Difference between revisions of "2019 AIME II Problems/Problem 8"

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\end{align*}</cmath>
 
\end{align*}</cmath>
  
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{53}</math>.
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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{053}</math>.
  
 
-scrabbler94
 
-scrabbler94

Revision as of 13:34, 15 March 2020

Problem 8

The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019$, and $f\left(\tfrac{1+\sqrt{3}i}{2}\right)=2015+2019\sqrt{3}i$. Find the remainder when $f(1)$ is divided by $1000$.

Solution

We have $\frac{1+\sqrt{3}i}{2} = \omega$ where $\omega = e^{\frac{i\pi}{3}}$ is a primitive 6th root of unity. Then we have

\begin{align*} f(\omega) &= a\omega^{2018} + b\omega^{2017} + c\omega^{2016}\\ &= a\omega^2 + b\omega + c \end{align*}

We wish to find $f(1) = a+b+c$. We first look at the real parts. As $\text{Re}(\omega^2) = -\frac{1}{2}$ and $\text{Re}(\omega) = \frac{1}{2}$, we have $-\frac{1}{2}a + \frac{1}{2}b + c = 2015 \implies -a+b + 2c = 4030$. Looking at imaginary parts, we have $\text{Im}(\omega^2) = \text{Im}(\omega) = \frac{\sqrt{3}}{2}$, so $\frac{\sqrt{3}}{2}(a+b) = 2019\sqrt{3} \implies a+b = 4038$. As $a$ and $b$ do not exceed 2019, we must have $a = 2019$ and $b = 2019$. Then $c = \frac{4030}{2} = 2015$, so $f(1) = 4038 + 2015 = 6053 \implies f(1) \pmod{1000} = \boxed{053}$.

-scrabbler94

See Also

2019 AIME II (ProblemsAnswer KeyResources)
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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