Let $f(x)=x^3 -x^2 -13x+24$. Find three pairs of $(x,y)$ such that if $y=f(x)$, then $x=f(y)$.
Each point of a circle is colored either red or blue.
(a) Prove that there always exists an isosceles triangle inscribed in this circle such that all its vertices are colored the same.
(b) Does there always exist an equilateral triangle inscribed in this circle such that all its vertices are colored the same?
The function $f$ satisfies $f(0)=0$, $f(1)=1$, and $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$ for all $x,y\in\mathbb{R}$. Show that $f(x)=x$ for all rational numbers $x$.
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 $a+b+c+d.$
Find the number of possible arrangements in Fisher Random Chess. The diagram below is one possible arrangement.
In a legal arrangement, the White's position must satisfy the following criteria:
- Eight pawns must be in the $2^{nd}$ row. (The same as regular chess)
- Two bishops must be in opposite colored squares (e.g. $b1$ and $e1$ in the above diagram)
- King must locate between two rooks (e.g. in the diagram above, King is at $c1$ and two rooks are at $a1$ and $g1$)
The Black's position will be mirroring to the White's.
Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.
Numbers $1,2,\cdots, 1974$ are written on a board. You are allowed to replace any two of these numbers by one number which is either the sum or the difference of these numbers. Show that after $1973$ times performing this operation, the only number left on the board cannot be $0$.
On an $8\times 8$ chess board, there are $32$ white pieces and $32$ black pieces, one piece in each square. If a player can change all the white pieces to black and all the black pieces to white in any row or column in a single move, then is it possible that after finitely many movies, there will be exactly one black piece left on the board?
There are three piles which contain $8$, $9$, and $19$ stones, respectively. You are allowed to choose two piles and transfer one stone from each of them to the third pile. Is it possible to make all piles all contain exactly $12$ stones after several such operations?
Let the lengths of five line segments be $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$, respectively, where $a_1 \ge a_2\ge a_3\ge a_4\ge a_5$. If any three of these five line segments can form a triangle, then prove at least one of such triangle is acute.
Given $n > 2$ points on a plane. Prove if any straight line passing two of these points, it must pass another one among these points, then all these $n$ points must be collinear.
If all sides of a convex pentagon $ABCDE$ are equal in length and $\angle{A}\ge\angle{B}\ge\angle{C}\ge\angle{D}\ge\angle{E}$, show that $ABCDE$ is a regular pentagon.
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
$$\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(b-a)^2}>2$$
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 $4\cos\frac{2\pi}{7}\cos\frac{\pi}{7}-2\cos\frac{2\pi}{7}$
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 $\binom{n}{0}+\binom{n}{1} + \binom{n}{2} + \cdots + \binom{n}{n} = 2^n$.