Olympiad problems

2019 India PRMO, 12

Let $N$ be the number of ways of choosing a subset of $5$ distinct numbers from the set $${10a+b:1\leq a\leq 5, 1\leq b\leq 5}$$ where $a,b$ are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when $N$ is divided by $73$?

1985 Austrian-Polish Competition, 5

Tags: combinatorics , enumeration

We are given a certain number of identical sets of weights; each set consists of four different weights expressed by natural numbers (of weight units). Using these weights we are able to weigh out every integer mass up to $1985$ (inclusive). How many ways are there to compose such a set of weight sets given that the joint mass of all weights is the least possible?

1955 Moscow Mathematical Olympiad, 309

Tags: convex polygon , combinatorial geometry , enumeration , number theory , combinatorics

A point $O$ inside a convex $n$-gon $A_1A_2 . . .A_n$ is connected with segments to its vertices. The sides of this $n$-gon are numbered $1$ to $n$ (distinct sides have distinct numbers). The segments $OA_1,OA_2, . . . ,OA_n$ are similarly numbered. a) For $n = 9$ find a numeration such that the sum of the sides’ numbers is the same for all triangles $A_1OA_2, A_2OA_3, . . . , A_nOA_1$. b) Prove that for $n = 10$ there is no such numeration.

2020 Iran MO (2nd Round), P1

Tags: combinatorics , set systems , enumeration

Let $S$ is a finite set with $n$ elements. We divided $AS$ to $m$ disjoint parts such that if $A$, $B$, $A \cup B$ are in the same part, then $A=B.$ Find the minimum value of $m$.

2005 Austrian-Polish Competition, 1

Tags: polygon , enumeration , combinatorics

For a convex $n$-gon $P_n$, we say that a convex quadrangle $Q$ is a [i]diagonal-quadrangle[/i] of $P_n$, if its vertices are vertices of $P_n$ and its sides are diagonals of $P_n$. Let $d_n$ be the number of diagonal-quadrangles of a convex $n$-gon. Determine $d_n$ for all $n\geq 8$.