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Stage 6 · Combinatorics

10 problems · National Olympiad, first round · mathsolympiadprep.com

The answer key prints on its own page at the end.

There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds.

China National Olympiad

Combinatorics Solution and answer checking →
In a game of Riemannian ball two teams face each other; at every moment, each team fields $k>1$ players. At the end of each play a point is awarded to one of the two teams; moreover, each team may make an arbitrary number of substitutions before the next play begins. Alice and Barbara are part of the Geodesics team. At the end of today's match, Alice observes that, while she was on the field, the Geodesics won 7 plays more than they lost. When Barbara was on the field, on the other hand, they lost 2 plays more than they won. Each of the other players of the Geodesics took part in exactly as many winning plays as losing plays. What is the value of $k$ ?
1. A3
2. B5
3. C6
4. D9
5. EThe given data are not sufficient to determine it
Gara di Febbraio

Combinatorics Solution and answer checking →
Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist?

Pan African MO

Combinatorics Solution and answer checking →
A $k \times k$ array contains each of the numbers $1, 2, \dots, m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k = 3^n$ ( $n \in \mathbb{N}^+$ )?

KoMaL

Combinatorics Solution and answer checking →
The equilateral triangle $ABC$ has sides of integer length $N$ . The triangle is completely divided (by drawing lines parallel to the sides of the triangle) into equilateral triangular cells of side length $1$ .
A continuous route is chosen, starting inside the cell with vertex $A$ and always crossing from one cell to another through an edge shared by the two cells. No cell is visited more than once. Find, with proof, the greatest number of cells which can be visited.

BMO Round 1 2006

Combinatorics Solution and answer checking →
$a)$ Two players play a cooperative game. They can discuss a strategy prior to the game, however, they cannot communicate and have no information about the other player during the game. The game master chooses one of the players in each round. The player on turn has to guess the number of the current round. Players keep note of the number of rounds they were chosen, however, they have no information about the other player's rounds. If the player's guess is correct, the players are awarded a point. Player's are not notified whether they've scored or not. The players win the game upon collecting 100 points. Does there exist a strategy with which they can surely win the game in a finite number of rounds?

KöMaL problem A 2024

Combinatorics Solution and answer checking →
Let $\mathcal{F}$ be a finite family of finite sets and let $A$ be an arbitrary finite set. We say that $\mathcal{F}$ shatters the set $A$ if for every $X\subseteq A$ there is a set $F\in \mathcal{F}$ such that $A\cap F=X$ . Show that $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets.
(5 pont)

KöMaL problem A 2014

Combinatorics Solution and answer checking →
Given a foundation that is in form of a square $4 \times 4$ , that is divided into smaller $1 \times 1$ squares. There is a gap of length $1$ between any two adjacent squares. The foundation is covered with several layers of bricks of size $2 \times 1$ . Every layer consists of $8$ bricks and each brick fully covers exactly one gap of length $1$ . Such cover is called strong, if every gap is covered by a brick at least in one of the layers. What is the minimum amount of layers that make a strong cover?
(Bogdan Rublyov)

Ukrainian National Mathematical Olympiad

Combinatorics Solution and answer checking →
Let $n$ , $k$ be integers greater than $1$ and satisfy $n < 2^k$ . Prove that there are $2k$ integers not divisible by $n$ , such that if we divide them into two groups, then there must exist a group in which the sum of some integers can be divided by $n$ .

China Mathematical Competition (Complementary Test)

Combinatorics Solution and answer checking →
On a $75 \times 75$ chessboard the rows and columns are numbered from 1 to 75. Chiara wants to place a checker on all and only the squares that have one coordinate even and the other a multiple of 3. How many checkers will she place in total on the chessboard?

Italian Mathematical Olympiad

Combinatorics Solution and answer checking →

Answer key — Stage 6 · Combinatorics

Worked solutions for every problem are on the site, one page per problem.

$9$ open
$B$ open
$2^{n-1}$ open
$3^{n+1} - 1$ open
Prove it — see the worked solution open
Prove it — see the worked solution open
Prove it — see the worked solution open
Prove it — see the worked solution open
Prove it — see the worked solution open
Prove it — see the worked solution open

Problems belong to the competitions that set them. Each question on this sheet names the competition it came from, and the licences the material is used under are listed below. Free to copy for classroom use.