Difference between revisions of "2021 Fall AMC 12A Problems/Problem 17"
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If a [[quadratic equation]] does not have two distinct real solutions, then its [[discriminant]] must be <math>\le0</math>. So, <math>b^2-4c\le0</math> and <math>c^2-4b\le0</math>. By inspection, there are <math>\boxed{\textbf{(B) } 6}</math> ordered pairs of positive integers that fulfill these criteria: <math>(1,1)</math>, <math>(1,2)</math>, <math>(2,1)</math>, <math>(2,2)</math>, <math>(3,3)</math>, and <math>(4,4)</math>. | If a [[quadratic equation]] does not have two distinct real solutions, then its [[discriminant]] must be <math>\le0</math>. So, <math>b^2-4c\le0</math> and <math>c^2-4b\le0</math>. By inspection, there are <math>\boxed{\textbf{(B) } 6}</math> ordered pairs of positive integers that fulfill these criteria: <math>(1,1)</math>, <math>(1,2)</math>, <math>(2,1)</math>, <math>(2,2)</math>, <math>(3,3)</math>, and <math>(4,4)</math>. | ||
− | {{AMC12 box|year=2021 Fall|ab=A|num-a= | + | {{AMC12 box|year=2021 Fall|ab=A|num-a=18|num-b=16}} |
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 21:48, 23 November 2021
Problem
For how many ordered pairs of positive integers does neither
nor
have two distinct real solutions?
Solution
If a quadratic equation does not have two distinct real solutions, then its discriminant must be . So,
and
. By inspection, there are
ordered pairs of positive integers that fulfill these criteria:
,
,
,
,
, and
.
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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All AMC 12 Problems and Solutions |
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