Practice (13)

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Suppose $x$ is in the interval $[0, \frac{\pi}{2}]$ and $\log_{24\sin x} (24\cos x)=\frac{3}{2}$. Find $24\cot^2 x$.

The solution of the equation $7^{x+7} = 8^x$ can be expressed in the form $x = \log_b 7^7$. What is $b$?

The graph of $y=x^6-10x^5+29x^4-4x^3+ax^2$ lies above the line $y=bx+c$ except at three values of $x$, where the graph and the line intersect. What is the largest of these values?

Assume $0 < r < 3$. Below are five equations for $x$. Which equation has the largest solution $x$?

What is the product of all the roots of the equation \[\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}.\]

What is the sum of all the solutions of $x = \left|2x-|60-2x|\right|$?

Let the sequence {$a_n$} satisfy $a_0=0$, $a_1=1$, $a_{n+2} = (n+3)a_{n+1} -(n+2)a_n$. Find whether the following equation is solvable in rational numbers:$$\sum_{i=1}^n\frac{x^i}{a_i-a_{i-1}}=-1\qquad\qquad(n \ge 2)$$

What is the sum of all real numbers $x$ such that $4^x - 6 \times 2^x + 8 = 0$?

Find the roots of $27x^3 + 9x^2 -30+8$

Let $S$ be the sum of all distinct real solutions of the equation $$\sqrt{x+2015}=x^2-2015$$ Compute $\lfloor 1/S \rfloor$.

Solve the following system in integers: $$ \left\{ \begin{array}{ll} x_1 + x_2 + \cdots + x_n &= n \\ x_1^2 + x_2^2 + \cdots + x_n^2 &= n \\ \cdots\\ x_1^n + x_2^n + \cdots + x_n^n &= n \end{array} \right. $$

Let $a_1, a_2, \cdots, a_{100}, b_1, b_2, \cdots, b_{100}$ be distinct real numbers. They are used to fill a $100 \times 100$ grids by putting the value of $(a_i + b_j)$ in the cell $(i, j)$ where $1 \le i, j \le 100$. Let $A_i$ be the product of all the numbers in column $i$, and $B_i$ be the product of all the numbers in row $i$. Show that if every $A_i$ equals to 1, then every $B_j$ equals to -1.

Solve the equation $$\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}=(a+1)\sqrt{\frac{x}{x+\sqrt{x}}}$$

What are all the ordered pairs of positive numbers $(x, y)$ for which $x=\sqrt{2y}$ and $y=\sqrt{x}$?

Solve the equation $x^4 -97x^3+2012x^2-97x+1=0$.

Solve the equation in real numbers $$\frac{2x}{2x^2-5x+3}+\frac{13x}{2x^2+x+3}=6$$

Find one real solution $(a, b, c, d)$ to the following system: $$ \left\{ \begin{array}{rcl} a+b+c+d&=&-2\\ ab+ac+ad+bc+bd+cd&=&-3\\ abc+abd+acd+bcd&=&4\\ abcd&=&3 \end{array} \right. $$

Solve equation $(6x+7)^2(3x+4)(x+1)=6$ in real numbers.

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}

Find one root to $\sqrt{3}x^7 + x^4 + 2=0$.

Solve this equation $(x-2)(x+1)(x+4)(x+7)=19$.

Let real numbers $x, y,$ and $z$ satisfy $$x+\frac{1}{y}=4\quad\text{,}\quad y+\frac{1}{z}=1\quad\text{,}\quad z +\frac{1}{x}=\frac{7}{3}$$ Find the value of $xyz$.

Find the range of real number $a$ if equation $\mid\frac{x^2}{x-1}\mid=a$ has exactly two distinct real roots.

Solve this equation $$\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$$

Let non-zero real numbers $a, b, c$ satisfy $a+b+c\ne 0$. If the following relations hold $$\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a}$$ Find the value of $$\frac{(a+b)(b+c)(c+a)}{abc}$$