How many positive integers $n$ satisfy the following condition:
$(130n)^{50} > n^{100} > 2^{200}$?
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is $6$. How many two-digit numbers have this property?
Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game?
Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?
How many numbers between $1$ and $2005$ are integer multiples of $3$ or $4$ but not $12$?
An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $\$20$ or more?
All of David's telephone numbers have the form $555 - abc - defg$, where $a$, $b$, $c$, $d$, $e$, $f$, and $g$ are distinct digits and in increasing order, and none is either $0$ or $1$. How many different telephone numbers can David have?
Forty slips are placed into a hat, each bearing a number $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, or $10$, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all four slips bear the same number. Let $q$ be the probability that two of the slips bear a number $a$ and the other two bear a number $b \neq a$. What is the value of $\frac{q}{p}$?
In how many distinguishable ways can the four letters in the word NINE be arranged?
How many distinct unit cubes are there with two faces painted red, two faces painted green and two faces painted blue? Two unit cubes are considered distinct if one unit cube cannot be obtained by rotating the other.
Quatro Airlines flies between four major cities. To provide direct flights from each city to the other three cities requires a total of six different direct routes, as shown. How many routes are needed to connect 15 cities, with exactly one route directly connecting each pair?
How many positive integers less than $1000$ do not have $7$ as any digit?
Using each of the digits 1 to 6, inclusive, exactly once, how many six-digit integers can be formed that are divisible by 6?
How many ways can all six numbers in the set $\{4, 3, 2, 12, 1, 6\}$ be ordered so that $a$ comes before $b$ whenever $a$ is a divisor of $b$?
A coin is flipped until it has either landed heads two times or tails two times, not necessarily in a row. If the first flip lands heads, what is the probability that a second head occurs before two tails? Express your answer as a common fraction.
A box contains $r$ red balls and $g$ green balls. When $r$ more red balls are added to the box, the probability of drawing a red ball at random from the box increases by $25\%$. What was the probability of randomly drawing a red ball from the box originally? Express your answer as a common fraction.
The game of Connex contains one 4-unit piece, two identical 3-unit pieces, three identical 2-unit pieces and four identical 1-unit pieces. How many different arrangements of pieces will make a 10-unit segment? The 10-unit segments consisting of the pieces 4-3-2-1 and 1-2-3-4 are two such arrangements to include.
The dart board shown here contains 20 uniquely numbered sectors. When Malaika aims for a particular number, she hits it half the time. The other half of the time, she randomly hits an adjacent number on either side with equal probability. The number in the sector that her dart hits is the number of points scored. Trying to earn the highest possible score, Malaika decides to aim for the same number for each of her next 20 throws. Based on the given information, for which number should Malaika aim?
How many positive two-digit integers have exactly $8$ positive factors?
After tossing a red, then a green and, finally, a white standard six-faced die, Patrick used the numbers showing on the upper faces of each die, in order, to create the incorrect equation below, such that red - green = white. By rotating each die a quarter turn in some direction so that the number on the top face moves to a lateral face, he finds that he can make a correct equation. Given that the opposite faces of a die have a sum of 7, how many correct equations are possible?
How many collections of six positive, odd integers have a sum of $18$? Note that $1 + 1 + 1 + 3 + 3 + 9$ and $9 + 1 + 3 + 1 + 3 + 1$ are considered to be the same collection.
In how many different ways can $15,015$ be represented as the sum of two or more consecutive positive integers written in ascending order?
At the school's carnival, one game featured this unique square dartboard with five smaller, shaded squares, shown here. The length of a side of the square dartboard is 4 times the length of a side of any of the five congruent, shaded squares. To win a prize, a player's dart has to land in a shaded region. If a player's dart randomly hits the dartboard, what is the probability of her winning a prize? Express your answer as a common fraction.
A state license plate contains the state logo in the center, preceded by three letters and followed by three digits. If the first two letters must both be consonants, excluding Y, how many different license plates are possible?