Practice (6)

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Let $a$, $b$ be real numbers such that $ab=-1$, $a+b=3$, compute $a^3+b^3$.

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

Denote $a$ and $b$ the roots of $(x-2)(x+4)+(x-3)(x+5)-(x-2)(x+5)=0$. Computer $a^3 + b^3 + \frac{11}{(a-1)(b-1)}$

Let $a$, $b$, $c$ be distinct nonzero real numbers, such that $$a+\frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}$$ Prove that $|abc|=1$.

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

Find all real solutions to the equations: $$ \begin{array}{rl} (x-y)(x^3+y^3)&=7\\ (x+y)(x^-y^3)&=3 \end{array} $$

Let $x, y$ and $z$ be real numbers such that $$ \begin{array}{rl} x^2 +2(y-1)(z-1) &=12 \\ y^2 +2(z-1)(x-1) &=6 \\ z^2 +2(x-1)(y-1) &=9 \end{array} $$ Find all the possible values of $x+y+z$.

Find all polynomials $f(x)$ such that $f(x^2) = f(x)f(x+1)$.

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 $$\frac{1}{\gamma_1 \overline{\gamma_1}}+\frac{1}{\gamma_2 \overline{\gamma_2}}+\cdots+\frac{1}{\gamma_5 \overline{\gamma_5}}$$

Let $x$ and $y$ be real numbers such that $$2 < \frac{x-y}{x+y} < 5$$ If $\frac{x}{y}$ is an integer, what is its value?

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

Find all pairs of positive integers $(a; b)$ such that $\frac{a}{b} + \frac{21b}{25a}$ is a positive integer.

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is composite.

There are 3 numbers A, B, and C, such that $1001C - 2002A = 4004$, and $1001B + 3003A = 5005$. What is the average of A, B, and C?

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

Let $a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$. What is $a + b + c + d$?

Solve in positive integers the equation $$m^2 - n^2 - 3n = 5$$


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 $P(x)$.

Solve the following question in integers $$x^6 + 3x^3 +1 = y^4$$

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.

Suppose $x, y, z \in \mathbb{R}^+$, such that \begin{align*} x^2 + y^2 + xy &= 9\\ y^2 + z^2 + yz &= 16\\ z^2 + x^2 + zx &= 25 \end{align*} Find the value of $xy+yz+zx$.

Suppose $P(x)$ is a monnic polynomial (meaning the leading coefficient is 1) with 20 roots, each distinct and of the form $\frac{1}{3^k}$ for $k=0, 1, \cdots, 19$. Find the coefficient of $x^{18}$ in $P(x)$.