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

2012 USAMO Problems/Problem 1

Revision as of 18:10, 24 April 2012 by Nsato (talk | contribs) (Created page with "==Problem== Find all integers <math>n \ge 3</math> such that among any <math>n</math> positive real numbers <math>a_1</math>, <math>a_2</math>, <math>\dots</math>, <math>a_n</mat...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

Find all integers $n \ge 3$ such that among any $n$ positive real numbers $a_1$, $a_2$, $\dots$, $a_n$ with \[\max(a_1, a_2, \dots, a_n) \le n \cdot \min(a_1, a_2, \dots, a_n),\] there exist three that are the side lengths of an acute triangle.

Solution

See also

2012 USAMO (Problems • Resources)
First Problem Followed by
Problem 2
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2012_USAMO_Problems/Problem_1&oldid=46570"