1995 AIME Problems/Problem 2

Problem

Find the last three digits of the product of the positive roots of $\sqrt{1995}x^{\log_{1995}x}=x^2$ .

Solution

Taking the $\log_{1995}$ (logarithm) of both sides and then moving to one side yields a quadratic equation: $2(\log_{1995}x)^2 - 4(\log_{1995}x) + 1 = 0$ . Applying the quadratic formula yields that $\log_{1995}x = \frac{2 \pm \sqrt{2}}{2}$ . Thus, the product of the two roots is $= 1995^2$ , making the solution $025$ .

1995 AIME (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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1995 AIME Problems/Problem 2

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