Difference between revisions of "2022 USAJMO Problems/Problem 6"
(Created page with "==Problem== Let <math>a_0,b_0,c_0</math> be complex numbers, and define <cmath>a_{n+1}=a_n^2+2b_nc_n</cmath> <cmath>b_{n+1}=b_n^2+2c_na_n</cmath> <cmath>c_{n+1}=c_n^2+2a_nb_n...") |
m |
||
| Line 11: | Line 11: | ||
==Solution== | ==Solution== | ||
| + | |||
| + | ==See Also== | ||
| + | {{USAJMO newbox|year=2022|num-b=5|after=Last Question}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 18:05, 6 October 2023
Problem
Let 
be complex numbers, and define
![\[a_{n+1}=a_n^2+2b_nc_n\]](http://latex.artofproblemsolving.com/c/a/4/ca4a66b31e3c693419b2771c78287d9c589fa2f5.png)
![\[b_{n+1}=b_n^2+2c_na_n\]](http://latex.artofproblemsolving.com/4/6/3/463e0666030d03bba947553e814096c91db2b329.png)
![\[c_{n+1}=c_n^2+2a_nb_n\]](http://latex.artofproblemsolving.com/5/e/c/5eceeed057fdd37716fb322b3962ee36433d75ab.png)
for all nonnegative integers 
.
Suppose that 
for all . Prove that
![\[|a_0|^2+|b_0|^2+|c_0|^2\leq 1.\]](http://latex.artofproblemsolving.com/f/a/d/fad6f3579e7bc550cc761eaa40aa09481da67745.png)
Solution
See Also
| 2022 USAJMO (Problems • Resources) | ||
| Preceded by Problem 5 |
Followed by Last Question | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAJMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
