2021 Fall AMC 12A Problems/Problem 17
Problem
For how many ordered pairs of positive integers does neither
nor
have two distinct real solutions?
Solution 1
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
.
Solution 2
We need to solve the following system of inequalities:
Feasible solutions are in the region formed between two parabolas and
.
Define and
.
Therefore, all feasible solutions are in the region formed between the graphs of these two functions.
For ,
and
.
Hence, the feasible
are 1, 2.
For ,
and
.
Hence, the feasible
are 1, 2.
For ,
and
.
Hence, the feasible
is 3.
For ,
and
.
Hence, the feasible
is 4.
For ,
. Hence, there is no feasible
.
Putting all cases together, the correct answer is .
~Steven Chen (www.professorchenedu.com)
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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.