Difference between revisions of "2020 AMC 12B Problems/Problem 8"
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==Solution== | ==Solution== | ||
| − | + | Set it up as a quadratic in terms of y: | |
| + | <cmath>y^2-2y+x^{2020}=0</cmath> | ||
| + | Then the discriminant is | ||
| + | <cmath>\Delta = 4-4x^{2020}</cmath> | ||
| + | This will clearly only yield real solutions when <math>x^{2020} \leq 1</math> | ||
==See Also== | ==See Also== | ||
Revision as of 19:21, 7 February 2020
Problem
How many ordered pairs of integers 
satisfy the equation ![\[x^{2020}+y^2=2y\]](http://latex.artofproblemsolving.com/6/7/5/675a2782e7f97942c051c125abe88fb075b0d07c.png)
Solution
Set it up as a quadratic in terms of y:
![\[y^2-2y+x^{2020}=0\]](http://latex.artofproblemsolving.com/5/7/3/57302fcd989d1abb3190666693fe5a475f8fdd1c.png)
Then the discriminant is
![\[\Delta = 4-4x^{2020}\]](http://latex.artofproblemsolving.com/2/f/a/2fa32c16b6896590c31f648e13db1aafd9439433.png)
This will clearly only yield real solutions when 
See Also
| 2020 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 7 |
Followed by Problem 9 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
