Let $a$, $b$, and $c$ be digits with $a\ne 0$. The three-digit integer $abc$ lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer $acb$ lies two thirds of the way between the same two squares. What is $a+b+c$?
How many pairs of positive integers $(a,b)$ are there such that $\gcd(a,b)=1$ and \[\frac{a}{b}+\frac{14b}{9a}\] is an integer?
A line passes through $A\ (1,1)$ and $B\ (100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$?
Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. The integers $x$ and $y$ satisfy $x^{2}-y^{2}=m^{2}$ for some positive integer $m$. What is $x+y+m$?
Soda is sold in packs of $6$, $12$ and $24$ cans. What is the minimum number of packs needed to buy exactly $90$ cans of soda?
Joe wants to measure $6$ liter water using just two containers whose capacities are $27$ liters and $15$ liters, respectively. Can you help him?
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?
A rectangular box measures $a \times b \times c$, where $a$, $b$, and $c$ are integers and $1\leq a \leq b \leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?
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 $BC$?
Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 10x - 6y - 34$. What is $x + y$?
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 $\frac {M}{6}$ = $\frac{6}{N}$
For any given positive integer $n$, prove $(n^2 +n +1)$ cannot be a perfect square.
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 right triangle has perimeter $32$ and area $20$. What is the length of its hypotenuse?
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?
How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+1$, where $b<100$?
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 $(a,b)$?
How many ordered pairs $(m,n)$ of positive integers, with $m \ge n$, have the property that their squares differ by $96$?
How many pairs of positive integers (a,b) are there such that $a$ and $b$ have no common factors greater than 1 and: $\frac{a}{b} + \frac{14b}{9a}$ is an integer?
Let $\triangle ABC$ be a right triangle whose three sides' lengths are all integers. Prove among its three sides' lengths, at lease one is a multiple of $3$, one is a multiple of $4$, and one is a multiple of $5$. (Note: they can be the same side. For example, in the $5-12-13$, $12$ is both a multiple of $3$ and $4$.)
Find all ordered pairs $(a, b)$ of positive integers such that $\sqrt{64a + b^2} + 8 = 8\sqrt{a} + b$.