Difference between revisions of "1959 IMO Problems/Problem 3"
(The "solution 2" was a complete balooney. The two polynimials are not equivalent by any means) |
m (→Solution) |
||
(7 intermediate revisions by 3 users not shown) | |||
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+ | + | <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 40: | Line 40: | ||
<center> | <center> | ||
− | <math>16 \cos^2 {2x} | + | <math>16 \cos^2 {2x} + 8 \cos {2x} - 4 = 0, </math> |
</center> | </center> | ||
− | The | + | 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}} | ||
− | + | == See Also == | |
{{IMO box|year=1959|num-b=2|num-a=4}} | {{IMO box|year=1959|num-b=2|num-a=4}} | ||
[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] |
Latest revision as of 22:47, 1 June 2024
Problem
Let be real numbers. Consider the quadratic equation in :
Using the numbers , form a quadratic equation in , whose roots are the same as those of the original equation. Compare the equations in and for .
Solution
Let the original equation be satisfied only for . Then we wish to construct a quadratic with roots .
Clearly, the sum of the roots of this quadratic must be
and the product of its roots must be
Thus the following quadratic fulfils the conditions:
Now, when we let , our equations are
and
This simplifies to previous equation. The first root of the first equation corresponds to and the second root of the first equation corresponds to . These are relatively well-known trigonometric values, yielding roots .
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 |