Practice (98)
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How many ordered pairs $(m,n)$ of positive integers, with $m \ge n$, have the property that their squares differ by $96$?
How many pairs of positive integers (a,b) are there such that $a$ and $b$ have no common factors greater than 1 and: $\frac{a}{b} + \frac{14b}{9a}$ is an integer?
Find all ordered pairs $(a, b)$ of positive integers such that $\sqrt{64a + b^2} + 8 = 8\sqrt{a} + b$.
Let $S_n$ be the minimal value of $\displaystyle\sum_{k=1}^n\sqrt{(2k-1)^2+a_k^2}$, where $n\in\mathbb{N}$, $a_1, a_2, \cdots, a_n\in\mathbb{R}^+$, and $a_1+a_2+\cdots a_n = 17$. If there exists a unique $n$ such that $S_n$ is also an integer, find $n$.
Find all nonnegative integers $x$ and $y$ such that $x^3+y^3 = (x+y)^2$.
Find the number of paris $(a, b)$ of nonnegative integers that satisfy $6a+7b=1000$
For how many positive integers $m$ is there at least 1 positive integer $n$ such that $mn \le m + n$?
For each positive integer $n$, let $s(n)$ denote the number of ordered positive integer pair $(x, y)$ for which $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ holds. Find all positive integers $n$ for which $s(n) = 5$.
For any given positive integer $n > 2$, show that there exists a right triangle with all sides' lengths are integers and one side's length equals $n$.
Solve this equation in positive integers $$x^3 - y^3 = xy + 61$$
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 equation in positive integers: $$x^2 +3x^2y^2 = 30y^2 + 517$$
Find all the integer solutions to the equation $xy - 10(x+ y)= 1$.
Solve in integers the equation $x^2 - xy +2x -3 y = 0$
How many integer solutions does the equation $(x+1)(y+1)=25$ have?
The sum of $n$ consecutive positive integers is 100. What is the greatest possible value of $n$?
For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. For how many positive integers $n$ is it true that both $P(n) = \sqrt{n}$ and $P(n+48) = \sqrt{n+48}$?
For how many positive integers $m$ does there exist at least one positive integer n such that $m \cdot n \le m + n$?
A regiment had 48 soldiers but only half of them had uniforms. During inspection, they form a 6 × 8 rectangle, and it was just enough to conceal in its interior everyone without a uniform. Later, some new soldiers joined the regiment, but again only half of them had uniforms. During the next inspection, they used a different rectangular formation, again just enough to conceal in its interior everyone without a uniform. How many new soldiers joined the regiment?
Determine all positive integers $m$ and $n$ such that $m^2+1$ is a prime number and $10(m^2 + 1) = n^2 + 1$.
There exist some integers, $a$, such that the equation $(a+1)x^2 -(a^2+1)x+2a^2-6=0$ is solvable in integers. Find the sum of all such $a$.
Find all positive integer $n$ such that $n^2 + 2^n$ is a perfect square.
In $\triangle{ABC}$, $AB = 33, AC=21,$ and $BC=m$ where $m$ is an integer. There exist points $D$ and $E$ on $AB$ and $AC$, respectively, such that $AD=DE=EC=n$ where $n$ is also an integer. Find all the possible values of $m$.
Let $S_n$ be the minimal value of $\displaystyle\sum_{k=1}^n\sqrt{a_k^2+b_k^2}$ where $\{a_k\}$ is an arithmetic sequence whose first term is $4$ and common difference is $8$. $b_1, b_2,\cdots, b_n$ are positive real numbers satisfying $\displaystyle\sum_{k=1}^nb_k=17$. If there exist a positive integer $n$ such that $S_n$ is also an integer, find $n$.
In a sports contest, there were $m$ medals awarded on $n$ successive days ($n > 1$). On the first day, one medal and $1/7$ of the remaining $m − 1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining
medals were awarded; and so on. On the $n^{th}$ and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?