Difference between revisions of "1962 IMO Problems/Problem 4"
(→Problem) |
|||
| Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
Solve the equation <math>\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1</math>. | Solve the equation <math>\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1</math>. | ||
| + | |||
| + | ==Video Solution== | ||
| + | |||
| + | https://youtu.be/bRza0zea-e0?si=_E-hIXLt05qvNY9r | ||
| + | [Video Solution by little-fermat] | ||
==Solution== | ==Solution== | ||
Latest revision as of 23:37, 3 September 2023
Contents
Problem
Solve the equation 
.
Video Solution
https://youtu.be/bRza0zea-e0?si=_E-hIXLt05qvNY9r [Video Solution by little-fermat]
Solution
First, note that we can write the left hand side as a cubic function of 
. So there are at most 
distinct values of that satisfy this equation. Therefore, if we find three values of 
that satisfy the equation and produce three different , then we found all solutions to this cubic equation (without expanding it, which is another viable option). Indeed, we find that 
, 
, and 
all satisfy the equation, and produce three different values of , namely 
, 
, and 
. So we solve 
. Therefore, our solutions are:
See Also
| 1962 IMO (Problems) • Resources | ||
| Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
| All IMO Problems and Solutions | ||
