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 $$\frac{1}{2}\cdot\frac{3}{4}\cdots\frac{2n-1}{2n} < \frac{1}{\sqrt{3n}}$$
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 $$\frac{2x}{2x^2-5x+3}+\frac{13x}{2x^2+x+3}=6$$
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 $k$.
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 $(\sqrt[3]{3}+\sqrt[3]{2})(\sqrt[3]{9}-\sqrt[3]{6}+\sqrt[3]{4})$
Simplify $\sqrt{1 + 1995\sqrt{4 + 1995 \cdot 1999}}$.
Evaluate $\sqrt{5+\sqrt{5^2+\sqrt{5^4+\sqrt{5^8+...}}}}$
Simplify $\sqrt{1-\frac{\sqrt{3}}{2}}$
Simplify $$\frac{1}{2+\frac{1}{2+\cdots}}$$
Simplify $$(\sqrt{2})^{(\sqrt{2})^{(\sqrt{2})^{\cdots}}}$$
Simplify $$2^{\sqrt{2^{\sqrt{2^{\sqrt{2}^{\cdots}}}}}}$$