Let $n$ be an integer greater than $2$, prove $n^{n+1} > (n+1)^n$.
Let $\{a_n\}$ be a sequence defined as $a_1=1$ and $a_n=\frac{a_{n-1}}{1+a_{n-1}}$ when $n\ge 2$. Find the general formula of $a_n$.
Prove a positive proper fraction $\frac{m}{n}$ must be a sum of some reciprocals of distinct integers.
Let $p$ be an odd prime number. For positive integer $k$ satisfying $1\le k\le p-1$, the number of divisors of $k p+1$ between $k$ and $p$ exclusive is $a_k$. Find the value of $a_1+a_2+\ldots + a_{p-1}$.
Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that
$$f(yf(x)-x)=f(x)f(y)+2x$$
for all $x,\ y\in{\mathbb{R}}$.
For pairwise distinct nonnegative reals $a,b,c$, prove that
$$\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(b-a)^2}>2$$
For each integer $a_0 >$ 1, define the sequence $a_0, a_1, a_2, \cdots$ by:
$$
a_{n+1} =
\left\{
\begin{array}{ll}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer}\\
a_n + 3 & \text{otherwise}
\end{array}
\right.
$$
For all $n \ge 0$. Determine all values of $a_0$ for which there is a number $A$ such that $a_n = A$ for infinitely many values of $n$.
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all real numbers $x$ and $y$,
$$f (f(x)f(y)) + f(x + y) = f(xy)$$
A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit’s starting point, $A_0$, and the hunter’s starting point, $B_0$, are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}$. In the nth round of the game, three things occur in order.
(i) The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1$.
(ii) A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most 1.
(iii) The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly 1.
Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after 109 rounds she can ensure that the distance between her and the rabbit is at most 100?
Language:
Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$. Line $AJ$ meets again
at $K$. Prove that the line $KT$ is tangent to $\Gamma$.
An integer $N > 2$ is given. A collection of $N(N + 1)$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $N(N-1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions hold:
(1) no one stands between the two tallest players,
(2) no one stands between the third and fourth tallest players,
...
(N) no one stands between the two shortest players.
Show that this is always possible.
An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is 1. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0$, $a_1$, $\cdots$, an such that, for each $(x, y)$ in $S$, we have:
$$a_0x^n + a_1x^{n-1}y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1$$
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 $\cot 70^\circ + 4\cos 70^\circ$
$\displaystyle\frac{2\cos 2^n A+1}{2\cos A+1}=\prod_{r=1}^{n} (2\cos 2^{r-1}A-1)$
Compute $4\cos\frac{2\pi}{7}\cos\frac{\pi}{7}-2\cos\frac{2\pi}{7}$
Let $P(x)$ be a monic cubic polynomial. The lines $y = 0$ and $y = m$ intersect $P(x)$ at points $A$, $C$, $E$ and $B$, $D$, $F$ from left to right for a positive real number $m$. If $AB = \sqrt{7}$, $CD = \sqrt{15}$, and $EF = \sqrt{10}$, what is the value of $m$?
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.)
Show that $\frac{1}{k+1}\binom{n}{k}=\frac{1}{n+1}\binom{n+1}{k+1}$.
Show that $\binom{n}{0}+\binom{n}{1} + \binom{n}{2} + \cdots + \binom{n}{n} = 2^n$.
Explain that $$\sin x +\sin (x+120^\circ) + \sin (x-120^\circ) = \cos x +\cos (x+120^\circ) + \cos (x-120^\circ) = 0$$ using at least two approaches.
Show that $$x+n=\sqrt{n^2 + x\sqrt{n^2+(x+n)\sqrt{n^2+(x+2n)\sqrt{\cdots}}}}$$
For $m=4k+1$ where $k$ is a positive integer. Show that $$\frac{1}{\sqrt{m}}\Big(\Big(\frac{1+\sqrt{m}}{2}\Big)^n-\Big(\frac{1-\sqrt{m}}{2}\Big)^n\Big)$$ must be an integer for any positive integer $n$.
(Sophie Germain's Identity) Prove $a^4 + 4b^4 = (a^2+2b^2-2ab)(a^2+2b^2+2ab)$.