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 "Mock AIME 1 2005-2006/Problem 1"

(→‎Solution)
(→‎Solution)
 
(7 intermediate revisions by the same user not shown)
Line 6: Line 6:
 
== Solution ==
 
== Solution ==
  
−
Number the points <math>p_1</math>, <math>p_2</math>, <math>\dots</math>, <math>p_2006</math>. Assume the center is <math>O</math> and the given point is <math>p_1</math>. Then <math>\anglep_nOp_(n+1)</math> = <math>\frac {\pi}{1003}</math>, and we need to find the maximum <math>n</math> such that <math>\anglep_1Op_n+1 \le 60</math> degrees (<math>n+1</math> is given so that there are <math>n</math> repetitions of <math>\frac {\pi}{1003}</math>). This can be done in <math>\frac {\pi}{3}</math> divided by <math>\frac {\pi}{1003} = </math>\frac {1003}{3}<math> = </math>334.333\dots<math>, so </math>n<math> + </math>1<math> = </math>335<math>. We can choose </math>p_2<math>, </math>p_3<math>, \dots, </math>p_335<math>, so </math>334<math> points. But we need to multiply by </math>2<math> to count the number of points on the other side of </math>p_1$, so the answer is \boxed{668}.
+
Number the points <math>p_1</math>, <math>p_2</math>, <math>\dots</math>, <math>p_{2006}</math>. Assume the center is <math>O</math> and the given point is <math>p_1</math>. Then <math>\angle p_nOp_{n+1}</math> = <math>\frac {\pi}{1003}</math>, and we need to find the maximum <math>n</math> such that <math>\angle p_1Op_{n+1} \le 60</math> degrees. This can be done in <math>\frac {\pi}{3}</math> divided by <math>\frac {\pi}{1003}</math> = <math>\frac {1003}{3}</math> = <math>334.333\dots</math>, so <math>n</math> + <math>1</math> = <math>335</math>. We can choose <math>p_2</math>, <math>p_3</math>, <math>\dots</math>, <math>p_{335}</math>, so <math>334</math> points. But we need to multiply by <math>2</math> to count the number of points on the other side of <math>p_1</math>, so the answer is <math>\boxed{668}</math>.

Latest revision as of 20:32, 17 April 2009

Problem 1

$2006$ points are evenly spaced on a circle. Given one point, find the maximum number of points that are less than one radius distance away from that point.

Solution

Number the points $p_1$, $p_2$, $\dots$, $p_{2006}$. Assume the center is $O$ and the given point is $p_1$. Then $\angle p_nOp_{n+1}$ = $\frac {\pi}{1003}$, and we need to find the maximum $n$ such that $\angle p_1Op_{n+1} \le 60$ degrees. This can be done in $\frac {\pi}{3}$ divided by $\frac {\pi}{1003}$ = $\frac {1003}{3}$ = $334.333\dots$, so $n$ + $1$ = $335$. We can choose $p_2$, $p_3$, $\dots$, $p_{335}$, so $334$ points. But we need to multiply by $2$ to count the number of points on the other side of $p_1$, so the answer is $\boxed{668}$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Mock_AIME_1_2005-2006/Problem_1&oldid=31343"