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 "1973 IMO Problems/Problem 3"
Line 6: Line 6:
 
Substitute <math>z=x+1/x</math> to change the original equation into <math>z^2+az+b-2=0</math>. This equation has solutions <math>z=\frac{-a \pm \sqrt{a^2+8-4b}}{2}</math>. We also know that <math>|z|=|x+1/x| \geq 2</math>. So,
 
Substitute <math>z=x+1/x</math> to change the original equation into <math>z^2+az+b-2=0</math>. This equation has solutions <math>z=\frac{-a \pm \sqrt{a^2+8-4b}}{2}</math>. We also know that <math>|z|=|x+1/x| \geq 2</math>. So,
  
−
<math>\left | \frac{-a \pm \sqrt{a^2+8-4b}}{2} \right | \geq 2</math>
+
<cmath>\left | \frac{-a \pm \sqrt{a^2+8-4b}}{2} \right | \geq 2</cmath>
  
−
<math>\frac{|a|+\sqrt{a^2+8-4b}}{2} \geq 2</math>
+
<cmath>\frac{|a|+\sqrt{a^2+8-4b}}{2} \geq 2</cmath>
  
−
<math>|a|+\sqrt{a^2+8-4b} \geq 4</math>
+
<cmath>|a|+\sqrt{a^2+8-4b} \geq 4</cmath>
  
 
Rearranging and squaring both sides,
 
Rearranging and squaring both sides,
  
−
<math>a^2+8-4b \geq a^2-16|a|+16</math>
+
<cmath>a^2+8-4b \geq a^2-16|a|+16</cmath>
  
−
<math>2|a|-b \geq 2</math>
+
<cmath>2|a|-b \geq 2</cmath>
  
 
So, <math>a^2+b^2 \geq a^2+(2-2|a|)^2 = 5a^2-8|a|+4 = 5(|a|-\frac{4}{5})^2+\frac{4}{5}</math>.
 
So, <math>a^2+b^2 \geq a^2+(2-2|a|)^2 = 5a^2-8|a|+4 = 5(|a|-\frac{4}{5})^2+\frac{4}{5}</math>.

Revision as of 14:48, 29 January 2021

Let $a$ and $b$ be real numbers for which the equation $x^4 + ax^3 + bx^2 + ax + 1 = 0$ has at least one real solution. For all such pairs $(a, b)$, find the minimum value of $a^2 + b^2$.

Solution

Substitute $z=x+1/x$ to change the original equation into $z^2+az+b-2=0$. This equation has solutions $z=\frac{-a \pm \sqrt{a^2+8-4b}}{2}$. We also know that $|z|=|x+1/x| \geq 2$. So,

\[\left | \frac{-a \pm \sqrt{a^2+8-4b}}{2} \right | \geq 2\]

\[\frac{|a|+\sqrt{a^2+8-4b}}{2} \geq 2\]

\[|a|+\sqrt{a^2+8-4b} \geq 4\]

Rearranging and squaring both sides,

\[a^2+8-4b \geq a^2-16|a|+16\]

\[2|a|-b \geq 2\]

So, $a^2+b^2 \geq a^2+(2-2|a|)^2 = 5a^2-8|a|+4 = 5(|a|-\frac{4}{5})^2+\frac{4}{5}$.

Therefore, the smallest possible value of $a^2+b^2$ is $\frac{4}{5}$, when $a=\pm \frac{4}{5}$ and $b=\frac{-2}{5}$.

Borrowed from [1]

See Also

1973 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=1973_IMO_Problems/Problem_3&oldid=143887"