Difference between revisions of "2020 AMC 12B Problems/Problem 8"
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==Problem== | ==Problem== | ||
− | How many ordered pairs of integers <math>(x,y)</math> satisfy the equation <cmath>x^{2020}+y^2=2y</cmath> | + | How many ordered pairs of integers <math>(x, y)</math> satisfy the equation<cmath>x^{2020}+y^2=2y?</cmath> |
+ | |||
+ | <math>\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{infinitely many}</math> | ||
==Solution== | ==Solution== |
Revision as of 22:59, 7 February 2020
Contents
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
Solution 2
Move the term to the other side to get
. Because
for all
, then
. If
or
, the right side is
and therefore
. When
, the right side become
, therefore
. Our solutions are
,
,
,
. There are
solutions, so the answer is
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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