2021 Fall AMC 10A Problems/Problem 20
Problem
How many ordered pairs of positive integers exist where both and do not have distinct, real solutions?
Solution 1 (Casework)
A quadratic equation does not have real solutions if and only if the discriminant is nonpositive. We conclude that:
- Since does not have real solutions, we have
- Since does not have real solutions, we have
Squaring the first inequality, we get Multiplying the second inequality by we get Combining these results, we get We apply casework to the value of
- If then from which
- If then from which
- If then from which
- If then from which
Together, there are ordered pairs namely and
~MRENTHUSIASM
Solution 2 (Oversimplified but risky)
A quadratic equation has one solution if and only if is Similarly, it is imaginary if and only if is less than one. We proceed as following:
We want both to be value or imaginary and to be value or imaginary. is one such case since is Also, are always imaginary for both and We also have along with since the latter has one solution, while the first one is imaginary. Therefore, we have total ordered pairs of integers, which is
~Arcticturn
See Also
2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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