Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2025 SSMO Accuracy Round Problems/Problem 8

Revision as of 11:26, 9 September 2025 by Pinkpig (talk | contribs) (Created page with "==Problem== We say that a permutation <math>(a_1, a_2, \dots ,a_{10})</math> of the integers <math>1</math> through <math>10</math> inclusive is \textit{peaked} if there do n...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

We say that a permutation $(a_1, a_2, \dots ,a_{10})$ of the integers $1$ through $10$ inclusive is \textit{peaked} if there do not exist three integers $1\le i < j < k \le 10$ such that $a_i > a_j$ and $a_j< a_k$. Let $\mathcal{S}$ be the set of all peaked permutations. If $a_p = 9$ and $a_q = 4$, the expected value of $|p-q|$ over all permutations in $\mathcal{S}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is the value of $m+n$?

Solution

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2025_SSMO_Accuracy_Round_Problems/Problem_8&oldid=260046"