number of turns one has to make in order to arrange these people by height, if: a) $n=5$; b) $n=6$; c) $n=7$?

2016 Irish Math Olympiad, 7

Tags: arrangement , combinatorics , array , counting

A rectangular array of positive integers has $4$ rows. The sum of the entries in each column is $20$. Within each row, all entries are distinct. What is the maximum possible number of columns?

2025 Kyiv City MO Round 1, Problem 2

Tags: arrangement , combinatorics

Can the numbers from $ 1 $ to $ 2025 $ be arranged in a circle such that the difference between any two adjacent numbers has the form $ 2^k $ for some non-negative integer $ k $? For different adjacent pairs of numbers, the values of $ k $ may be different. [i]Proposed by Anton Trygub[/i]

2014 BAMO, 1

Tags: arrangement , cube , 3d geometry , counting , combinatorics , geometry

The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.

2005 USAMO, 1

Tags: induction , relatively prime , number theory , divisor , arrangement , prime number , combinatorics

Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

1966 IMO Shortlist, 14

Tags: circles , counting , arrangement , combinatorics , combinatorial geometry

What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ? [i]Posted already on the board I think...[/i]

2009 Romania National Olympiad, 3

Tags: inequalities , permutation , arrangement

Let be a natural number $ n, $ a permutation $ \sigma $ of order $ n, $ and $ n $ nonnegative real numbers $ a_1,a_2,\ldots , a_n. $ Prove the following inequality. $$ \left( a_1^2+a_{\sigma (1)} \right)\left( a_2^2+a_{\sigma (2)} \right)\cdots \left( a_n^2+a_{\sigma (n)} \right)\ge \left( a_1^2+a_1 \right)\left( a_2^2+a_{2} \right)\cdots \left( a_n^2+a_n \right) $$

2014 Contests, 1

Tags: geometry , counting , combinatorics , cube , 3d geometry , arrangement

The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.

2022 India National Olympiad, 3

Tags: arrangement

For a positive integer $N$, let $T(N)$ denote the number of arrangements of the integers $1, 2, \cdots N$ into a sequence $a_1, a_2, \cdots a_N$ such that $a_i > a_{2i}$ for all $i$, $1 \le i < 2i \le N$ and $a_i > a_{2i+1}$ for all $i$, $1 \le i < 2i+1 \le N$. For example, $T(3)$ is $2$, since the possible arrangements are $321$ and $312$ (a) Find $T(7)$ (b) If $K$ is the largest non-negative integer so that $2^K$ divides $T(2^n - 1)$, show that $K = 2^n - n - 1$. (c) Find the largest non-negative integer $K$ so that $2^K$ divides $T(2^n + 1)$

1978 Bundeswettbewerb Mathematik, 2

Tags: combinatorics , line , arrangement

A set of $n^2$ counters are labeled with $1,2,\ldots, n$, each label appearing $n$ times. Can one arrange the counters on a line in such a way that for all $x \in \{1,2,\ldots, n\}$, between any two successive counters with the label $x$ there are exactly $x$ counters (with labels different from $x$)?

1966 IMO Longlists, 14

Tags: combinatorics , combinatorial geometry , counting , circles , arrangement

What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ? [i]Posted already on the board I think...[/i]