All positive integers relatively prime to 2015 are written in increasing order. Let the twentieth number be $p$. The value of $\frac{2015}{p}\u2212 1$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $a + b$.
For how many ordered pairs $(x, y)$ of integers satisfying $0 \le x$, $y \le 10$, and $(x + y)^2 + (xy - 1)^2$ is a prime number?
Find the least composite positive integer that is not divisible by any of 3, 4, and 5.
Let the positive divisors of $n$ be $d_1, d_2, \dots$ in increasing order. If $d_6 = 35$, determine the minimum possible value of $n$.
Nicky is studying biology and has a tank of $17$ lizards. In one day, he can either remove $5$ lizards or add $2$ lizards to his tank. What is the minimum number of days necessary for Nicky to get rid of all of the lizards from his tank?
Find all ordered pairs $(a, b)$ of positive integers such that $\sqrt{64a + b^2} + 8 = 8\sqrt{a} + b$.
Compute the smallest positive integer with at least four two-digit positive divisors.
Find the maximum possible value of the greatest common divisor of $MOO$ and $MOOSE$, given that $S$, $O$, $M$, and $E$ are some nonzero digits. (The digits $S$, $O$, $M$, and $E$ are not necessarily pairwise distinct.)
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 $n$.
Can a five-digit number consisting of 5 distinct even digits a perfect square?
Find the smallest $n$ such that $\frac{1}{n}(1^2 + 2^2 + \cdots + n^2)$ is a square of an integer.
Solve in integers $y^2=x^4 + x^3 + x^2 +x +1$.
Solve in integers $x^3 + (x+1)^3 + \cdots + (x+7)^3 = y ^3$
Solve in positive integers $y^2 = x^2 + x + 1$
Solve in integers $\frac{1}{x}+\frac{1}{y} + \frac{1}{z} = \frac{3}{5}$
$N$ delegates attend a round-table meeting, where $N$ is an even number. After a break, these delegates randomly pick a seat to sit down again to continue the meeting. Prove that there must exist two delegates so that the number of people sitting between them is the same before and after the break.
Find the largest multiple of 99 among the nine-digit integers, whose digits are all distinct.
Prove there is no integer solutions to $x^2 = y^5 - 4$.
Find all positive integer solutions to: $x^2 + 3y^2 = 1998x$.
Prove that there exist infinite many triples of consecutive integers each of which is a sum of two squares. For example: $8 = 2^2 + 2^2$, $9 = 3^2 + 0^2$, and $10=3^1 + 1^2$
Find all triangles whose sides are consecutive integers and areas are also integers.
Find all positive integers $k$, $m$ such that $k < m$ and
$$1+ 2 +\cdots+ k = (k +1) + (k + 2) +\cdots+ m$$
Prove that there are infinitely many positive integers $n$ such that $(n^2+1)$ divides $n!$.
There are $100$ tigers, $100$ foxes, and $100$ monkeys in the animal kingdom. Tigers always tell the truth; Foxes always tell lies; and monkeys sometimes tell the truth but sometimes not. These $300$ animals are divided into $100$ groups, each of which has exactly two of the same kind and one of another kind. Now comes the Kong Fu Panda. He asks every animal: "is there a tiger in your group?" and receives $138$ "yes" answers. Then he asks everyone: "is there a fox in your group?" and receives $188$ positive answer this time. Find the number of monkeys who tell the truth both time.
Find the smallest positive integer $n$ such that the last $3$ digits of $n^3$ is $888$.