2021 Fall AMC 12A Problems/Problem 17: Difference between revisions
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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 20:48, 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) | |
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