Practice (8)

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Let $a + ar_1 + ar_1^2 + ar_1^3 + \cdots$ and $a + ar_2 + ar_2^2 + ar_2^3 + \cdots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $r_1$, and the sum of the second series is $r_2$. What is $r_1 + r_2$?

The zeroes of the function $f(x)=x^2-ax+2a$ are integers .What is the sum of the possible values of a?

The polynomial $x^3-ax^2+bx-2010$ has three positive integer roots. What is the smallest possible value of $a$?

A quadratic equation $ax^2 - 2ax + b = 0$ has two real solutions. What is the average of these two solutions?

Let $a$ and $b$ be the roots of the equation $x^2-mx+2=0$. Suppose that $a+\frac1b$ and $b+\frac1a$ are the roots of the equation $x^2-px+q=0$. What is $q$?

The quadratic equation $x^2+mx+n=0$ has roots twice those of $x^2+px+m=0$, and none of $m,n,$ and $p$ is zero. What is the value of $\frac{n}{p}$?

Let $\alpha$ and $\beta$ be two real roots of the equation $x^2 + x - 4=0$. Find the value of $\alpha^2 - 5\beta + 10$ without computing the value of $\alpha$ and $\beta$.

Solve the following equations for all real numbers $r, s, t$: $$ \begin{array}{rl} rst &=30\\ rs+st+tr &=-11\\ r+s+t &=-4 \end{array} $$

Let $\gamma_i$ and $\overline{\gamma_i}$ be the 10 zeros of $x^{10}+(13x-1)^{10}$, where $i=1, 2, 3, 4, 5$. Compute $$\frac{1}{\gamma_1 \overline{\gamma_1}}+\frac{1}{\gamma_2 \overline{\gamma_2}}+\cdots+\frac{1}{\gamma_5 \overline{\gamma_5}}$$

What is the sum of all of the roots of $(2x + 3) (x - 4) + (2x + 3) (x - 6) = 0$?

Show that the equation $$x^2 + y^2 -19xy - 19 =0$$ is not solvable in integers.

Solve in positive integers $$x^3 + y^3 + z^3 = 3xyz$$

Let $P(x) = kx^3 + 2k^2x^2 + k^3$. Find the sum of all real numbers $k$ for which $x - 2$ is a factor of $P(x)$.

If one root of the equation $x^2 -6x+m^2-2m+5=0$ is $2$. Find the value of the other root and $m$.

If the equation $x^2+2(m-2)x + m^2 + 4 = 0 $ has two real roots, and the sum of their square is 21 more than their product, find the value of $m$.

Let $\alpha$ and $\beta$ be the two roots of $x^2 + 2x -5=0$. Evaluate $\alpha^2 + \alpha\beta + 2\alpha$.

If at least one real root of equation $x^2 - mx +5+m=0$ equals one root of $x^2 - (7m+1)x+13m+7=0$, compute the product of the four roots of these two equations.

If the difference of the two roots of the equation $x^2 + 6x + k=0$ is 2, what is the value of $k$?

If the two roots of $(a^2 -1)x^2 -(a+1)x+1=0$ are reciprocal, find the value of $a$.

Let $x_1$ and $x_2$ be the two roots of $x^2 - 3mx +2(m-1)=0$. If $\frac{1}{x_1}+\frac{1}{x_2}=\frac{3}{4}$, what is the value of $m$?

Let $x_1$ and $x_2$ be the two roots of $2x^2 -7x -4=0$, compute the values of the following expressions using as many different ways as possible. (1) $x_1^2 + x_2^2$ (2) $(x_1+1)(x_2+1)$ (3) $\mid x_1 - x_2 \mid$

If one root of $x^2 + \sqrt{2}x + a = 0$ is $1-\sqrt{2}$, find the other root as well as the value of $a$.

Consider the equation $x^2 +(m-2)x + \frac{1}{2}m-3=0$. (1) Show that this equation always have two distinct real roots (2) Let $x_1$ and $x_2$ be its roots. If $x_1+x_2=m+1$, what is the value of $m$?

If $n>0$ and $x^2 -(m-2n)x + \frac{1}{4}mn=0$ has two equal positive real roots, what is the value of $\frac{m}{n}$?

If real number $m$ and $n$ satisfy $mn\ne 1$ and $19m^2+99m+1=0$ and $19+99n+n^2=0$, what is the value of $\frac{mn+4m+1}{n}$?