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Given three congruent circles with centers $P$, $Q$, and $R$ tangent to the sides of rectangle $ABCD$, the circle centered at $Q$ has a diameter of $6$ and passes through points $P$ and $R$. Compute the area of rectangle $ABCD$. | 72 | 0.333333 |
Given that the probability of a biased coin landing heads is $\frac{1}{3}$ and tails is $\frac{2}{3}$, and Tom, Dick, and Harry flip the coin repeatedly until they get their first head, calculate the probability that all three flip their coins the same number of times. | \frac{1}{19} | 0.916667 |
Given that $3x - 2y - 2z = 0$ and $x - 4y + 8z = 0$, where $z \neq 0$, calculate the value of $\frac{3x^2 - 2xy}{y^2 + 4z^2}$. | \frac{120}{269} | 0.833333 |
Find the ratio of AD:DC in triangle ABC, where AB=6, BC=8, AC=10, and D is a point on AC such that BD=6. | \frac{18}{7} | 0.666667 |
Students from Arlington school worked for $4$ days, students from Bradford school worked for $7$ days, and students from Clinton school worked for $8$ days. If a total of eight students, six students, and seven students from Arlington, Bradford, and Clinton schools respectively earned a total of $1,456, calculate the total earnings from Clinton school. | 627.20 | 0.166667 |
Given each piece of candy in the store costs a whole number of cents, and a piece of yellow candy costs $15$ cents, determine the smallest possible value of $n$ if Joe has exactly enough money to buy $10$ pieces of red candy, $16$ pieces of green candy, $18$ pieces of blue candy, or $n$ pieces of yellow candy. | 48 | 0.833333 |
Given $d$ is a digit, determine for how many values of $d$ the number $3.014d$ is greater than $3.015$. | 0 | 0.333333 |
An ellipse with a horizontal major axis and center at $O$ has a pair of parallel tangents that are vertical and equidistant from $O$. Find the number of points equidistant from the ellipse and these two parallel tangents. | 2 | 0.75 |
Given the repeating decimal $5.1717171717\ldots$, express the number as a fraction and calculate the sum of the numerator and denominator of this fraction when reduced to lowest terms. | 611 | 0.916667 |
Let $n$ be the largest integer that is the product of exactly three distinct prime numbers $d$, $e$, and $d^2 + e^2$, where $d$ and $e$ are distinct single-digit primes such that $d < e$. Find the sum of the digits of $n$. | 13 | 0.666667 |
Given that Ms. Demeanor's class consists of 50 students, more than half of her students bought crayons from the school bookstore, each buying the same number of crayons, with each crayon costing more than the number of crayons bought by each student, and the total cost for all crayons was $19.98, determine the cost of each crayon in cents. | 37 | 0.25 |
A group of students is organizing a fundraiser. Initially, $60\%$ of the group are girls. Later on, three girls leave and three boys join, after which $50\%$ of the group are girls. Determine the number of girls initially in the group. | 18 | 0.916667 |
A triangle $\bigtriangleup ABC$ has vertices lying on the parabola defined by $y = x^2 + 4$. Vertices $B$ and $C$ are symmetric about the $y$-axis and the line $\overline{BC}$ is parallel to the $x$-axis. The area of $\bigtriangleup ABC$ is $100$. $A$ is the point $(2,8)$. Determine the length of $\overline{BC}$. | 10 | 0.583333 |
Find the ratio of $AE:EC$ in $\triangle ABC$ given that $AB=6$, $BC=8$, $AC=10$, and $E$ is on $\overline{AC}$ with $BE=6$. | \frac{18}{7} | 0.333333 |
Min-jun's video had a score of $120$, and $75\%$ of the votes were likes. Determine how many total votes were cast on his video at that point. | 240 | 0.75 |
A wooden cube $n$ units on each side is painted red on four of its six faces and then cut into $n^3$ unit cubes. Exactly one-third of the total number of faces of the unit cubes are red. What is $n$? | 2 | 0.5 |
Two numbers have a difference, sum, and product that are in the ratio \(1:8:15\). Find the product of the two numbers. | \frac{100}{7} | 0.5 |
The product of the three ages of Kiana and her twin brothers is $72$. Express their ages as $x$, $x$, and $y$ and determine their sum. | 14 | 0.916667 |
The ratio of $a$ to $b$ is $5:2$, the ratio of $c$ to $d$ is $2:1$, and the ratio of $d$ to $b$ is $2:5$. Determine the ratio of $a$ to $c$. | \frac{25}{8} | 0.916667 |
