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Let $f$ be the unique polynomial of degree at most $2026$ such that for all $n \in\{1,2,3, \ldots, 2027\}$,
$$
f(n)= \begin{cases}1 & \text { if } n \text { is a perfect square } \\ 0 & \text { otherwise }\end{cases}
$$
Suppose that $\frac{a}{b}$ is the coefficient of $x^{2025}$ in $f$, where $a$ and $b$ are integers such that $\operatorname{gcd}(a, b)=1$. Compute the unique integer $r$ between $0$ and $2026$ (inclusive) such that $a-r b$ is divisible by $2027$. (Note that $2027$ is prime.)
|
1037
|
Let $a, b$, and $c$ be pairwise distinct complex numbers such that
$$
a^{2}=b+6, \quad b^{2}=c+6, \quad \text { and } \quad c^{2}=a+6
$$
Compute the two possible values of $a+b+c$. In your answer, list the two values in a comma-separated list of two valid \LaTeX expressions.
|
\frac{-1+\sqrt{17}}{2}, \frac{-1-\sqrt{17}}{2}
|
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