If $17!=355687ab8096000$ where $a$ and $b$ are two missing single digits. Find $a$ and $b$.
We define the Fibonaccie numbers by $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n}$. Find the greatest common divisor $(F_{100}, F_{99})$, and $(F_{100}, F_{96})$.
The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, find the probability that Sharon can discard one of her cards and $\textit{still}$ have at least one card of each color and at least one card with each number.
Let $f(x)=x^3 -x^2 -13x+24$. Find three pairs of $(x,y)$ such that if $y=f(x)$, then $x=f(y)$.
Let $f$ be a function such that $$ \sqrt {x - \sqrt { x + f(x) } } = f(x) , $$for $x > 1$. In that domain, $f(x)$ has the form $\frac{a+\sqrt{cx+d}}{b},$ where $a,b,c,d$ are integers and $a,b$ are relatively prime. Find $a+b+c+d.$
Prove the triple angle formulas: $$\sin 3\theta = 3\sin\theta -4\sin^3\theta$$ and $$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$$
Show that $\sin\alpha + \sin\beta + \sin\gamma - \sin(\alpha + \beta+\gamma) = 4\sin\frac{\alpha+\beta}{2}\sin\frac{\beta+\gamma}{2}\sin\frac{\gamma+\alpha}{2}$
Compute $\cot 70^\circ + 4\cos 70^\circ$
Compute $4\cos\frac{2\pi}{7}\cos\frac{\pi}{7}-2\cos\frac{2\pi}{7}$
Find an acute angle $\alpha$ such that $\sqrt{15-12\cos\alpha} + \sqrt{7-4\sqrt{3}\sin\alpha}=4$. (Find at least two different solutions.)
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 $\triangle{ABC}$, 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 $\cos A +\cos B + \cos C \le\frac{3}{2}$.
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.
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
$$\sqrt{x^2+y^2-xy}+\sqrt{y^2 + z^2 - yz}\ge\sqrt{x^2+z^2+xz}$$
Let $m$ be a positive integer. Show that $$\frac{1}{\sqrt{m+1}}< \sin\frac{1}{\sqrt{m}}$$
Let $x\in(0, \pi/2)$ be expressed in radian. Explain why the relation $\sin x < x < \tan x$ hold?
Solve this equation $$2\sqrt{2}x^2 + x -\sqrt{1-x ^2}-\sqrt{2}=0$$