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 "1991 AIME Problems/Problem 15"

(solutions by (1) chess64 (3) calc rulz)
(→‎Solution 4 (Vector))
 
(10 intermediate revisions by 6 users not shown)
Line 5: Line 5:
  
 
__TOC__
 
__TOC__
−
== Solution ==
+
 
−
=== Solution 1 ===
+
== Solution 1 (Geometric Interpretation)==
−
Interpret the problem geometrically. Consider <math>n</math> right triangles joined at their vertices, with bases <math>a_1,a_2,\ldots,a_n</math> and heights <math>1,3,\ldots, 2n - 1</math>. The sum of their hypotenuses is the value of <math>S_n</math>. The minimum value of <math>S_n</math>, then, is the length of the straight line connecting the bottom vertex of the first right triangle and the top vertex of the last right triangle, so
+
Consider <math>n</math> right triangles joined at their vertices, with bases <math>a_1,a_2,\ldots,a_n</math> and heights <math>1,3,\ldots, 2n - 1</math>. The sum of their hypotenuses is the value of <math>S_n</math>. The minimum value of <math>S_n</math>, then, is the length of the straight line connecting the bottom vertex of the first right triangle and the top vertex of the last right triangle, so
 
<cmath>
 
<cmath>
 
S_n \ge \sqrt {\left(\sum_{k = 1}^n (2k - 1)\right)^2 + \left(\sum_{k = 1}^n a_k\right)^2}.
 
S_n \ge \sqrt {\left(\sum_{k = 1}^n (2k - 1)\right)^2 + \left(\sum_{k = 1}^n a_k\right)^2}.
 
</cmath>
 
</cmath>
−
Since the sum of the first <math>n</math> integers is <math>n^2</math> and the sum of <math>a_1,a_2,\ldots,a_n</math> is 17, we get
+
Since the sum of the first <math>n</math> odd integers is <math>n^2</math> and the sum of <math>a_1,a_2,\ldots,a_n</math> is 17, we get
 
<cmath>
 
<cmath>
 
S_n \ge \sqrt {17^2 + n^4}.
 
S_n \ge \sqrt {17^2 + n^4}.
 
</cmath>
 
</cmath>
−
If this is integer, we can write <math>17^2 + n^4 = m^2</math>, for an integer <math>m</math>. Thus, <math>(m - n^2)(m + n^2) = 289\cdot 1 = 17\cdot 17 = 1\cdot 289.</math> The only possible value, then, for <math>m</math> is <math>145</math>, in which case <math>n^2 = 144</math>, and <math>n = \boxed {012}</math>.
+
If this is an integer, we can write <math>17^2 + n^4 = m^2</math>, for an integer <math>m</math>. Thus, <math>(m - n^2)(m + n^2) = 289\cdot 1 = 17\cdot 17 = 1\cdot 289.</math> The only possible value, then, for <math>m</math> is <math>145</math>, in which case <math>n^2 = 144</math>, and <math>n = \boxed {012}</math>.
  
−
=== Solution 2 ===
+
== Solution 2 ==
 
The inequality  
 
The inequality  
 
<cmath>
 
<cmath>
Line 24: Line 24:
 
is a direct result of the [[Minkowski Inequality]]. Continue as above.
 
is a direct result of the [[Minkowski Inequality]]. Continue as above.
  
−
=== Solution 3 ===
+
== Solution 3 ==
 
Let <math>a_{i} = (2i - 1) \tan{\theta_{i}}</math> for <math>1 \le i \le n</math> and <math>0 \le \theta_{i} < \frac {\pi}{2}</math>. We then have that
 
Let <math>a_{i} = (2i - 1) \tan{\theta_{i}}</math> for <math>1 \le i \le n</math> and <math>0 \le \theta_{i} < \frac {\pi}{2}</math>. We then have that
 
<cmath>
 
<cmath>
Line 35: Line 35:
 
</cmath>
 
</cmath>
 
This is thus the minimum value, with equality when all the tangents are equal. The only value for which <math>\sqrt {289 + n^{4}}</math> is an integer is <math>n = 12</math> (see above solutions for details).
 
This is thus the minimum value, with equality when all the tangents are equal. The only value for which <math>\sqrt {289 + n^{4}}</math> is an integer is <math>n = 12</math> (see above solutions for details).
 +
 +
== Solution 4 (Vector) ==
 +
Observe the sum and apply it to the Cartesian plane, we can see that it is basically calculating distance between the origin and two points. Therefore we can assume every <math>\sqrt{(2k-1)^2+a_k^2}</math> to be a vector from the origin to point <math>(2k-1, a_k)</math>.
 +
Now we can do sum inside the vector so we get <math>((1+3+5+\ldots+2k-1,  a_1+a_2+a_3+\ldots+a_k)</math>, with <math>a_1+\ldots+a_k</math> to be <math>17</math> and <math>1+3+\ldots+2k-1</math> to be <math>k^2</math>. Then we calculate the length of the vector to be <math>\sqrt{k^4+289}</math>. Since the sum needs to be an integer, we assume <math>k^4+289</math> equals to <math>s^2</math>. Applying difference between squares, we get that <math>k^2-s=289</math> and <math>k^2+s=-1</math>. Therfore <math>2k^2</math> is <math>288</math> and <math>k=n=12</math> to be the final answer.
 +
 +
----JINZHENQIAN
  
