PEN Q Problems

Part of PEN Problems

Subcontests

(13)

1

Q 13

On Christmas Eve, 1983, Dean Jixon, the famous seer who had made startling predictions of the events of the preceding year that the volcanic and seismic activities of

1980

1981

were connected with mathematics. The diminishing of this geological activity depended upon the existence of an elementary proof of the irreducibility of the polynomial

P(x)=x^{1981}+x^{1980}+12x^{2}+24x+1983.

Is there such a proof?

1

Q 12

Prove that if the integers

a_{1}

a_{2}

\cdots

a_{n}

are all distinct, then the polynomial

(x-a_{1})^{2}(x-a_{2})^{2}\cdots (x-a_{n})^{2}+1

cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

1

Q 11

Show that the polynomial

x^{8} +98 x^{4}+1

can be expressed as the product of two nonconstant polynomials with integer coefficients.

1

Q 10

Suppose that the integers

a_{1}

a_{2}

\cdots

a_{n}

are distinct. Show that

(x-a_{1})(x-a_{2}) \cdots (x-a_{n})-1

cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

1

Q 9

For non-negative integers

n

k

P_{n, k}(x)

denote the rational function

\frac{(x^{n}-1)(x^{n}-x) \cdots (x^{n}-x^{k-1})}{(x^{k}-1)(x^{k}-x) \cdots (x^{k}-x^{k-1})}.

P_{n, k}(x)

is actually a polynomial for all

n, k \in \mathbb{N}

1

Q 8

Show that a polynomial of odd degree

2m+1

\mathbb{Z}

f(x)=c_{2m+1}x^{2m+1}+\cdots+c_{1}x+c_{0},

is irreducible if there exists a prime

p

p \not\vert c_{2m+1}, p \vert c_{m+1}, c_{m+2}, \cdots, c_{2m}, p^{2}\vert c_{0}, c_{1}, \cdots, c_{m}, \; \text{and}\; p^{3}\not\vert c_{0}.

1

Q 7

f(x)=x^{n}+5x^{n-1}+3

n>1

is an integer. Prove that

f(x)

cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

1

Q 6

Prove that for a prime

p

x^{p-1}+x^{p-2}+ \cdots +x+1

is irreducible in

\mathbb{Q}[x]

1

Q 5

(Eisentein's Criterion) Let

f(x)=a_{n}x^{n} +\cdots +a_{1}x+a_{0}

be a nonconstant polynomial with integer coefficients. If there is a prime

p

p

divides each of

a_{0}

a_{1}

\cdots

a_{n-1}

p

does not divide

a_{n}

p^2

does not divide

a_{0}

f(x)

is irreducible in

\mathbb{Q}[x]

1

Q 4

p

has decimal digits

p_{n}p_{n-1} \cdots p_0

p_{n}>1

. Show that the polynomial

p_{n}x^{n} + p_{n-1}x^{n-1}+\cdots+ p_{1}x + p_0

cannot be represented as a product of two nonconstant polynomials with integer coefficients

1

Q 3

n \ge 2

be an integer. Prove that if

k^2 + k + n

is prime for all integers

k

0 \leq k \leq \sqrt{\frac{n}{3}}

k^2 +k + n

is prime for all integers

k

0 \leq k \leq n - 2

1

Q 2

Prove that there is no nonconstant polynomial

f(x)

with integral coefficients such that

f(n)

is prime for all

n \in \mathbb{N}

1

Q 1

p(x) \in \mathbb{Z}[x]

P(a)P(b)=-(a-b)^2

for some distinct

a, b \in \mathbb{Z}

P(a)+P(b)=0