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

Difference between revisions of "2008 USAMO Problems/Problem 5"

m
 
Line 4: Line 4:
 
== Solutions ==
 
== Solutions ==
  
−
=== Solution 1 ===
+
=== Solution ===
 
Every time we perform an operation on the numbers on the blackboard <math>R = \left < r_1, r_2, r_3 \right ></math>, we perform the corresponding operation on the integers <math>A = \left < a_1, a_2, a_3 \right ></math> so that <math>R \cdot A = 0</math> continues to hold.  (For example, if we replace <math>r_1</math> with <math>r_1 - r_2</math> then we replace <math>a_2</math> with <math>a_1 + a_2</math>.)
 
Every time we perform an operation on the numbers on the blackboard <math>R = \left < r_1, r_2, r_3 \right ></math>, we perform the corresponding operation on the integers <math>A = \left < a_1, a_2, a_3 \right ></math> so that <math>R \cdot A = 0</math> continues to hold.  (For example, if we replace <math>r_1</math> with <math>r_1 - r_2</math> then we replace <math>a_2</math> with <math>a_1 + a_2</math>.)
  
Line 11: Line 11:
  
  
−
{{alternate solutions}}
+
 
  
 
== See also ==
 
== See also ==

Latest revision as of 22:37, 21 July 2020

Problem

(Kiran Kedlaya) Three nonnegative real numbers $r_1$, $r_2$, $r_3$ are written on a blackboard. These numbers have the property that there exist integers $a_1$, $a_2$, $a_3$, not all zero, satisfying $a_1r_1 + a_2r_2 + a_3r_3 = 0$. We are permitted to perform the following operation: find two numbers $x$, $y$ on the blackboard with $x \le y$, then erase $y$ and write $y - x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $0$ on the blackboard.

Solutions

Solution

Every time we perform an operation on the numbers on the blackboard $R = \left < r_1, r_2, r_3 \right >$, we perform the corresponding operation on the integers $A = \left < a_1, a_2, a_3 \right >$ so that $R \cdot A = 0$ continues to hold. (For example, if we replace $r_1$ with $r_1 - r_2$ then we replace $a_2$ with $a_1 + a_2$.)

It's possible to show we can always pick an operation so that $|A|^2$ is strictly decreasing. Without loss of generality, let $r_3 > r_2 > r_1$ and $a_3$ be positive. Then it cannot be true that both $a_1$ and $a_2$ are at least $\frac { - a_3}{2}$, or else $a_1r_1 + a_2r_2 + a_3r_3 > 0$. Without loss of generality, let $a_1 < \frac { - a_3}{2}$. Then we can replace $a_1$ with $a_1 + a_3$ and $r_3$ with $r_3 - r_1$ to make $|A|$ smaller. Since it is a strictly decreasing sequence of positive integers, after a finite number of operations we have $a_3 = 0$. We can now see that this result holds for $(r_1,r_2,0)$ if and only if it holds for $(1,\frac{r_2}{r_1},0)$. We can see that $\frac{r_2}{r_1}$ is a rational number given that $a_3$ = 0. It is a well known result of the euclidean algorithm that if we continue to perform these operations, $r_1$ or $r_2$ will eventually be 0.



See also

  • <url>viewtopic.php?t=202910 Discussion on AoPS/MathLinks</url>
2008 USAMO (Problems • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2008_USAMO_Problems/Problem_5&oldid=128847"