Diophantine Equations

Integers $x$ and $y$ with $x > y > 0$ satisfy $x + y + xy = 80$. What is $x$?

How many ordered pairs of integers $(x, y)$ satisfy the equation

There is a prime number $p$ such that $16p + 1$ is the cube of a positive integer. Find $p$.

Find $3x^2y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2y^2 = 30x^2 + 517$.

If the positive integer $c$ has positive integer divisors $a$ and $b$ with $c = ab$, then $a$ and $b$ are said to be complementary divisors of $c$. Suppose that $N$ is a positive integer that has one complementary pair of divisors that differ by $20$ and another pair of complementary divisors that differ by $23$. What is the sum of the digits of $N$?

How many ordered pairs of integers $(m, n)$ satisfy the equation $m^2 + mn + n^2 = m^2n^2$?

Determine all non-negative integral solutions $(n_1, n_2, \dots, n_{14})$ if any, apart from permutations, of the Diophantine Equation

Two farmers agree that pigs are worth $300$ dollars and that goats are worth $210$ dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a $390$ dollar debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?

Let $a$ and $b$ be relatively prime positive integers with $a > b > 0$ and $\dfrac{a^3 - b^3}{(a - b)^3} = \dfrac{73}{3}$. What is $a - b$?

Two distinct numbers are selected from the set $\{1, 2, 3, 4, \dots, 36, 37\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers?

How many triples $(a, b, c)$ of positive integers satisfy the simultaneous equations

An arithmetic sequence (of positive integers) has $n \ge 3$ terms, initial term $a$ and common difference $d > 1$. Carl wrote down all the terms in this sequence correctly except for one term which was off by $1$. The sum of the terms was $222$. What was $a + d + n$?

How many ordered pairs of positive integers $(b, c)$ exist where both $x^2 + bx + c = 0$ and $x^2 + cx + b = 0$ do not have distinct, real solutions?

A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are $57$, $60$, and $91$. What is the fourth term of this sequence?

There is a unique ordered triple $(a, k, m)$ of nonnegative integers such that

What is $a + k + m$?

A rectangular grid of squares has $141$ rows and $91$ columns. Each square has room for two numbers. Horace and Vera each fill in the grid by putting the numbers from $1$ through $141 \times 91 = 12{,}831$ into the squares. Horace fills the grid horizontally: he puts $1$ through $91$ in order from left to right into row $1$, puts $92$ through $182$ into row $2$ in order from left to right, and continues similarly through row $141$. Vera fills the grid vertically: she puts $1$ through $141$ in order from top to bottom into column $1$, then $142$ through $282$ into column $2$ in order from top to bottom, and continues similarly through column $91$. How many squares get two copies of the same number?

Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly $19$. How many sheets were borrowed?

One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that

Find the value of $n$.

Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.

For some integer $m$, the polynomial $x^3 - 2011x + m$ has the three integer roots $a$, $b$, and $c$. Find $|a| + |b| + |c|$.

Let $m \ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1 \le a_i \le m$ for all $i$ such that $m$ divides $a_1 + a_2 + a_3 + a_4$. There is a polynomial

such that $D(m) = q(m)$ for all odd integers $m \ge 5$. What is $c_1$?