Let $b$ and $c$ be two positive integers, and $a$ be a prime number. If $a^2 + b^2 = c^2$, prove $a < b$ and $b+1=c$.
How many ordered pairs of positive integers $(a, b, c)$ that can satisfy $$\left\{\begin{array}{ll}ab + bc &= 44\\ ac + bc &=23\end{array}\right.$$
Solve in integers the equation $2(x+y)=xy+7$.
Solve in integers the question $x+y=x^2 -xy + y^2$.
Find the ordered pair of positive integers $(x, y)$ with the largest possible $y$ such that $\frac{1}{x} - \frac{1}{y}=\frac{1}{12}$ holds.
How many ordered pairs of integers $(x, y)$ satisfy $0 < x < y$ and $\sqrt{1984} = \sqrt{x} + \sqrt{y}$?
Find the number of positive integers solutions to $x^2 - y^2 = 105$.
Find any positive integer solution to $x^2 - 51y^2 = 1$.
Solve in positive integers $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{4}{5}$
Find all ordered pairs of integers $(x, y)$ that satisfy the equation $$\sqrt{y-\frac{1}{5}} + \sqrt{x-\frac{1}{5}} = \sqrt{5}$$
What is the remainder when $\left(8888^{2222} + 7777^{3333}\right)$ is divided by $37$?
Solve in integers $\frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$
Let the lengths of two sides of a right triangle be $l$ and $m$, respectively, and the length of the hypotenuse be $n$. If both $m$ and $n$ are positive integers, and $l$ is a prime number, show that $2(l+m+n)$ must be a perfect square.
If $p$, $q$, $\frac{2p-1}{q}$, and $\frac{2q-1}{p}$ are all integers, and $p>1$, $q>1$, find the value of $(p+q)$
Let positive integer $d$ is a divisor of $2n^2$, where $n$ is also a positive integer. Prove $(n^2 + d)$ cannot be a perfect square.
Prove the product of $4$ consecutive positive integers is a perfect square minus $1$.
For any arithmetic sequence whose terms are all positive integers, show that if one term is a perfect square, this sequence must have infinite number of terms which are perfect squares.
Find all prime number $p$ such that both $(4p^2+1)$ and $(6p^2+1)$ are prime numbers.
Prove there exist infinite number of positive integer $a$ such that for any positive integer $n$, $n^4 + a$ is not a prime number.
What is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?
Find that largest integer $A$ that satisfies the following property: in any permutation of the sequence $1001$, $1002$, $1003$, $\cdots$, $2000$, it is always possible to find $10$ consecutive terms whose sum is no less than $A$.
Find all positive integer $n$ such that $(3^{2n+1} -2^{2n+1}- 6^n)$ is a composite number.
Find $n$ different positive integers such that any two of them are relatively prime, but the sum of any $k$ ($k < n$) of them is a composite number.
The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?
Find the number of five-digit positive integers, $n$, that satisfy the following conditions:
- the number $n$ is divisible by $5$,
- the first and last digits of $n$ are equal, and
- the sum of the digits of $n$ is divisible by $5$.