(Bezout's theorem) Show that two positive integers $a$ and $b$ are co-prime if there exist integer $x$ and $y$ satisfying $ax+by=1$.
(Cauchy–Schwarz inequality) Show that if $u$ and $v$ a two vectors, then $|\langle u, v\rangle|^2\le \langle u, u\rangle\cdot\langle v, v\rangle$. This inequality can also be written as $$|u_1v_1+u_2v_2+\cdots +u_nv_n|^2 \le (|u_1|^2+|u_2|^2+\cdots|u_n|^2)(|v_1|^2+|v_2|^2+\cdots|v_n|^2)$$
(Apollonius’ Theorem) Let $AD$ be one median of $\triangle{ABC}$ where point $D$ lies on side $BC$. Show that the following relation holds:
$$AB^2 +AC^2 = 2\times(AD^2 +BD^2)$$
Compute the value of $\sin 1^\circ \sin 2^\circ \cdots \sin 89^\circ$.
If $0 < \alpha < \beta < \frac{\pi}{2}$, show $$\frac{\cot\beta}{\cot\alpha}<\frac{\cos\beta}{\cos\alpha}<\frac{\beta}{\alpha}$$
Evaluate $\cos\frac{\pi}{2n+1}+\cos\frac{3\pi}{2n+1}+\cdots+\cos\frac{(2n-1)\pi}{2n+1}$.
Let $p$ be an odd prime divisor of number $(a^2+1)$ where $a$ is an integer. Show that $p\equiv 1\pmod{4}$.
Show there exist infinite many primes in the form of $(4k+1)$ where $k$ is a positive integer.
In $\triangle{ABC}$ show that $$\tan\frac{A}{2}\tan\frac{B}{2}+\tan\frac{B}{2}\tan\frac{C}{2}+\tan\frac{C}{2}\tan\frac{A}{2}=1$$
In $\triangle{ABC}$, show that
$$\tan\frac{A}{2}\tan\frac{B}{2}\tan\frac{C}{2}\le\frac{\sqrt{3}}{9}$$
In $\triangle{ABC}$, $\angle{C}=\angle{A}+60^\circ$. If $BC=1$, $AC=r$ and $AB=r^2$, where $r > 1$, prove $r \le\sqrt{2}$.
In $\triangle{ABC}$, show that
\begin{equation}
\sin A + \sin B + \sin C = 4\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}
\end{equation}
Try to use at least two different approaches.
(Euler's theorem) In $\triangle{ABC}$, let $R$ and $r$ be its circumradius and inradius, respectively, show that $$|OI|^2 = R^2 - 2Rr$$ where $O$ is the circumcenter and $I$ is the incenter.
This relation can also be rewritten as $$\frac{1}{R-d}+\frac{1}{R+d}=\frac{1}{r}$$
In $\triangle{ABC}$ show that $\cos A +\cos B + \cos C \le\frac{3}{2}$.
Given any $\triangle{ABC}$, show that $$\cos A + \cos B + \cos C = 1+4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$$
In $\triangle{ABC}$, show that $$\sin\frac{A}{2}=\sqrt{\frac{(p-b)(p-c)}{bc}}$$ where $p=\frac{a+b+c}{2}$ is the semi-perimeter.
Show that $$S_{\triangle{ABC}}=\frac{abc}{4R}$$
Find the value of $$\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$$
Show that
$$\sec^2\alpha = 1 + \tan^2\alpha$$
$$\csc^2\alpha = 1 + \cot^2\alpha$$
Prove the identity: $\tan^2 x - \sin^2 x = \tan^2 x \sin^2 x$.
If $\cos x - \sin x = \sqrt{2}\sin x$, prove $\cos x +\sin x = \sqrt{2}\cos x$.
Let $x$, $y$, $z$ be three positive real numbers satisfying $xyz+x+z=y$. Find the maximum value of $$P=\frac{2}{x^2 + 1}-\frac{2}{y^2+1}+\frac{3}{z^2+1}$$
Let real numbers $x$ and $y$ satisfy the relation $4x^2-5xy+4y^2=5$. Find the maximum and minimal value of $x^2+y^2$.
Given non-negative real numbers $x$, $y$ and $z$, prove
$$\sqrt{x^2+y^2-xy}+\sqrt{y^2 + z^2 - yz}\ge\sqrt{x^2+z^2+xz}$$
Given any five real numbers, show that at least two of them $x$ and $y$ satisfy the condition $|xy+1|>|x-y|$.