Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2010\}$. What is the probability that $abc + ab + a$ is divisible by $3$?
A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$?
Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?
What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by $8$?
Suppose that \[\frac{2x}{3}-\frac{x}{6}\] is an integer. Which of the following statements must be true about $x$?
A right triangle has perimeter $32$ and area $20$. What is the length of its hypotenuse?
Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?
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 $(a,b)$?
Integers $a, b, c,$ and $d$, not necessarily distinct, are chosen independently and at random from $0$ to $2019$, inclusive. What is the probability that $(ad-bc)$ is even?
Suppose that $m$ and $n$ are positive integers such that $75m = n^{3}$. What is the minimum possible value of $m + n$?
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let $S$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $S$?
How many ordered pairs $(m,n)$ of positive integers, with $m \ge n$, have the property that their squares differ by $96$?
For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n.$ For how many values of $n$ is $n + S(n) + S(S(n)) = 2007?$
Let $n$ denote the smallest positive integer that is divisible by both $4$ and $9,$ and whose base-$10$ representation consists of only $4$'s and $9$'s, with at least one of each. What are the last four digits of $n?$
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?
How many sets of two or more consecutive positive integers have a sum of $15$?
For how many real values of $x$ is $\sqrt{120-\sqrt{x}}$ an integer?
Six distinct positive integers are randomly chosen between $1$ and $2020$, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of $5$?
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?
What is the tens digit in the sum $7!+8!+9!+...+2018!$
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
How many positive integers $n$ satisfy the following condition:
$(130n)^{50} > n^{100} > 2^{200}$?
How many positive cubes divide $3! \cdot 5! \cdot 7!$ ?