Olympiad problems

2000 Dutch Mathematical Olympiad, 2

Three boxes contain 600 balls each. The first box contains 600 identical red balls, the second box contains 600 identical white balls and the third box contains 600 identical blue balls. From these three boxes, 900 balls are chosen. In how many ways can the balls be chosen? For example, one can choose 250 red balls, 187 white balls and 463 balls, or one can choose 360 red balls and 540 blue balls.

1992 IMO Longlists, 72

Tags: combinatorics , counting , inclusion-exclusion

In a school six different courses are taught: mathematics, physics, biology, music, history, geography. The students were required to rank these courses according to their preferences, where equal preferences were allowed. It turned out that: [list] [*][b](i)[/b] mathematics was ranked among the most preferred courses by all students; [*][b](ii)[/b] no student ranked music among the least preferred ones; [*][b](iii) [/b]all students preferred history to geography and physics to biology; and [*][b](iv)[/b] no two rankings were the same. [/list] Find the greatest possible value for the number of students in this school.

2009 Serbia National Math Olympiad, 3

Tags: combinatorics , graph theory , set , set systems , set theory , inclusion-exclusion

Determine the largest positive integer $n$ for which there exist pairwise different sets $\mathbb{S}_1 , ..., \mathbb{S}_n$ with the following properties: $1$) $|\mathbb{S}_i \cup \mathbb{S}_j | \leq 2004$ for any two indices $1 \leq i, j\leq n$, and $2$) $\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1,2,...,2008 \}$ for any $1 \leq i < j < k \leq n$ [i]Proposed by Ivan Matic[/i]

2024 Belarus Team Selection Test, 1.1

Tags: inclusion-exclusion , combinatorics

Find the minimal positive integer $n$ such that no matter what $n$ distinct numbers from $1$ to $1000$ you choose, such that no two are divisible by a square of the same prime, one of the chosen numbers is a square of prime. [i]D. Zmiaikou[/i]

2022 Bulgaria JBMO TST, 3

Tags: combinatorics , counting , inclusion-exclusion

For a positive integer $n$ let $t_n$ be the number of unordered triples of non-empty and pairwise disjoint subsets of a given set with $n$ elements. For example, $t_3 = 1$. Find a closed form formula for $t_n$ and determine the last digit of $t_{2022}$. (I also give here that $t_4 = 10$, for a reader to check his/her understanding of the problem statement.)