Practice (TheColoringMethod)

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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 $ax^5+by^5$.

Find the range of function $y=x+\sqrt{x^2 -3x+2}$.

Solve $$\Big|\frac{1}{\log_{\frac{1}{2}}x+2}\Big|> \frac{3}{2}$$

If for any non-negative real numbers $x$ and $y$, function $f(x)$ satisfies the properties that $f(x)\ge 0$, $f(1)\ne 0$, and $f(x+y^2)=f(x)+2f^2(y)$ , compute the value of $f(2+\sqrt{3})$.

If the minimal and maximum values of function $$f(x)=-\frac{1}{2}x^2 + \frac{13}{2}$$ in the domain $[a, b]$ are $2a$ and $2b$, respectively, determine the values of $a$ and $b$.

Is function $f(x)=\lg(x+\sqrt{x^2+1})$ an odd or even function?

If real number $x$ satisfies $x^4 - 2x^3 -7x^2 + 8x +12\le 0$, find the max value of $|x+\frac{4}{x}|$


For any real numbers $x$ and $y$, the following holds $$[f(x+y)]^2 = [f(x)]^2 + [f(y)]^2$$ Find the exact form of $f(x)$.

Let $f(x)$ be a polynomial with respect to $x$ and $$f(x+1)+f(x-1)=2x^2-4x$$ Find $f(x)$.

Find the function $f(x)$ such that $f(0)=1$, $f(\frac{\pi}{2})=2$, and for any $x, y\in\mathbb{R}$, $$f(x+y)+f(x-y)=2f(x)\cos y$$

Let the domain of function $f(n)$ be $\mathbb{N}$, $f(1)=1$, and for any $m, n\in\mathbb{N}$, $$f(m+n)=f(m)+f(n)+mn$$ Determine $f(n)$.

Let the domain of function $f(n)$ be $\mathbb{N}$, $f(1)=1$, and for any integer $n \ge 2$, $$f(n)=f(n-1) + 2^{n-1}$$ Determine $f(n)$.

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 $$c-\sqrt{c^2-ab} < a < c +\sqrt{c^2-ab}$$

    Suppose the graph of $f(x)=x^4 + ax^3 + bx^2 + cd + d$, where $a$, $b$, $c$, $d$ are all real constants, passes through three points $A \big(2,\frac{1}{2}\big)$, $B \big(3, \frac{1}{3}\big)$, and $C \big(4, \frac{1}{4}\big)$. Find the value of $f(1) + f(5)$.

    Find a quadratic polynomial $f(x)=x^2 + mx +n$ such that $$f(a)=bc,\quad f(b) = ca,\quad f(c) = ab$$ where $a$, $b$, $c$ are three distinct real numbers.

    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 $$\sqrt[3]{2+\frac{10}{3\sqrt{3}}}\quad\text{and}\quad\sqrt[3]{2-\frac{10}{3\sqrt{3}}}$$

    Prove there cannot exist a $998$-degree polynomial with real number coefficients $P(x)$ such that $$[P(x)]^2-1=P(x^2+1)$$ holds for any $x\in\mathbb{C}$.

    If $abc=1$, solve this equation $$\frac{2ax}{ab+a+1}+\frac{2bx}{bc+b+1}+\frac{2cx}{ca+c+1}=1$$

    How many pairs of ordered real numbers $(x, y)$ are there such that $$ \left\{ \begin{array}{ccl} \mid x\mid + y &=& 12\\ x + \mid y \mid &=&6 \end{array} \right. $$

    Let $a$, $b$, and $c$ be three distinct numbers such that $$\frac{a+b}{a-b}=\frac{b+c}{2(b-c)}=\frac{c+a}{3(c-a)}$$ Prove that $8a + 9b + 5c = 0$.

    Solve this equation in real numbers: $$\sqrt{x}+\sqrt{y-1}+\sqrt{z-2}=\frac{1}{2}\times(x+y+z)$$