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 |
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