Polynomials - Factorization and modification

SOURCE:COMPETITION
Number of Problems: 4. : (Book)

1.

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Let $a$ , $b$ , and $c$ be positive integers with $age$ $bge$ $c$ such that $a^2-b^2-c^2+ab=2011$ and $a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$ .

What is $a$ ?

$extbf{(A)} 249qquad extbf{(B)} 250qquad extbf{(C)} 251qquad extbf{(D)} 252qquad extbf{(E)} 253$

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2.

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Let a, b, and c be real numbers such that $a-7b+8c=4$ and $8a+4b-c=7$ . Then $a^2-b^2+c^2$ is

$mathrm{(A) }0qquadmathrm{(B) }1qquadmathrm{(C) }4qquadmathrm{(D) }7qquadmathrm{(E) }8$

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3.

4.

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Let $a > 0$ , and let $P(x)$ be a polynomial with integer coefficients such that

$P(1) = P(3) = P(5) = P(7) = a$ , and
$P(2) = P(4) = P(6) = P(8) = -a$ .

What is the smallest possible value of $a$ ?

$extbf{(A)} 105 qquad extbf{(B)} 315 qquad extbf{(C)} 945 qquad extbf{(D)} 7! qquad extbf{(E)} 8!$

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