 
== See also ==
 
== See also ==
Line 40: Line 46:
  
 
[[Category:Intermediate Algebra Problems]]
 
[[Category:Intermediate Algebra Problems]]
 +
{{MAA Notice}}

Latest revision as of 15:58, 10 November 2023

Problem

For positive integer $n_{}^{}$, define $S_n^{}$ to be the minimum value of the sum $\sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2},$ where $a_1,a_2,\ldots,a_n^{}$ are positive real numbers whose sum is 17. There is a unique positive integer $n^{}_{}$ for which $S_n^{}$ is also an integer. Find this $n^{}_{}$.

Solution 1 (Geometric Interpretation)

Consider $n$ right triangles joined at their vertices, with bases $a_1,a_2,\ldots,a_n$ and heights $1,3,\ldots, 2n - 1$. The sum of their hypotenuses is the value of $S_n$. The minimum value of $S_n$, then, is the length of the straight line connecting the bottom vertex of the first right triangle and the top vertex of the last right triangle, so \[S_n \ge \sqrt {\left(\sum_{k = 1}^n (2k - 1)\right)^2 + \left(\sum_{k = 1}^n a_k\right)^2}.\] Since the sum of the first $n$ odd integers is $n^2$ and the sum of $a_1,a_2,\ldots,a_n$ is 17, we get \[S_n \ge \sqrt {17^2 + n^4}.\] If this is an integer, we can write $17^2 + n^4 = m^2$, for an integer $m$. Thus, $(m - n^2)(m + n^2) = 289\cdot 1 = 17\cdot 17 = 1\cdot 289.$ The only possible value, then, for $m$ is $145$, in which case $n^2 = 144$, and $n = \boxed {012}$.

Solution 2

The inequality \[S_n \ge \sqrt {\left(\sum_{k = 1}^n (2k - 1)\right)^2 + \left(\sum_{k = 1}^n a_k\right)^2}.\] is a direct result of the Minkowski Inequality. Continue as above.

Solution 3

Let $a_{i} = (2i - 1) \tan{\theta_{i}}$ for $1 \le i \le n$ and $0 \le \theta_{i} < \frac {\pi}{2}$. We then have that \[S_{n} = \sum_{k = 1}^{n}\sqrt {(2k - 1)^{2} + a_{k}^{2}} = \sum_{k = 1}^{n}(2k - 1) \sec{\theta_{k}}\] Note that that $S_{n} + 17 = \sum_{k = 1}^{n}(2k - 1)(\sec{\theta_{k}} + \tan{\theta_{k}})$. Note that for any angle $\theta$, it is true that $\sec{\theta} + \tan{\theta}$ and $\sec{\theta} - \tan{\theta}$ are reciprocals. We thus have that $S_{n} - 17 = \sum_{k = 1}^{n}(2k - 1)(\sec{\theta_{k}} - \tan{\theta_{k}}) = \sum_{k = 1}^{n}\frac {2k - 1}{\sec{\theta_{k}} + \tan{\theta_{k}}}$. By the AM-HM inequality on these $n^{2}$ values, we have that: \[\frac {S_{n} + 17}{n^{2}}\ge \frac {n^{2}}{S_{n} - 17}\rightarrow S_{n}^{2}\ge 289 + n^{4}\] This is thus the minimum value, with equality when all the tangents are equal. The only value for which $\sqrt {289 + n^{4}}$ is an integer is $n = 12$ (see above solutions for details).

Solution 4 (Vector)

Observe the sum and apply it to the Cartesian plane, we can see that it is basically calculating distance between the origin and two points. Therefore we can assume every $\sqrt{(2k-1)^2+a_k^2}$ to be a vector from the origin to point $(2k-1, a_k)$. Now we can do sum inside the vector so we get $((1+3+5+\ldots+2k-1, a_1+a_2+a_3+\ldots+a_k)$, with $a_1+\ldots+a_k$ to be $17$ and $1+3+\ldots+2k-1$ to be $k^2$. Then we calculate the length of the vector to be $\sqrt{k^4+289}$. Since the sum needs to be an integer, we assume $k^4+289$ equals to $s^2$. Applying difference between squares, we get that $k^2-s=289$ and $k^2+s=-1$. Therfore $2k^2$ is $288$ and $k=n=12$ to be the final answer.

JINZHENQIAN

See also

1991 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Last question
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME 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=1991_AIME_Problems/Problem_15&oldid=201976"