Difference between revisions of "2020 AMC 12B Problems/Problem 8"
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Then the discriminant is | Then the discriminant is | ||
<cmath>\Delta = 4-4x^{2020}</cmath> | <cmath>\Delta = 4-4x^{2020}</cmath> | ||
− | This will clearly only yield real solutions when <math>x^{2020} \leq 1</math> | + | This will clearly only yield real solutions when <math>x^{2020} \leq 1</math>, because it is always positive. |
+ | Then <math>x=-1,0,1</math>. Checking each one: | ||
+ | <math>-1</math> and <math>1</math> are the same when raised to the 2020th power: | ||
+ | <cmath>y^2-2y+1=(y-1)^2=0</cmath> | ||
+ | This has only has solutions <math>1</math>, so <math>(\pm 1,1)</math> are solutions. | ||
+ | Next, if <math>x=0</math>: | ||
+ | <cmath>y^2-2y=0</cmath> | ||
+ | Which has 2 solutions, so <math>(0,2)</math> and <math>(0,0)</math> | ||
+ | |||
+ | These are the only 4 solutions, so <math>\boxed{D}</math> | ||
==See Also== | ==See Also== |
Revision as of 19:28, 7 February 2020
Problem
How many ordered pairs of integers satisfy the equation
Solution
Set it up as a quadratic in terms of y: Then the discriminant is This will clearly only yield real solutions when , because it is always positive. Then . Checking each one: and are the same when raised to the 2020th power: This has only has solutions , so are solutions. Next, if : Which has 2 solutions, so and
These are the only 4 solutions, so
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 |
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