Practice (98)

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Solve in integers the equation $$x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3.$$

How many ordered pairs of positive integers $(x, y)$ can satisfy the equation $x^2 + y^2 = x^3$?

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 $2(x+y)=xy+7$.

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 $x^2 - 51y^2 = 1$.


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.

Solve in integer: $36((xy+1)z+x)=475(yz+1)$

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

A restaurant sells three sizes of drinks: small for \$1.20, medium for \$1.30 and large for \$1.80. Each person at a table of ten ordered one drink, for a total cost of \$14.90, before sales tax. How many people ordered a large drink?

A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $17.71$. What was the cost of a pencil in cents?

Arithmetic sequences $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1 < a_2 \le b_2$ and $a_n b_n = 2010$ for some $n$. What is the largest possible value of $n$?

Each morning of her five-day workweek, Jane bought either a 50-cent muffin or a 75-cent bagel. Her total cost for the week was a whole number of dollars, How many bagels did she buy?

Consider the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10\%$?

The zeroes of the function $f(x)=x^2-ax+2a$ are integers .What is the sum of the possible values of a?

In $\triangle ABC$, $AB = 86$, and $AC=97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is $BC$?

The sum of the first $m$ positive odd integers is 212 more than the sum of the first $n$ positive even integers. What is the sum of all possible values of $n$?

Let $a$, $b$, and $c$ be positive integers with $a\ge$ $b\ge$ $c$ such that $a^2-b^2-c^2+ab=2011$ and $a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$. What is $a$?

How many ordered pairs of positive integers (M,N) satisfy the equation $\frac {M}{6}$ = $\frac{6}{N}$

In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town's population during this twenty-year period?

Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?

A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers and $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half the area of the whole floor. How many possibilities are there for the ordered pair $(a,b)$?