Let $x_1$ and $x_2$ be two real roots of $x^2-x-1=0$. Find the value of $2x_1^5 + 5x_2^3$.
Find integer $m$ such that the equation $x^2+mx-m+1=0$ has two positive integer roots.
Let $\alpha$ and $\beta$ be two real roots of $x^4 +k=3x^2$ and also satisfy $\alpha + \beta = 2$. Find the value of $k$.
Determine all roots, real or complex, of the following system
\begin{align}
x+y+z &= 3\\
x^2+y^2+z^2 &= 3\\
x^3+y^3+z^3 &= 3
\end{align}
Let $r_1, \cdots, r_5$ be the roots of the polynomial $x^5 + 5x^4 - 79x^3 +64x^2 + 60x+144$. What is $r_1^2 +\cdots + r_5^2$?
Find all pairs of real numbers $(a, b)$ so that there exists a polynomial $P(x)$ with real coefficients and $P(P(x))=x^4-8x^3+ax^2+bx+40$.
Let real numbers $a, b, c, d$ satisfy
$$
\left\{
\begin{array}{ccl}
ax+by&=3\\
ax^2+by^2&=7\\
ax^3+by^3&=16\\
ax^4 + by^4 &=42
\end{array}
\right.
$$
Find $ax^5+by^5$.
Let real numbers $a$, $b$, and $c$ satisfy $a+b+c=2$ and $abc=4$. Find
the minimal value of the largest among $a$, $b$, and $c$.
the minimal value of $\mid a\mid +\mid b \mid +\mid c \mid$.
If $a\ne 0$ and $\frac{1}{4}(b-c)^2=(a-b)(c-a)$, compute $\frac{b+c}{a}$.
If all roots of the equation $$x^4-16x^3+(81-2a)x^2 +(16a-142)x+(a^2-21a+68)=0$$ are integers, find the value of $a$ and solve this equation.
Let real numbers $a, b, c$ satisfy $a > 0$, $b>0$, $2c>a+b$, and $c^2>ab$. Prove $$c-\sqrt{c^2-ab} < a < c +\sqrt{c^2-ab}$$
If all coefficients of the polynomial $$f(x)=a_nx^n + a_{n-1}x^{n-1}+\cdots+a_3x^3+x^2+x+1=0$$ are real numbers, prove that its roots cannot be all real.
Compute the value of $$\sqrt[3]{2+\frac{10}{3\sqrt{3}}}+\sqrt[3]{2-\frac{10}{3\sqrt{3}}}$$
and simplify $$\sqrt[3]{2+\frac{10}{3\sqrt{3}}}\quad\text{and}\quad\sqrt[3]{2-\frac{10}{3\sqrt{3}}}$$
Let $P(x)$ be a monic cubic polynomial. The lines $y = 0$ and $y = m$ intersect $P(x)$ at points $A$, $C$, $E$ and $B$, $D$, $F$ from left to right for a positive real number $m$. If $AB = \sqrt{7}$, $CD = \sqrt{15}$, and $EF = \sqrt{10}$, what is the value of $m$?
The sum and product of two numbers are equal to $y$. For which values of $y$ are these two numbers real?
Let $m$ and $n$ be the roots of $P(x)=ax^2+bx+c$. Find the coefficients of the quadratic polynomial whose roots are $m^2-n$ and $n^2-m$.
Let $\alpha$ and $\beta$ be the roots of $x^2+px+1$, and let $\gamma$ and $\sigma$ be the roots of $x^2+qx+1$.
Show
$$(\alpha - \gamma)(\beta-\gamma)(\alpha+\sigma)(\beta+\sigma) = q^2 - p^2$$
Let $b \ge 0$ be a real number. The product of the four real roots of the equations $x^2+2bx+c=0$ and $x^2+2cx+b=0$ is equal to $1$. Find the values of $b$ and $c$.
Show that if $a$, $b$, $c$ are the lengths of the sides of a triangle, then the equation
$$b^2x^2+(b^2+c^2-a^2)x + c^2=0$$
does not have any real roots.
Let $x_1$ and $x_2$ be the two roots of equation $x^2 − 3x + 2 = 0$. Find the following values without
computing $x_1$ and $x_2$ directly.
i) $x_1^4 + x_2^4$
ii) $x_1 - x_2$
(Note: for (i) above, how many different solutions can you find?)
Find the sum of all possible integer values of $a$ such that the equation $(a + 1)x^2-(a^2 + 1)x + (2a^2 − 6) = 0$ is solvable in integers.
Let $f(x)=a_0+a_1x+a_2x^2+\cdots +a_nx^n$ be a $n$-degree polynomial and all its coefficients $a_i$ $(0\le i\le n)$ be either $1$ or $-1$. If $f(x)$ has only real roots, what is the maximum value of $n$?
Prove that, if $|\alpha| < 2\sqrt{2}$, then there is no value of $x$ for which $$x^2-\alpha|x| + 2 < 0\qquad\qquad(*)$$
Find the solution set of (*) for $\alpha=3$.
For $\alpha > 2\sqrt{2}$, then the sum of the lengths of the intervals in which $x$ satisfies (*) is denoted by $S$. Find $S$ in terns of $\alpha$ and deduce that $S < 2\alpha$.