Compute $\sin 25^\circ \sin 35^\circ \sin 85^\circ$.
Prove $\tan 20^\circ \tan 40^\circ \tan 60^\circ \tan 80^\circ=3$.
Show that $\cot 70^\circ + 4\cos 70^\circ = \sqrt{3}$.
Prove that $\cos 1^\circ$ is irrational.
In $\triangle{ABC}$, find the measurement of $C$ if
$$3\sin A + 4\cos B = 6\quad\text{and}\quad 4\sin B + 3\cos A = 1$$
Compute the value of $(\sqrt{3}\tan 18^\circ + \tan 18^\circ\tan 12^\circ +\sqrt{3}\tan 12^\circ)$.
Compute the value of $\cos 6^\circ\cos 42^\circ \cos 66^\circ\cos 78^\circ$.
Show that for any positive integer: $$\tan x \tan 2x +\tan 2x \tan 3x +\cdots + \tan(n-1)x\tan nx=\frac{\tan nx}{\tan x}-n$$
Show that $$\tan x + 2\tan 2x + 2^2\tan 2^2x +\cdots + 2^n\tan 2^nx = \cot x - 2^{n+1}\cot 2^{n+1}x$$
Show that $$\frac{1}{\sin 1^\circ\sin 2^\circ}+\frac{1}{\sin 2^\circ\sin 3^\circ}+\cdots+\frac{1}{\sin 89^\circ\sin 90^\circ}=\cos 1^\circ\csc^2 1^\circ$$
(Pascal Identity) Show $\binom{n}{k} + \binom{n}{k+1}=\binom{n+1}{k+1}$.
$\textbf{Prisoners' Problem}$
One hundred prisoners will be lined up. Each one will be assigned either a red hat or a blue hat. No one can see the color of his or her own hat. However, each person is able to see the color of the hat worn by every person in front of him or her. That is, for example, the last person in line can see the colors of the hats on $99$ people in front of him or her; and the first person, who is at the front of the line, cannot see the color of any hat.
Beginning with the last person in line, and then moving to the $99^{th}$ person, the $98^{th}$, etc., each will be asked to name the color of his or her own hat. He or she can only answer "red" or "blue". If the color is correctly named, the person lives; if not, the person is shot dead on the spot. Everyone in line is able to hear all the responses but not the gunshots.
Before being lined up, the $100$ prisoners are allowed to discuss a strategy aiming to save as many of them as possible. How many people can be saved if they can agree on a good strategy?
As a more challenging question, what if the hats can have $100$ known different colors instead of $2$?
$\textbf{Flip the Grid}$
Given two grids shown below, is it possible to transform ($a$) to ($b$) after a series of operations? In each operation, one can change all the signs in either one entire row or one entire column.
John uses the equation method to evaluate the following expression:$$S=1-1+1-1+1-\cdots$$ and get $$S=1-S \implies \boxed{S=\frac{1}{2}}$$
However, $S$ clearly cannot be a fraction. Can you point out what is wrong here?
Is it possible to use twenty seven $1\times 2\times 4$ blocks to construct a $6\times 6\times 6$ cube?
Find all pairs of positive integers $(a, b)$ satisfying $a! + b! = a^b + b^a$.
Find the remainder when $x^{2017}$ is divided by $(x+1)^2$.
Let real numbers $a$ and $b$ satisfy $0 < a < a +\frac{1}{2} \le b$ and $a^{40}+b^{40}=1$. Show that all the twelve digits after the decimal point are $9$ if $b$ is expressed in decimal.
Let $f(x)=2016x - 2015$. Solve this equation $$\underbrace{f(f(f(\cdots f(x))))}_{2017\text{ iterations}}=f(x)$$
In $\triangle{ABC}$, let $\angle{A}=120^\circ$. If $A'$, $B'$ and $C'$ are feet of the three interior angle bisectors as shown, prove $A'B'\perp A'C'$.
Take a list of positive integers $1$, $2$, $3$, $\cdots$, $2017$. At each step, pick up two of the numbers on the list, say $a$ and $b$, cross them out and replace them by the single number $(ab+a+b)$. Keep doing this until only a single number is left. What is (are) the possible value(s) of this last number?
As shown, both $ABCD$ and $OPRQ$ are squares. Additionally, $O$ is the center of $ABCD$, $OP=1$, $BP=\sqrt{2}$, and $CQ=\sqrt{5}$. Find the length of $DR$.
Does the expression $x+\sqrt{2x^2-2x+1}$ has either maximum or minimal value?
In trapezoid $ABCD$, $AD\parallel BC$ and $AD:BC=1:2$. Point $F$ lies on $AB$ and point $E$ is on $CF$. If $S_{\triangle{AOF}}:S_{\triangle{DOE}}=1:3$ and $S_{\triangle{BEF}}=24$, find the area of $\triangle{AOF}$.
Let integers $u$ and $v$ be two integral roots to the quadratic equation $x^2 + bx+c=0$ where $b+c=298$. If $u < v$, find the smallest possible value of $v-u$.