- Solve the following diophantine equation in natural numbers: $$y^2 = 1 + x + x^2 + x^3 + x^4$$
Given that $9^{4000}$ has $3817$ digits and has a leftmost digit $9$ (base $10$). How many of the number $9^0, 9^1, 9^2, \cdots, 9^{4000}$ have leftmost digit $9$.
Prove that $\frac{5^{125}-1}{5^{25}-1}$ is composite.
Find all integers $a$, $b$, $c$ with $1 < a < b < c$ such that the number $(a-1)(b-1)(c-1)$ is a divisor of $(abc-1)$.
Solve in positive integers $\big(1+\frac{1}{x}\big)\big(1+\frac{1}{y}\big)\big(1+\frac{1}{z}\big)=2$
Find all positive integers $n$ and $k_i$ $(1\le i \le n)$ such that $$k_1 + k_2 + \cdots + k_n = 5n-4$$ and $$\frac{1}{k_1} + \frac{1}{k_2} + \cdots + \frac{1}{k_n}=1$$
Solve in positive integers the equation $$3(xy+yz+zx)=4xyz$$
There are $2015$ people standing in a circle, counting $1$ and $2$ in turn continuously. Those who count $2$ will be out. For example, people who stand at initial positions of $2, 4, \dots, 2014, 1, 3, \dots$ etc will be out. The game goes on until there is only one person remaining in the circle. What is his initial position?
Let complex number $z_1=2-i\cos\theta$, $z_2=2-i\sin\theta$. Find the maximum value of $|z_1z_2|$.
Show that the difference of two squares of odd numbers must be a multiple of $8$.
If the circle \(x^2 + y^2 = k^2\) covers at least one maximum and one minimal of the curve \(f(x)=\sqrt{3}\sin\frac{\pi x}{k}\), find the range of \(k\).
Let $x, y \in [-\frac{\pi}{4}, \frac{\pi}{4}], a \in \mathbb{Z}^+$, and
$$
\left\{
\begin{array}{rl}
x^3 + \sin x - 2a &= 0 \\
4y^3 +\frac{1}{2}\sin 2y +a &=0
\end{array}
\right.
$$
Compute the value of $\cos(x+2y)$
Prove the following identities
\begin{align}
\sin (3\alpha) &= 4\cdot \sin(60-\alpha)\cdot \sin\alpha\cdot \sin(60+\alpha)\\
\cos (3\alpha) &= 4 \cdot\cos(60-\alpha)\cdot \cos\alpha\cdot \cos(60+\alpha)\\
\tan (3\alpha) &= \tan(60-\alpha) \cdot\tan\alpha \cdot\tan(60+\alpha)
\end{align}
Show that
$$\sin^2\alpha - \sin^2\beta = \sin(\alpha + \beta)\sin(\alpha-\beta)$$
$$\cos^2\alpha - \cos^2\beta = - \sin(\alpha + \beta)\sin(\alpha-\beta)$$
Compute $$\sin^410^{\circ} +\sin^450^{\circ}+\sin^470^\circ$$
Simplify $$\sin^2\alpha + \sin^2\Big(\alpha + \frac{\pi}{3}\Big)+\sin^2\Big(\alpha - \frac{\pi}{3}\Big)$$
Let $A (x_1, y_1)$, $B (x_2, y_2)$, and $C (x_3, y_3)$ be three points on the unit circle, and $$x_1 + x_2 + x_3 = y_1+y_2+y_3=0$$ Prove $$x_1^2 +x_2^2+x_3^2=y_1^2+y_2^2+y_3^2=\frac{3}{2}$$
Compute $(1+\tan 1^\circ)(1+\tan 2^\circ)\cdots(1+\tan 44^\circ)(1+\tan 45^\circ)$
Compute $$\cos\frac{2\pi}{7} \cdot \cos \frac{4\pi}{7}\cdot \cos \frac{8\pi}{7} $$
Compute $$\cos\frac{\pi}{2n+1}\cdot\cos\frac{2\pi}{2n+1}\cdots\cos\frac{n\pi}{2n+1}$$
Compute $$\Big(1+\cos\frac{\pi}{5}\Big)\Big(1+\cos\frac{3\pi}{5}\Big)$$
Compute $$\sin^2 10^\circ + \cos^2 40^\circ + \sin 10^\circ \cos 40^\circ$$
Compute $$\sin^2 20^\circ -\sin 5^\circ (\sin 5^\circ +\frac{\sqrt{6}-\sqrt{2}}{2}\cos 20^\circ)$$
Let $\alpha, \beta \in (0, \frac{\pi}{2})$. Show that $\alpha + \beta = \frac{\pi}{2}$ if and only if $$\frac{\sin^4 \alpha}{\cos^2 \beta} + \frac{\cos^4\alpha}{\sin^2\beta} = 1$$
If $\sin\alpha + \sin\beta = \frac{3}{5}$ and $\cos\alpha+\cos\beta=\frac{4}{5}$, compute $\cos(\alpha -\beta)$ and $\sin(\alpha+\beta)$.