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 "1959 IMO Problems/Problem 3"

 
m (→‎Solution)
 
(12 intermediate revisions by 6 users not shown)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
  
−
Let <math>\displaystyle a,b,c</math> be real numbers. Consider the quadratic equation in <math>\displaystyle \cos{x}</math> :
+
Let <math>a,b,c</math> be real numbers. Consider the quadratic equation in <math>\cos{x}</math> :
  
 
<center>
 
<center>
−
<math>\displaystyle a\cos ^{2}x + b\cos{x} + c = 0.</math>
+
<math>a\cos ^{2}x + b\cos{x} + c = 0.</math>
 
</center>
 
</center>
  
−
Using the numbers <math>\displaystyle a,b,c</math>, form a quadratic equation in <math>\displaystyle \cos{2x}</math>, whose roots are the same as those of the original equation. Compare the equations in <math>\displaystyle \cos{x}</math> and <math>\displaystyle \cos{2x}</math> for <math>\displaystyle a=4, b=2, c=-1</math>.
+
Using the numbers <math>a,b,c</math>, form a quadratic equation in <math>\cos{2x}</math>, whose roots are the same as those of the original equation. Compare the equations in <math>\cos{x}</math> and <math>\cos{2x}</math> for <math>a=4, b=2, c=-1</math>.
  
 
== Solution ==
 
== Solution ==
  
−
Let the original equation be satisfied only for <math>\displaystyle \cos{x}=m, \cos{x}=n </math>.  Then we wish to construct a quadratic with roots <math>\displaystyle 2m^2 -1, 2n^2 -1</math>.
+
Let the original equation be satisfied only for <math>\cos{x}=m, \cos{x}=n </math>.  Then we wish to construct a quadratic with roots <math>2m^2 -1, 2n^2 -1</math>.
  
 
Clearly, the sum of the roots of this quadratic must be
 
Clearly, the sum of the roots of this quadratic must be
Line 22: Line 22:
  
 
<center>
 
<center>
−
<math> 4m^2 n^2 - 2(m^2 + n^2) + 1 = \frac{4c^2}{a^2}+\frac{4ac - 2b^2}{a^2} + \frac{a^2}{a^2} = \frac{(a+2)^2 - 2b^2}{a^2} </math>
+
<math> 4m^2 n^2 - 2(m^2 + n^2) + 1 = \frac{4c^2}{a^2}+\frac{4ac - 2b^2}{a^2} + \frac{a^2}{a^2} = \frac{(a+2c)^2 - 2b^2}{a^2} </math>
 
</center>
 
</center>
  
Line 28: Line 28:
  
 
<center>
 
<center>
−
<math> \displaystyle a^2 \cos ^2 {2x} + (2a^2 + 4ac - 2b^2)\cos{2x} + (a+2c)^2 - 2b^2 = 0 </math>
+
<math>a^2 \cos ^2 {2x} + (2a^2 + 4ac - 2b^2)\cos{2x} + (a+2c)^2 - 2b^2 = 0 </math>
 
</center>
 
</center>
  
−
Now, when we let <math>\displaystyle a=4, b=2, c= -1</math>, our equations are  
+
Now, when we let <math>a=4, b=2, c= -1</math>, our equations are  
  
 
<center>
 
<center>
−
<math>\displaystyle 4 \cos^2 {x} + 2 \cos {x} - 1 = 0 </math>
+
<math>4 \cos^2 {x} + 2 \cos {x} - 1 = 0 </math>
 
</center>
 
</center>
  
Line 40: Line 40:
  
 
<center>
 
<center>
−
<math>\displaystyle 16 \cos^2 {2x} + 8 \cos {2x} - 4 = 0, </math>
+
<math>16 \cos^2 {2x} + 8 \cos {2x} - 4 = 0, </math>
 
</center>
 
</center>
  
−
i.e., they are multiples of each other.  The reason behind this is that the roots of the first equation are <math> \cos {x} = \frac{-1 \pm \sqrt{5}}{4}</math>, which implies that <math>\displaystyle x</math> is one of two certain multiples of <math>\frac{\pi}{5}</math>, and when <math> \displaystyle 5 \nmid n</math>, <math> \left|\cos\left(\frac{n\pi}{5}\right)\right|</math> can only assume two distinct values.  Q.E.D.
+
This simplifies to previous equation.  
−
 
+
The first root of the first equation <math> \cos {x} = \frac{-1 + \sqrt{5}}{4}</math> corresponds to <math> \cos {2x} = \frac{-1 - \sqrt{5}}{4}</math> and the second root of the first equation <math> \cos {x} = \frac{-1 - \sqrt{5}}{4}</math> corresponds to <math> \cos {2x} = \frac{-1 + \sqrt{5}}{4}</math>. These are relatively well-known trigonometric values, yielding roots <math>x = \boxed{72^\circ, 144^\circ}</math>.
  
 
{{alternate solutions}}
 
{{alternate solutions}}
  
−
== Resources ==
+
== See Also ==
−
 
+
{{IMO box|year=1959|num-b=2|num-a=4}}
−
* [[1959 IMO Problems]]
 
−
* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=341512#p341512 Discussion on AoPS/MathLinks]
 
−
 
 
  
 
[[Category:Olympiad Algebra Problems]]
 
[[Category:Olympiad Algebra Problems]]

Latest revision as of 22:47, 1 June 2024

Problem

Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ :

$a\cos ^{2}x + b\cos{x} + c = 0.$

Using the numbers $a,b,c$, form a quadratic equation in $\cos{2x}$, whose roots are the same as those of the original equation. Compare the equations in $\cos{x}$ and $\cos{2x}$ for $a=4, b=2, c=-1$.

Solution

Let the original equation be satisfied only for $\cos{x}=m, \cos{x}=n$. Then we wish to construct a quadratic with roots $2m^2 -1, 2n^2 -1$.

Clearly, the sum of the roots of this quadratic must be

$2 (m^2 + n^2) - 2 = 2 \left(\frac{b^2-2ac}{a^2}\right) - 2 = \frac{2b^2 - 4ac - 2a^2}{a^2},$

and the product of its roots must be

$4m^2 n^2 - 2(m^2 + n^2) + 1 = \frac{4c^2}{a^2}+\frac{4ac - 2b^2}{a^2} + \frac{a^2}{a^2} = \frac{(a+2c)^2 - 2b^2}{a^2}$

Thus the following quadratic fulfils the conditions:

$a^2 \cos ^2 {2x} + (2a^2 + 4ac - 2b^2)\cos{2x} + (a+2c)^2 - 2b^2 = 0$

Now, when we let $a=4, b=2, c= -1$, our equations are

$4 \cos^2 {x} + 2 \cos {x} - 1 = 0$

and

$16 \cos^2 {2x} + 8 \cos {2x} - 4 = 0,$

This simplifies to previous equation. The first root of the first equation $\cos {x} = \frac{-1 + \sqrt{5}}{4}$ corresponds to $\cos {2x} = \frac{-1 - \sqrt{5}}{4}$ and the second root of the first equation $\cos {x} = \frac{-1 - \sqrt{5}}{4}$ corresponds to $\cos {2x} = \frac{-1 + \sqrt{5}}{4}$. These are relatively well-known trigonometric values, yielding roots $x = \boxed{72^\circ, 144^\circ}$.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

See Also

1959 IMO (Problems) • Resources
Preceded by
Problem 2 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 4
All IMO Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1959_IMO_Problems/Problem_3&oldid=219965"