# 2022-10 Polynomial with root 1

Prove or disprove the following:

For any positive integer $$n$$, there exists a polynomial $$P_n$$ of degree $$n^2$$ such that

(1) all coefficients of $$P_n$$ are integers with absolute value at most $$n^2$$, and

(2) $$1$$ is a root of $$P_n =0$$ with multiplicity at least $$n$$.

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# 2018-02 Impossible to squeeze

For $$n\ge 1$$, let $$f(x)=x^n+\sum_{k=0}^{n-1} a_k x^k$$ be a polynomial with real coefficients. Prove that if $$f(x)>0$$ for all $$x\in [-2,2]$$, then $$f(x)\ge 4$$ for some $$x\in [-2,2]$$.

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# 2014-17 Zeros of a polynomial

Let $p(x)=x^n+x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_1 x_1 + a_0$ be a polynomial. Prove that if $$p(z)=0$$ for a complex number $$z$$, then $|z| \le 1+ \sqrt{\max (|a_0|,|a_1|,|a_2|,\ldots,|a_{n-2}|)}.$

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# 2013-23 Polynomials with rational zeros

Find all polynomials $$P(x) = a_n x^n + \cdots + a_1 x + a_0$$ satisfying (i) $$a_n \neq 0$$, (ii) $$(a_0, a_1, \cdots, a_n)$$ is a permutation of $$(0, 1, \cdots, n)$$, and (iii) all zeros of $$P(x)$$ are rational.

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# 2013-11 Integer coefficient complex-valued polynomials

Determine all polynomials $$P(z)$$ with integer coefficients such that, for any complex number $$z$$ with $$|z| = 1$$, $$| P(z) | \leq 2$$.

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# 2013-08 Minimum of a set involving polynomials with integer coefficients

Let $$p$$ be a prime number. Let $$S_p$$ be the set of all positive integers $$n$$ satisfying
$x^n – 1 = (x^p – x + 1) f(x) + p g(x)$
for some polynomials $$f$$ and $$g$$ with integer coefficients. Find all $$p$$ for which $$p^p -1$$ is the minimum of $$S_p$$.

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# 2011-19 Irreducible polynomial

Find all n≥2 such that the polynomial xn-xn-1-xn-2-…-x-1 is irreducible over the rationals.

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# 2011-4 A polynomial with distinct real zeros

Let n>2. Let f (x) be a degree-n polynomial with real coefficients. If f (x) has n distinct real zeros r1<r2<…<rn, then Rolle’s theorem implies that the largest real zero q of (x) is between rn-1 and rn. Prove that q>(rn-1+rn)/2.

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