Let $a_1, a_2, \cdots, a_n$ be a sequence of positive numbers, and $b_1, b_2, \cdots, b_n$ be any permutation of the first sequence. Prove that $$\frac{a_1}{b_1}+\frac{a_2}{b_2}+\cdots \frac{a_n}{b_n}\ge n$$
The Fibonacci numbers are defined by $F_1=1, F_2=1$, and $F_n=F_{n-1} + F_{n-2}$ for $n=3, 4, \cdots$. Find and prove a formula for the sum of the first $n$ Fibonacci numbers, i.e. $F_1 + F_2 + \cdots +F_n$.
Find the number of paris $(a, b)$ of nonnegative integers that satisfy $6a+7b=1000$
Solve the congruence $5x \equiv 21 \pmod{37}$.
Show that $n^{13} \equiv n\pmod{2730}$ for all integers $n$.
Prove that the sum of $n$ consecutive perfect squares cannot be a perfect square for $n=3, 4, 5,$ and $6$.
Let $n$ be a positive integer that is one less than a multiple of 24. Prove that if $a$ and $b$ are positive integers such that $ab=n$, then $a+b$ is a multiple of 24.
Solve the following equations for all real numbers $r, s, t$:
$$
\begin{array}{rl}
rst &=30\\
rs+st+tr &=-11\\
r+s+t &=-4
\end{array}
$$
Find all real solutions to the equations:
$$
\begin{array}{rl}
(x-y)(x^3+y^3)&=7\\
(x+y)(x^-y^3)&=3
\end{array}
$$
Let $x, y$ and $z$ be real numbers such that
$$
\begin{array}{rl}
x^2 +2(y-1)(z-1) &=12 \\
y^2 +2(z-1)(x-1) &=6 \\
z^2 +2(x-1)(y-1) &=9
\end{array}
$$
Find all the possible values of $x+y+z$.
For all real numbers $a, b, c$, prove that $$\frac{a^2+b^2}{a+b}+\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}\ge a+b+c$$
For all positive real numbers, prove the following inequalities:
a) $x^5 + y^5 + z^5 \ge x^4y + y^4z + z^4x$
b) $x^5 + y^5 + z^5 \ge x^2y^2z + y^2z^2x + z^2x^2y$
c) $x^3y^2 +y^3z^2 + z^3x^2 \ge x^2y^2z+y^2z^2x+z^2x^2y$
Let $x, y, z$ be positive real numbers, prove that $$\Large(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\Large)(\sqrt{x}+\sqrt{y}+\sqrt{z})^4\ge 9\sqrt{3}$$
Let $a, b, c$ be positive real numbers such that $a^2 + b^2 +c^2 +(a+b+c)^2 \le 4$. Prove that $$\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\ge 3$$
Let complex number $z$ satisfy $|z|=1$. If $f(z)=|z+1+i|$ reaches its maximum and minimal values when $z=z_1$ and $z=z_2$, respectively. Compute $z_1-z_2$.
Let complex number $z$ satisfy $|z|=1$, $w = z^4-z^3-3z^2i-z+1$. Find the minimal value of $|w|$.
Let unit vectors $a$, $b$, and $c$ satisfy $a+b+c=0$, prove the angles between these vectors are all $120^\circ$.
Let complex numbers $a$, $b$, and $c$ satisfy $a|bc| + b|ca| + c|ab| = 0$. Show that $$|(a-b)(b-c)(c-a)|\ge 3\sqrt{3}|abc|$$
Let $x, y \in \big(0, \frac{\pi}{2}\big)$. Show that if the equation $(\cos x + i \sin y)^n = \cos nx + i \sin ny$ holds for two consecutive positive integers, then it will hold for all positive integers.
Find all polynomials $f(x)$ such that $f(x^2) = f(x)f(x+1)$.
Let complex number $z$ and $w$ satisfy $w=z+\frac{1}{z}$ and $-1 < w < 2$. Find the range of $Re(z)$
On the complex plane, the vertices of a square are $Z_1, Z_2, Z_3, O$ anti-clockwise, where $O$ is the origin. If $Z_2 = 1+\sqrt{3}i$, find $Z_1\cdot Z_3$.
Let $z$ be a complex number and $k$ be a known real number. Find the maximum value of $|z^2 +kz+1|$ if $|z|=1$.
Let $\theta, a \in \mathbb{R}$ and complex number $z=(a+\cos\theta)+(2a-\sin\theta)i$. If $|z|\le 2$, find the range of $a$.
If $\sin t+\cos t=1$, and $s=\cos t +i\sin t$, compute $f(s)=1+s+s^2+\cdots +s^n$