Library / /555 of 740

, 2022

Combinatorics Difficulty 5.2 AIME, harder Prove it United States

Problem:
Find the number of ordered pairs $(A, B)$ such that the following conditions hold:
- $A$ and $B$ are disjoint subsets of $\{1,2, \ldots, 50\}$ .
- $|A|=|B|=25$
- The median of $B$ is 1 more than the median of $A$ .

Solution

Solution:
The median of both sets, which we will call $a$ and $b$ respectively, are more than exactly 12 of the members in their own set. Since $a$ and $b$ are consecutive, they must also be higher than the lower half of the other set and lower than the higher half of the other set, meaning that they are both higher than exactly 24 numbers in $\{1,2, \ldots, 50\}-\{a, b\}$ . Thus, $a=25$ and $b=26$ .

The 24 lower numbers can be divided into the two groups (with 12 in each group) in $\binom{24}{12}$ ways. Similarly, the 24 higher numbers can be divided into the two groups in $\binom{24}{12}$ ways. Thus, the answer is $\binom{24}{12}^{2}$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.