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

2023 SSMO Relay Round 5 Problems

Revision as of 20:36, 15 December 2023 by Pinkpig (talk | contribs) (Created page with "==Problem 1== Let <math>S_n</math> be the set of all rational numbers of the form <math>0.\overline{a_1a_2a_3\dots a_n},</math> where <math>n</math> is an integer satisfying...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem 1

Let $S_n$ be the set of all rational numbers of the form $0.\overline{a_1a_2a_3\dots a_n},$ where $n$ is an integer satisfying $n\geq 1$ and $a_1,a_2,\dots,a_n$ are nonzero integers. If \[n = 5\left(\sum_{n=1}^{\infty}\left(\sum_{a\in S_n}\frac{a}{10^{n}}\right)\right),\] find $n.$

Solution

Problem 2

Let $T=$ TNYWR. Let $a_n = \text{lcm} \{1, 2, \dots n\}$ for positive integers $n$.Compute \[\sum_{i=1}^N a_i \pmod {720}.\]

Solution

Problem 3

Let $T=$ TNYWR. Suppose that $x^2+y^2 = N.$ Find the remainder when the expected value of the square of the maximum value $ax+by$ is divided by $100,$ where $a$ and $b$ distinct members from the set $\{1,2,\dots,N\}.$

Solution

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_SSMO_Relay_Round_5_Problems&oldid=207616"