Let $x_1$ and $x_2$ be the two real roots of the equation $x^2 - 2(k+1)x+k^2 + 2 = 0$. If $(x_1+1)(x_2+1) =8$, find the value of $k$

Let $x_1$ and $x_2$ be the two real roots of the equation $x^2 - 2mx + (m^2+2m+3)=0$. Find the minimal value of $x_1^2 + x_2^2$.

If for any integer $k\ne 27$ and $\big(a-k^{2015}\big)$ is divisible by $(27-k)$, what is the last two digits of $a$?

Let $P(x)$ be a polynomial with integer coefficients. Show that $P(7)=5$ and $P(15)=9$ cannot hold simultaneously.

Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. The number of possible values of $k$ is

Two different positive numbers $a$ and $b$ each differ from their reciprocals by $1$. What is $a+b$?

For all positive integers $n$, let $f(n)=\log_{2002} n^2$. Let $N=f(11)+f(13)+f(14)$. Which of the following relations is true?

Find a polynomial with integral coefficients whose zeros include $\sqrt{2}+\sqrt{5}$.

Let $p(x)$ be a polynomial with integer coefficients. Assume that $p(a) = p(b) = p(c) = -1$, where $a, b, c$ are three different integers. Prove that $p(x)$ has no integral zeros.

Prove that the sum $$\sqrt{1001^2 + 1} + \sqrt{1002^2 + 1} + \cdots + \sqrt{2000^2 + 1}$$ is irrational.

If $P(x)$ denotes a polynomial of degree $n$ such that $P(k) = k/(k +1)$ for $k = 0, 1, 2, \dots n$, determine $P(n + 1)$.

The product of two of the four zeros of the quartic equation $$x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$$ is $-32$. Find $k$.

Let $n$ be an even positive integer, and let $p(x)$ be an $n$-degree polynomial such that $p(-k) = p(k)$ for $k = 1, 2, \dots , n$. Prove that there is a polynomial $q(x)$ such that $p(x) = q(x^2)$.

Let $a, b, c$ be distinct integers. Can the polynomial $(x - a)(x - b)(x - c) - 1$ be factored into the product of two polynomials with integer coefficients?

Let $p_1, p_2, \cdots, p_n$ be distinct integers and let $f(x)$ be the polynomial of degree $n$ given by $$f(x) = (x - p_1)(x - p_2)\cdots (x -p_n)$$ Prove that the polynomial $g(x) = (f(x))^2 + 1$ cannot be expressed as the product of two non-constant polynomials with integral
coefficients.

Does there exist a polynomial $f(x)$ for which $xf(x - 1) = (x + 1)f(x)$

Is it possible to write the polynomial $f(x) = x^{105}-9$ as the product of two polynomials of degree less than 105 with integer coefficients?

Let $P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ be a polynomial with integral coefficients. Suppose that there exist four distinct integers $a, b, c, d$ with $P(a) = P(b) = P(c) = P(d) = 5$. Prove that there is no integer $k$ satisfying $P(k) = 8$.

Consider the lines that meet the graph $y = 2x^4 + 7x^3 + 3x - 5$ in four distinct points $P_i = (x_i, y_i), i = 1, 2, 3, 4$. Prove that
$$\frac{x_1 + x_2 + x_3 + x_44}{4}$$
is independent of the line, and compute its value.

Find the value of $(2 + \sqrt{5})^{1/3} - (-2 + \sqrt{5})^{1/3}$.

There exist some integers, $a$, such that the equation $(a+1)x^2 -(a^2+1)x+2a^2-6=0$ is solvable in integers. Find the sum of all such $a$.

Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1)=(x+2)P(x)$ for every real $x$, and $(P(2))^2 = P(3)$. Then $P\big(\frac72\big)=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

The roots of $x^2 + ax + b+1$ are positive integers. Show that $a^2+b^2$ is not a prime number.

Let $\alpha$ and $\beta$ be the roots of $x^2+px+1$, and let $\gamma$ and $\delta$ be the roots of $x^2+qx+1$. Show $$(\alpha-\gamma)(\beta-\gamma)(\alpha+\delta)(\beta+\delta)=q^2-p^2$$

Find the real root of the polynomial $p(x)=8x^3 -3x^2 -3x -1$.