Suppose no term in an arithmetic sequence $\{a_n\}$ equals $0$. Let $S_n$ be the sum of its first $n$ terms. If $S_{2n-1} = a_n^2$, find the expression for its $n^{th}$ term $a_n$.
Prove that $7\mid 8^n-1$ for $n\ge 1$.
Show that $5\mid 4^{2n}-1$ for $n\ge 1$.
Prove that $15\mid 4^{2n}-1$ for $n\ge 1$.
Let $x$ be a real number, show that if the value of $\left(x+\frac{1}{x}\right)$ is an integer, then the value of $\left(x^n+\frac{1}{x^n}\right)$ is an integer too.
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 $$\frac{1}{2}(a+b)+\frac{1}{4}\ge \sqrt{\frac{a+b}{2}}$$
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 $$6a+4b+5c\ge 5\sqrt{ab} + 3\sqrt{bc} + 7\sqrt{ca}$$
(Nesbitt's Inequality) Let $a, b, c$ be positive numbers. Show that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac{3}{2}$$
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 $$\sqrt{10+4\sqrt{3-2\sqrt{2}}}$$
Simplify $\sqrt{1 + 1995\sqrt{4 + 1995 \cdot 1999}}$.
Simplify $\sqrt{1-\frac{\sqrt{3}}{2}}$
Simplify $$\frac{1}{2+\frac{1}{2+\cdots}}$$
Compute $$\frac{1}{\frac{1}{\frac{1}{\cdots}+1+\frac{1}{\cdots}}+1+\frac{1}{\frac{1}{\cdots}+1+\frac{1}{\cdots}}}$$
Simplify $\sqrt{\sqrt[3]{9}+6\sqrt[3]{3}+9}$
Solve $x^2 +6x - 4\sqrt{5}=0$.
Determine all pairs $(a, b)$ of real numbers such that $10, a, b, ab$ is an arithmetic progression.
Let $\alpha\in\Big(\frac{3\pi}{2}, 2\pi\Big)$. Simplify $$\sqrt{\frac{1}{2}-\frac{1}{2}\sqrt{\frac{1}{2}+\frac{1}{2}\cdot\cos 2\alpha}}$$
Simplify $$\binom{n}{0} - \frac{1}{2}\binom{n}{1} +\binom{n}{2} - \frac{1}{2}\binom{n}{3} + \cdots $$
Show that $1\cdot 1! + 2\cdot 2! + \cdots + n\cdot n! = (n+1)!-1$
Given two segments $AB$ and $MN$, show that $$MN\perp AB \Leftrightarrow AM^2 - BM^2 = AN^2 - BN^2$$
Three of the roots of $x^4 + ax^2 + bx + c = 0$ are $2$, $−3$, and $5$. Find the value of $a + b + c$.