Diophantine Equations: all problems

Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win?

The positive integers $x$ and $y$ are the two smallest positive integers for which the product of 360 and $x$ is a square and the product of 360 and $y$ is a cube. What is the sum of $x$ and $y$?

How many digits are in the product $4^5 \cdot 5^{10}$?

For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3n$ three-digit whole numbers?

A 2-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number?

In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A + B$?

What is the correct ordering of the three numbers, $10^8$, $5^{12}$, and $2^{24}$?

Suppose $a$, $b$, and $c$ are nonzero real numbers, and $a + b + c = 0$. What are the possible value(s) for $\frac{a}{|a|} + \frac{b}{|b|} + \frac{c}{|c|} + \frac{abc}{|abc|}$?

(A) 0 (B) 1 and $-1$ (C) 2 and $-2$ (D) 0, 2, and $-2$ (E) 0, 1, and $-1$

After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored 15 points. None of the other 7 team members scored more than 2 points. What was the total number of points scored by the other 7 team members?

A baseball league consists of two four-team divisions. Each team plays every other team in its division N games. Each team plays every team in the other division M games with $N > 2M$ and $M > 4$. Each team plays a 76 game schedule. How many games does a team play within its own division?

What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is not a divisor of the product of the first $n$ positive integers?

How many perfect cubes lie between $2^8 + 1$ and $2^{18} + 1$, inclusive?

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

After the digits of a 3 digit number are reversed, the number is 297 more than the original number. What's the largest possible value for the original number?

The hundreds digit of a three-digit number is 2 more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?

In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His erroneous product was 161. What is the correct value of the product $a$ and $b$?

A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers $x$ and $x + 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$?

A two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?

What is the largest positive 2-digit factor of $3^{2^{2011}} - 2^{2^{2011}}$?

If $m$ and $n$ are positive integers where $m^2 + 14m - 32 = 3^n$, what is the value of $m + n$?

Which of the following is equivalent to $\sqrt{\dfrac{x}{1 - \frac{x-1}{x}}}$ when $x < 0$?

(A) $-x$ (B) $x$ (C) $1$ (D) $\sqrt{\frac{x}{2}}$ (E) $x\sqrt{-1}$

In the United States, coins have the following thicknesses: penny, 1.55 mm; nickel, 1.95 mm; dime, 1.35 mm; quarter, 1.75 mm. If a stack of these coins is exactly 14 mm high, how many coins are in the stack?

Boris has an incredible coin-changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?

(A) $3.63 (B) $5.13 (C) $6.30 (D) $7.45 (E) $9.07