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Stage 3 · Combinatorics

10 problems · AMC 10/12, early questions · mathsolympiadprep.com

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

A tetrahedron of spheres is formed with thirteen layers and each sphere has a number written on it. The top sphere has a 1 written on it and each of the other spheres has written on it the number equal to the sum of the numbers on the spheres in the layer above with which it is in contact. What is the sum of the numbers on all of the internal spheres?

CEMC Fermat

Combinatorics Solution and answer checking →
In the $5 \times 5$ grid shown, 15 cells contain X’s and 10 cells are empty.
X X X X
X X X X
X X
X X X
X X
Any X may be moved to any empty cell. What is the smallest number of X’s that must be moved so that each row and each column contains exactly three X’s?
$1$
$2$
$3$
$4$
$5$

CEMC Fermat 2020

Combinatorics Solution and answer checking →
There are four spade cards with numbers $1$ , $2$ , $3$ , $4$ , six heart cards with numbers $1$ , $2$ , $3$ , $\ldots$ , $6$ and eight diamond cards with numbers $1$ , $2$ , $3$ , $\ldots$ , $8$ . Suppose you choose three cards, one from each group. How many possible choices are there if the total of the numbers on the chosen cards must be a multiple of $7$ ?

Japan Junior Mathematical Olympiad

Combinatorics Solution and answer checking →
Zebadiah has 3 red shirts, 3 blue shirts, and 3 green shirts in a
drawer. Without looking, he randomly pulls shirts from his drawer one at
a time. He would like a set of shirts that includes either 3 of the same
colour or 3 of different colours. What is the minimum number of shirts
that Zebadiah has to pull out to guarantee that he has such a
set?
1. A $4$
2. B $3$
3. C $6$
4. D $5$
5. E $7$
CEMC Fermat 2022

Combinatorics Solution and answer checking →
Alvin, Bingyi and Cheska play a two-player game that never ends
in a tie. In a recent tournament between the three players, a total of
60 games were played and each pair of players played the same number of
games.
When Alvin and Bingyi played, Alvin won 20% of the
games.
When Bingyi and Cheska played, Bingyi won 60% of the
games.
When Cheska and Alvin played, Cheska won 40% of the
games.
How many games did Bingyi win?
1. A $12$
2. B $24$
3. C $28$
4. D $30$
5. E $36$
CEMC Pascal 2022

Combinatorics Solution and answer checking →
Francesca put one of the integers $1$ , $2$ , $3$ , $4$ , $5$ , $6$ , $7$ , $8$ , $9$
in each of the nine squares of the $3$ by $3$ grid shown. No integer is used twice. She calculated the product of the three
integers in each row and wrote the products to the right of the
corresponding rows. She calculated the product of the integers in each
column and wrote the products below the corresponding columns. Finally,
she erased the integers from the nine squares.

Which integer was in the
square marked $N\,$ ?
1. A $3$
2. B $4$
3. C $6$
4. D $8$
5. E $9$
CEMC Fermat 2024

Combinatorics Solution and answer checking →
How many solutions does the equation $x+y+z=2016$ have, where $x$ , $y$ and $z$ are integers with $x>1000$ , $y>600$ , and $z>400$ ?

PMO National Stage Oral Phase

Combinatorics Solution and answer checking →
The Gaussbot factory assembles robots. Each robot comes in one of
three colours: red, blue, or green. Each robot also has a number stamped
on its head: $1$ , $2$ , $3$ , or $4$ . The $n$ th robot assembled is the first robot to
have the same colour and the same number as a previously assembled
robot. What is the greatest possible value of $n$ ?
1. A $11$
2. B $12$
3. C $13$
4. D $7$
5. E $8$
CEMC Gauss (Grade 8) 2024

Combinatorics Solution and answer checking →
A tennis tournament starts with $8$ players. Francesca is equally likely to
play against any of the other $7$
players in her first match. If Francesca plays against Dominique or
Estella, the probability that Francesca wins is $\frac{2}{5}$ . If Francesca plays against
any of the other $5$ players, the
probability that she wins is $\frac{3}{4}$ . What is the probability that
Francesca wins her first match?
1. A $\dfrac{23}{50}$
2. B $\dfrac{29}{50}$
3. C $\dfrac{3}{5}$
4. D $\dfrac{13}{20}$
5. E $\dfrac{2}{3}$
CEMC Fermat 2025

Combinatorics Solution and answer checking →
Four teams play in a tournament in which each team plays exactly one game against each of the other three teams. At the end of each game, either the two teams tie or one team wins and the other team loses. A team is awarded 3 points for a win, 0 points for a loss, and 1 point for a tie. If $S$ is the sum of the points of the four teams after the tournament is complete, which of the following values can $S$ not equal?
$13$
$17$
$11$
$16$
$15$

CEMC Fermat 2020

Combinatorics Solution and answer checking →

X	X	X	X
X	X	X		X
X	X
X	X		X
		X	X

Answer key — Stage 3 · Combinatorics

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

$772626$ open
$B$ open
$28$ open
$D$ open
$C$ open
$C$ open
$105$ open
$C$ open
$D$ open
$C$ 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.

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