Difference between revisions of "2019 AIME II Problems/Problem 8"
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− | We wish to find <math>f(1) = a+b+c</math>. We first look at the real parts. As <math>\text{Re}(\omega^2) = -\frac{1}{2}</math> and <math>\text{Re}(\omega) = \frac{1}{2}</math>, we have <math>-\frac{1}{2}a + \frac{1}{2}b + c = 2015 \implies -a+b + 2c = 4030</math>. Looking at imaginary parts, we have <math>\text{Im}(\omega^2) = \text{Im}(\omega) = \frac{\sqrt{3}}{2}</math>, so <math>\frac{\sqrt{3}}{2}(a+b) = 2019\sqrt{3} \implies a+b = 4038</math>. As <math>a</math> and <math>b</math> do not exceed 2019, we must have <math>a = 2019</math> and <math>b = 2019</math>. Then <math>c = \frac{4030}{2} = 2015</math>, so <math>f(1) = 4038 + 2015 = 6053 \implies f(1) \pmod{1000} = \boxed{ | + | We wish to find <math>f(1) = a+b+c</math>. We first look at the real parts. As <math>\text{Re}(\omega^2) = -\frac{1}{2}</math> and <math>\text{Re}(\omega) = \frac{1}{2}</math>, we have <math>-\frac{1}{2}a + \frac{1}{2}b + c = 2015 \implies -a+b + 2c = 4030</math>. Looking at imaginary parts, we have <math>\text{Im}(\omega^2) = \text{Im}(\omega) = \frac{\sqrt{3}}{2}</math>, so <math>\frac{\sqrt{3}}{2}(a+b) = 2019\sqrt{3} \implies a+b = 4038</math>. As <math>a</math> and <math>b</math> do not exceed 2019, we must have <math>a = 2019</math> and <math>b = 2019</math>. Then <math>c = \frac{4030}{2} = 2015</math>, so <math>f(1) = 4038 + 2015 = 6053 \implies f(1) \pmod{1000} = \boxed{053}</math>. |
-scrabbler94 | -scrabbler94 |
Revision as of 13:34, 15 March 2020
Problem 8
The polynomial has real coefficients not exceeding , and . Find the remainder when is divided by .
Solution
We have where is a primitive 6th root of unity. Then we have
We wish to find . We first look at the real parts. As and , we have . Looking at imaginary parts, we have , so . As and do not exceed 2019, we must have and . Then , so .
-scrabbler94
See Also
2019 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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