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 that for any $m$, $n \in \mathbb{N}$, we have
$$F_{m+n+1}=F_{m+1}F_{n+1}+F_{m}F_{n}.$$
Deduce from here that $F_{2n+1}=F^2_{n+1}+F^2_{n}$ for any $n \in \mathbb{N}$
Show
$$\frac{1}{2} \cdot \frac{3}{4} \cdots \frac{2n-1}{2n} < \frac{1}{\sqrt{3n}}$$
The number $n$ is called nice if $n$ can be represeted as a sum of positive integers $n=a_1+\cdots +a_k$ such that the sum of their reciprocals $\frac{1}{a_1} + \cdots + \frac{1}{a_k}=1$. It is known that the numbers from 33 to 73 are nice. Prove that n is nice for all $n\ge 33$.
Prove that for every positive integer $n$ there exists a $n$-digit number divisible by $5^n$ all of whose digits are odd.
Let $f$ be a function from $\mathbb{N}$ to $\mathbb{N}$ such that
(i) $f(1)=0$
(ii) $f(2n)=2f(n)+1)$
(iii) $f(2x+1)=2f(n)$
Find the least value of $n$ such that $f(n)=2016$.
Let $n$ be a positive integer. Each point $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers with $x+y < n$, is colored red or blue, subject to the following condition:if a point $(x,y)$ is red, then so are all the points $(x',y')$ with $x'\le x$ and $y'\le y$ Let $A$ be the number of ways choose $n$ blue points with distinct $x$-coordinates. and let B be the number of ways to choose $n$ blue points with distinct $y$-coordinates. Prove that $A=B$
Let $z=\cos{\theta} + i\sin{\theta} $. Show $z^{-1} = \cos{\theta} - i\sin{\theta}$.
Find the value of the following expression: $$\binom{2020}{0}-\binom{2020}{2}+\binom{2020}{4}-\cdots+\binom{2020}{2020}$$
Given integers $a$, $b$, $n$. Show that there exist integers $x$, $y$, such that $$(a^2+b^2)^n = x^2 + y^2$$.
Solve the equation $z^4+1=0$.
The points $(0,0)$, $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.
Find $c$ if $a$, $b$, and $c$ and positive integers which satisfy $c = (a+bi)^3 - 107i$
Find the number of ordered pairs $(a, b)$ of real numbers such that
$$(a+bi)^{2016}=a-bi$$.
Let $z_1$, $z_2$, $z_3$ be complex numbers with nonzero imaginary parts such that $|z_1| = |z_2| = |z_3|$. Show that if $z_1+z_2z_3$, $z_2+z_1z_3$, $z_3+z_1z_2$ are real, then $z_1z_2z_3 = 1$.
Let $(x^{2017}+x^{2019}+2)^{2018} = a_0+a_1x+\cdots+a_nx^n$. Find
$$a_0-\frac{a_1}{2}-\frac{a_2}{2}+a_3-\frac{a_4}{2}-\frac{a_5}{2}+a_6-\cdots$$
The sum and product of two numbers are equal to $y$. For which values of $y$ are these two numbers real?
Let $m$ and $n$ be the roots of $P(x)=ax^2+bx+c$. Find the coefficients of the quadratic polynomial whose roots are $m^2-n$ and $n^2-m$.
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 $\sigma$ be the roots of $x^2+qx+1$.
Show
$$(\alpha - \gamma)(\beta-\gamma)(\alpha+\sigma)(\beta+\sigma) = q^2 - p^2$$
Let $a$, $b$, $c$ be distinct real numbers. Show that there is a real number $x$ such that
$$x^2+2(a+b+c)x+3(ab+bc+ac)$$
is negative.
Consider the quadratic equation $ax^2-bx+c=0$ where $a$, $b$, $c$ are real numbers and $a \ne 0$. Find the values of $a$, $b$, $c$ such that $a$ and $b$ are the roots of the equation and $c$ is it's discriminant.
Let $b \ge 0$ be a real number. The product of the four real roots of the equations $x^2+2bx+c=0$ and $x^2+2cx+b=0$ is equal to $1$. Find the values of $b$ and $c$.
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 any real roots.