If the product of two roots of polynomial $x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$ is $- 32$. Find the value of $k$.
Show that for any integer $n\ge 2$: $n! < \Big(\frac{n+1}{2}\Big)^n$
Let $a_1, a_2, \cdots, a_n$ be positive real numbers such that $a_1\cdot a_2\cdots a_n=1$. Show that $$(1+a_1)(1+a_2)\cdots(1+a_n)\ge 2^n$$
Let ${{a}_{2}}, {{a}_{3}}, \cdots, {{a}_{n}}$ be positive real numbers that satisfy ${{a}_{2}}\cdot {{a}_{3}}\cdots {{a}_{n}}=1$ . Prove that $$(a_2+1)^2\cdot (a_3+1)^3\cdots (a_n+1)^n\ge n^n$$
Let $a, b$ be positive real numbers. Prove $$(a+b)\sqrt{\frac{a+b}{2}} \ge a\sqrt{b} + b \sqrt{a}$$.
Let $a, b$ be positive numbers, show that $$\frac{1}{2}(a+b)+\frac{1}{4}\ge \sqrt{\frac{a+b}{2}}$$
If $a>1$, then $$\frac{1}{a-1}+\frac{1}{a} + \frac{1}{a+1} > \frac{3}{a}$$ holds.
Let $a, b$ be positive numbers such that $a+b=1$. Show that $$\Big(a+\frac{1}{a}\Big)^2 +\Big(b+\frac{1}{b}\Big)^2\ge \frac{25}{2}$$
Let $a, b, c$ be positive real numbers. Show that $$6a+4b+5c\ge 5\sqrt{ab} + 3\sqrt{bc} + 7\sqrt{ca}$$
Let $a, b, c$ be the lengths of the sides of triangle $ABC$. Show that
$$\sqrt{a}(c+a-b) + \sqrt{b}(a+b-c)+\sqrt{c}(b+c-a)\le\sqrt{(a^2 + b^2 + c^2)(a+b+c)}$$
Let $a, b, c$ be positive numbers such that $a+b+c=1$. Prove $$\Big(1+\frac{1}{a}\Big)\Big(1+\frac{1}{b}\Big)\Big(1+\frac{1}{c}\Big)\ge 64$$
(Nesbitt's Inequality) Let $a, b, c$ be positive numbers. Show that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac{3}{2}$$
Let $a, b, c$ be positive numbers lying in the interval $(0, 1]$. Show that $$\frac{a}{1+b+ca}+\frac{b}{1+c+ab}+\frac{c}{1+a+bc}\le 1$$
Let $x, y, z$ be strictly positive real numbers. Prove that $$\Big(\frac{x}{y}+\frac{z}{\sqrt[3]{xyz}}\Big)^2+\Big(\frac{y}{z}+\frac{x}{\sqrt[3]{xyz}}\Big)^2+\Big(\frac{z}{x}+\frac{y}{\sqrt[3]{xyz}}\Big)^2 \ge 12$$
Let $x, y, z$ be three distinct positive real numbers such that $x+\sqrt{y+\sqrt{z}}=z+\sqrt{y+\sqrt{x}}$. Show that $40xz<1$
(Rearrangement Theorem) Let $a_1, a_2, \cdots, a_n$ and $b_1, b_2, \cdots, b_n$ be sequences of positive real numbers, and let $c_1, c_2, \cdots, c_n$ be a permutation of $b_1, b_2, \cdots, b_n$. The sum $S=a_1b_1+a_2b_2+\cdots+a_nb_n$ is maximal if the two sequences $a_1, a_2, \cdots, a_n$ and $b_1, b_2, \cdots, b_n$ are sorted in the same way and minimal if the two sequences are sorted oppositely, one increasing and the other decreasing.
Find the value of $x^3+x^2y+xy^2+y^3$ if $x=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and $y=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$.
Simplify $$\sqrt{10+4\sqrt{3-2\sqrt{2}}}$$
Let $\sqrt{1+\sqrt{21+12\sqrt{3}}}=\sqrt{a}+\sqrt{b}$. Find $a+b$.
Let $a\ge 0, n\ge 0,$ and $m > 0$. Show that $\sqrt{a+m}+\sqrt{a+m+n} > \sqrt{a} + \sqrt{a+2m+n}$.
Simplify $(\sqrt{10}+\sqrt{11}+\sqrt{12})(\sqrt{10}+\sqrt{11}-\sqrt{12})(\sqrt{10}-\sqrt{11}+\sqrt{12})(-\sqrt{10}+\sqrt{11}+\sqrt{12})$
Simplify $(\sqrt[3]{3}+\sqrt[3]{2})(\sqrt[3]{9}-\sqrt[3]{6}+\sqrt[3]{4})$
The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Which positive integer?
Simplify $\sqrt{1 + 1995\sqrt{4 + 1995 \cdot 1999}}$.