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Stage 5 · Combinatorics

10 problems · AIME late · mathsolympiadprep.com

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

$\frac{1}{2} n(n+1)$ distinct numbers are randomly arranged in a triangle:
Let $M_{\mathrm{k}}$ be the maximum number in the $k$ -th row (counting from the top), find the probability that $M_{1}<M_{2}<M_{3}<\cdots<M_{\mathrm{n}}$ holds.

Combinatorics Solution and answer checking →
199 people have registered to participate in a tennis tournament. In the first round, pairs of opponents are selected by lottery. The same process is used to select pairs in the second, third, and all subsequent rounds. After each match, one of the two opponents is eliminated, and whenever the number of participants in the tournament is odd, one of them skips the current round.
Assume that in each match between two tennis players, a new can of balls is used. How many cans of balls will be needed for the entire tournament?

Combinatorics Solution and answer checking →
In a chess tournament, there are an odd number of participants, and each participant plays one game against every other participant. A win earns 1 point, a draw earns 0.5 points, and a loss earns 0 points; it is known that two of the participants scored a total of 8 points, and the average score of the others is an integer. How many participants are there in the tournament?

Combinatorics Solution and answer checking →
Fill the 12 natural numbers from $11 \sim 22$ into the small circles in the figure, each number must be used, and the sum of the four numbers on each side must be equal. The difference between the minimum and maximum value of this sum is $\qquad$ .

Combinatorics Solution and answer checking →
There is a water tap and two containers: a three-liter and a five-liter. How can you get 4 liters of water in the larger one?

Combinatorics Solution and answer checking →
Ayeesha had 12 guests aged $6,7,8,9$ and 10 at her birthday party. Four of the guests were 6 years old. The most common age was 8 years old. What was the mean age of the guests?
A 6
В 6.5
C 7
D 7.5
E 8

Combinatorics Solution and answer checking →
Ivana has two identical dice and on the faces of each are the numbers $-3,-2,-1,0,1,2$ . If she throws her dice and multiplies the results, what is the probability that their product is negative?
A $\frac{1}{4}$
B $\frac{11}{36}$
C $\frac{1}{3}$
D $\frac{13}{36}$
E $\frac{1}{2}$

Combinatorics Solution and answer checking →
$A_{1}, A_{2}, \cdots, A_{t}$ are all $r$ -sets, $X=\bigcup_{i=1}^{t} A_{i}$ , find $\min |X|$ . Here the minimum is over all $A_{1}, A_{2}, \cdots, A_{i}$ of $|X|$ .

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1.83 Mark 10 points on a circle. How many different convex polygons can be constructed using some of these points as vertices? (Polygons are considered the same only if all their vertices coincide)

Combinatorics Solution and answer checking →
Given the sets: $A=\{0,1,2,3\}$ and $B=\left\{2^{x} \mid x \in A\right\}$
a) Determine the elements of the set $A \cap B$ ;
b) Determine $\operatorname{card} M$ , where $M=\{\overline{a b c} \mid a, b, c \in A$ and $a \neq b \neq c\}$ .

Combinatorics Solution and answer checking →

Answer key — Stage 5 · Combinatorics

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

$\frac{2^{n}}{(n+1)!}$ open
$198$ open
$9$ open
$8$ open
$4$ open
$7.5$ open
$\frac{1}{3}$ open
$n$ open
$968$ open
$18$ open

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