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
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 Note that:
- If then from which
- If then from which
- If then from which
- If then from which
Together, there are
~MRENTHUSIASM
Solution 1(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 b and c. 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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All AMC 10 Problems and Solutions |
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