Practice (TheColoringMethod)

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(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 $|xy+1|>|x-y|$.