Practice (97)

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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 $x^2 + y^2 = x^3$?

Solve in positive integers $x^2 - 4xy + 5y^2 = 169$.

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

Solve in integers the question $x+y=x^2 -xy + y^2$.

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

Solve in positive integers $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{4}{5}$

Find all ordered pairs of integers $(x, y)$ that satisfy the equation $$\sqrt{y-\frac{1}{5}} + \sqrt{x-\frac{1}{5}} = \sqrt{5}$$


Solve in integers $\frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$


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.


If $p$, $q$, $\frac{2p-1}{q}$, and $\frac{2q-1}{p}$ are all integers, and $p>1$, $q>1$, find the value of $(p+q)$


Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:

  • If $16$, $15$, or $14$ students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and
  • There are three integers $0 < x < y < z < 14$ such that when $x$, $y$, or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.

Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.

Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$.

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?

A class collects $50$ dollars to buy flowers for a classmate who is in the hospital. Roses cost $3$ dollars each, and carnations cost $2$ dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50$ dollars?