Olympiad problems

2016 Irish Math Olympiad, 7

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?

2014 Contests, 1

Tags: B8 , cube , geometry , 3D geometry , counting , combinatorics , Arrangements

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 , number theory , prime numbers , Arrangements , relatively prime , Divisors , 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.

2014 BAMO, 1

Tags: B8 , cube , geometry , 3D geometry , counting , combinatorics , Arrangements

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: INMO 2022 , Arrangements

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 $T(2^n + 1)$