Olym 10A. 다항식 심화
- 실전문제
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Problem n. Title Here
Let $P$ be a non-constant polynomial with integer coefficients such that if $n$ is a perfect power, so is $P(n)$. Prove that $P(x) = x$ or $P$ is a perfect power of a polynomial with integer coefficients.
A perfect power is an integer $n^k$, where $n \in \mathbb Z$ and $k \ge 2$. A perfect power of a polynomial is a polynomial $P(x)^k$, where $P$ has integer coefficients and $k \ge 2$.
USMCA 2020
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Problem n. Title Here
A mirrored polynomial is a polynomial $f$ of degree $100$ with real coefficients such that the $x^{50}$ coefficient of $f$ is $1$, and $f(x) = x^{100} f(1/x)$ holds for all real nonzero $x$. Find the smallest real constant $C$ such that any mirrored polynomial $f$ satisfying $f(1) \ge C$ has a complex root $z$ obeying $|z| = 1$.
USMCA 2019
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