Show that $|\sin(nx)|\le n|\sin(x)|$ for any positive integer $n$.
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.
$n$ straight lines are drawn in the plane in such a way that not two of them are parallel and not three of them meet at one point. Show that the number of regions in which these lines divide the plane is $\frac{n(n+1)}{2}+1$.
The Fibonacci sequence $(F_n)$, $n\ge 0$ is defined by the recurrence relation $F_{n+2}=F_{n+1}+F_{n}$ with $F_0=0$ and $F_1=1$. Prove for any $m, n \in\mathbb{N}$, we have $$F_{m+n+1}=F_{m+1}{n+1}+F_mF_n$$
Deduce from here that $F_{2n+1}=F_{n+1}^2 +F_n^2$ for any $n\in\mathbb{N}$.
Show that $$\frac{1}{2}\cdot\frac{3}{4}\cdots\frac{2n-1}{2n} < \frac{1}{\sqrt{3n}}$$
Prove that for every positive integer $n$, there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
The roots of $x^2 + ax + b+1$ are positive integers. Show that $a^2+b^2$ is not a prime number.
Let $\alpha$ and $\beta$ be the roots of $x^2+px+1$, and let $\gamma$ and $\delta$ be the roots of $x^2+qx+1$. Show $$(\alpha-\gamma)(\beta-\gamma)(\alpha+\delta)(\beta+\delta)=q^2-p^2$$
Let $a, b, c$ be distinct real numbers. Show that there is a real number $x$ such that $$x^2 +2(a+b+c)x+3(ab+bc+ca)$$ is negative.
Solve the equation $x^4 -97x^3+2012x^2-97x+1=0$.
Show that if $a, b, c$ are the lengths of the sides of a triangle, then the equation $$b^2x^2 +(b^2+c^2-a^2)x+c^2=0$$ does not have real roots.
Solve the equation in real numbers $$\frac{2x}{2x^2-5x+3}+\frac{13x}{2x^2+x+3}=6$$
Let $(x^{2014} + x^{2016} +2)^{2015} = a_0 + a_1 x + \cdots + a_nx^n$. Find $$a_0 -\frac{a_1}{2} -\frac{a_2}{2} + a3 - \frac{a_4}{2}-\frac{a_5}{2} + a_6 - \cdots$$
Find $a$ and $b$ so that $(x-1)^2$ divides $ax^4 + bx^3+1$.
Find all pairs of real numbers $a, b$, such that the polynomial $$p(x)=(a+b)x^5 + abx^2 +1$$ is divisible by $x^2 - 3x+2$.
Find the real root of the polynomial $p(x)=8x^3 -3x^2 -3x -1$.
Let $n$ be a positive integer, and for $1\le k\le n$, let $S_k$ be the sum of the products of $1, \frac{1}{2}, \cdots, \frac{1}{n}$, taken $k$ a time ($k^{th}$ elementary symmetric polynomial). Find $S_1 + S_2 + \cdots +S_n$.
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 $$\Big(a+\frac{1}{a}\Big)^2 +\Big(b+\frac{1}{b}\Big)^2\ge \frac{25}{2}$$