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Prove that if $p$ and $(p^2 + 8)$ are prime, then $(p^3 + 8p + 2)$ is prime.

Find all positive integers $n$ such that for all odd integers $a$. If $a^2\le n$, then $a|n$.

Given that $9^{4000}$ has $3817$ digits and has a leftmost digit $9$ (base $10$). How many of the number $9^0, 9^1, 9^2, \cdots, 9^{4000}$ have leftmost digit $9$.

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is composite.

Find all positive integers $n$ and $k_i$ $(1\le i \le n)$ such that $$k_1 + k_2 + \cdots + k_n = 5n-4$$ and $$\frac{1}{k_1} + \frac{1}{k_2} + \cdots + \frac{1}{k_n}=1$$

There are $2015$ people standing in a circle, counting $1$ and $2$ in turn continuously. Those who count $2$ will be out. For example, people who stand at initial positions of $2, 4, \dots, 2014, 1, 3, \dots$ etc will be out. The game goes on until there is only one person remaining in the circle. What is his initial position?

Let $x, y \in [-\frac{\pi}{4}, \frac{\pi}{4}], a \in \mathbb{Z}^+$, and $$\left\{ \begin{array}{rl} x^3 + \sin x - 2a &= 0 \\ 4y^3 +\frac{1}{2}\sin 2y +a &=0 \end{array} \right.$$ Compute the value of $\cos(x+2y)$

Compute $$\Big(1+\cos\frac{\pi}{5}\Big)\Big(1+\cos\frac{3\pi}{5}\Big)$$

Solve in $\textit{rational}$ numbers the equation $x^2 - dy^2 = 1$ where $d$ is an integer.

Let $x$ be a positive real number, and $\lfloor{x}\rfloor$ be the largest integer that not exceeding $x$. Prove that there exist infinity number of positive integers, $n$, such that $\lfloor{\sqrt{2}}\ n\rfloor$ is a perfect square.

Show that there are infinitely many integers $n$ such that $2n + 1$ and $3n + 1$ are perfect squares, and that such $n$ must be multiples of $40$.

Prove that if $m=2+2\sqrt{28n^2 +1}$ is an integer for some $n\in\mathbb{N}$, then $m$ is a perfect square.

Prove that if the difference of two consecutive cubes is $n^2$, $n\in\mathbb{N}$, then $(2n-1)$ is a square.

If $n$ is an integer such that the values of $(3n+1)$ and $(4n+1)$ are both squares, prove that $n$ is a multiple of $56$.

Show that if $\frac{x^2+1}{y^2}+4$ is a perfect square, then this square equals $9$.

Solve in nonnegative integers the equation $$2^x -1 = xy$$

Solve in integers the equation $x^2+y^2+z^2-2xyz=0$

Show that $x^4 + y^4 = z^2$ is not solvable in positive integers.

Find all primes $p$ for which there exist positive integers $x$, $y$, and $n$ such that $$p^n = x^3+y^3$$

Let $n$ be a positive integer and $[ \ n ] = a.$ Find the largest integer $n$ such that the following two conditions are satisfied: $(1)$ $n$ is not a perfect square; $(2)$ $a^{3}$ divides $n^{2}$.

Joe is playing with a set of $6$ masses: $1$g, $2$g, $4$g, $8$g, $16$g, and $32$g. He found that some weights can be measured in more than one way. For example, $7$g can be measure by putting $1$g, $2$g, and $4$g on one side of a balance. It can also be achieved by putting $1$g and $8$g on different sides of a balance. He therefore wonder which weight can be measured using these masses in the most number of different ways? Can you help him to find it out? Describe how will you approach this problem. The final answer is optional.

Solve this equation in positive integers $$x^3 - y^3 = xy + 61$$

Let $a_1, a_2, \cdots, a_{100}, b_1, b_2, \cdots, b_{100}$ be distinct real numbers. They are used to fill a $100 \times 100$ grids by putting the value of $(a_i + b_j)$ in the cell $(i, j)$ where $1 \le i, j \le 100$. Let $A_i$ be the product of all the numbers in column $i$, and $B_i$ be the product of all the numbers in row $i$. Show that if every $A_i$ equals to 1, then every $B_j$ equals to -1.

Let $O$ be the incenter of $\triangle{ABC}$. Connect $AO$, $BO$, and $CO$ and extends so that they intersect with $\triangle{ABC}$'s circumcircle at $D$, $E$, and $F$, respectively. Let $DE$ intersect $AC$ at $G$, and $DF$ intersects $AB$ at $H$. Show that $G$, $H$ and $O$ are collinear.

For any non-negative real numbers $x$ and $y$, the function $f(x+y^2)=f(x) + 2[f(y)]^2$ always holds, $f(x)\ge 0$, $f(1)\ne 0$. Find the value of $f(2+\sqrt{3})$.