Find all pairs of positive integers $(a; b)$ such that $\frac{a}{b} + \frac{21b}{25a}$ is a positive integer.
Prove that $\frac{5^{125}-1}{5^{25}-1}$ is composite.
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 $\cos(x+2y)$
How many among the first $1000$ Fibonacci numbers are multiples of $11$?
Let $F(1)=1, F(2)=1, F(n+2)= F(n+1)+F(n)$ be the Fibonacci sequence. Prove if $i | j$, then $F(i) | F(j)$. In another word, every $k^{th}$ element is a multiple of $F(k)$.
The ratio $\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ is closest to which of the following numbers?
According to the standard convention for exponentiation,
$$2^{2^{2^2}} = 2^{\left(2^{\left(2^2\right)}\right)} = 2^{16} = 65,536$$
If the order in which the exponentiations are performed is changed, how many other values are possible?
There are 3 numbers A, B, and C, such that $1001C - 2002A = 4004$, and $1001B + 3003A = 5005$. What is the average of A, B, and C?
What is the sum of all of the roots of $(2x + 3) (x - 4) + (2x + 3) (x - 6) = 0$?
Let $a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$. What is $a + b + c + d$?
Let $a$, $b$, and $c$ form a geometric sequence. Can the last two digits of $N=a^3+b^3+c^3-3abc$ be 20?
Prove that $5x^2\pm 4$ is a perfect square if and only if $x$ is a term in the Fibonacci sequence.
Find the maximal value of $m^2+n^2$ if $m$ and $n$ are integers between $1$ and $1981$ satisfying $(n^2-mn-m^2)^2=1$.
Solve in positive integers the equation $$m^2 - n^2 - 3n = 5$$
Sequence $ \{a_{n}\}$ is defined by $ a_{1}= 2007,\, a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1}$ for $ n \ge 1.$ Prove that $ [a_{n}] =2007-n$ for $ 0 \le n \le 1004,$ where $ [x]$ denotes the largest integer no larger than $ x.$
Let $ f$ be a function given by $ f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x$.
a) Solve the equation $ f(x) = 0$.
b) Find the number of the subsets of the set \[ \{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.\]
Show that the equation $$x^2 + y^2 -19xy - 19 =0$$ is not solvable in integers.
Solve in positive integers $$x^3 + y^3 + z^3 = 3xyz$$
How many ordered triples of positive integers $(x, y, z)$ satisfy $(x^y)^z = 64$?
Let $P(x) = kx^3 + 2k^2x^2 + k^3$. Find the sum of all real numbers $k$ for which $x - 2$ is a factor of $P(x)$.
Let $f$ be a real-valued function such that $f(x) + 2f(\frac{2002}{x}) = 3x$ for all $x > 0$. Find $f(2)$.
What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1$, $k_2$, $\dots$ , $k_n$ for which $$k_1^2 + k_2^2 + \cdots +k_n^2 = 2002$$
Solve the following question in integers $$x^6 + 3x^3 +1 = y^4$$
Let both $A$ and $B$ be two-digit numbers, and their difference is $14$. If the last two digits of $A^2$ and $B^2$ are the same, what are all the possible values of $A$ and $B$.
A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?