Given that a circle is divided into 18 sectors with central angles forming an arithmetic sequence, find the degree measure of the smallest possible sector angle. | 3 | 0.5 |
How many positive factors of 48 are also multiples of 8? | 4 | 0.833333 |
A cell phone plan costs $30$ each month, plus $4$ cents per text message, and $15$ cents for each minute used over $40$ hours. Given that Tim sent 200 text messages and talked for 42 hours in March, calculate the total cost of his plan for the month. | 56.00 | 0.916667 |
Determine how many different prime numbers are factors of $N$ if $\log_2 ( \log_3 ( \log_5 (\log_ 7 (\log_{11} N)))) = 16$. | 1 | 0.75 |
Given nine squares are arranged in a 3x3 grid and numbered from 1 to 9 from left to right, top to bottom. The paper is folded three times in sequence: fold the right third over to the middle third, fold the left third over the previous fold covering the right and middle thirds, and fold the bottom third up to the top third. Determine the numbered square that ends up on top after these folds. | 7 | 0.333333 |
If $5(3x + 7\pi) = Q$, solve for the value of $10(6x + 14\pi)$. | 4Q | 0.416667 |
A triangle with integral sides has a perimeter of 12. Determine the area of this triangle. | 6 | 0.833333 |
Cameron has $90$ red tokens and $60$ blue tokens. He can exchange three red tokens for one gold token and two blue tokens, or two blue tokens for one gold token and one red token. What is the total number of gold tokens Cameron ends up with after exchanging his tokens until no more exchanges can be made? | 148 | 0.083333 |
There are several sets of three different numbers whose sum is 21, which can be chosen from {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. How many of these sets contain a 7? | 5 | 0.583333 |
Given \(3 \cdot 2^{2000} - 5 \cdot 2^{1999} + 4 \cdot 2^{1998} - 2^{1997} = m \cdot 2^{1997}\), find the value of \(m\). | 11 | 0.333333 |
Given $(1296^{\log_6 1728})^{\frac{1}{4}}$, evaluate the resulting expression. | 1728 | 0.083333 |
Lucy and her five friends are ages $4$, $6$, $8$, $10$, $12$, and $14$. Given that two of her friends whose ages sum to $18$ went to the skatepark, two friends less than $12$ went to the swimming pool, and Lucy and the $6$-year-old went to the library, determine Lucy's age. | 12 | 0.416667 |
Calculate the average of the shortest distances from the lemming to each side of a square field with a side length of $12$ meters. | 6 | 0.75 |
Let $x$ be the smallest real number greater than $5$ such that $\sin(x) = \cos(2x)$, where the arguments are in degrees. Find the value of $x$. | 30 | 0.333333 |
Evaluate the expression $\frac{(0.5)^4}{(0.05)^3} + 3$. | 503 | 0.75 |
The sum of the numerical coefficients in the expansion of $(2x^2 - 3xy + y^2)^6$ is 64. | 0 | 0.333333 |
Given a round-robin tournament with 7 teams, each team plays one game against each other team, and each game results in one team winning and one team losing, what is the maximum number of teams that could be tied for the most wins at the end of the tournament? | 7 | 0.25 |
For how many positive integer values of $n$ are both $\frac{n}{4}$ and $4n$ three-digit whole numbers? | 0 | 0.916667 |
Given the four-digit number $5056$, find the number of different numbers that can be formed by rearranging its digits. | 9 | 0.833333 |
Given that Alice sells an item at $15 less than the list price and receives $15\%$ of her selling price as her commission, while Bob sells the same item at $25 less than the list price and receives $25\%$ of his selling price as his commission, determine the list price of the item. | 40 | 0.916667 |
Evaluate the double summation $\sum^{50}_{i=1} \sum^{50}_{j=1} 2(i+j)$. | 255000 | 0.916667 |
Determine the number of distinct terms in the expansion of $\left[(a+2b)^2 (a-2b)^2\right]^3$. | 7 | 0.666667 |
Given a rectangular track consisting of two straight sections of length $80$ meters each and two semicircular ends, the track has a width of $10$ meters, and it takes $60$ seconds longer to jog around the outer edge than around the inner edge, determine Mira's speed in meters per second. | \frac{\pi}{3} | 0.583333 |
Given the set $\{2,3,5,7,11,13\}$, add one of the numbers twice to another number, and then multiply the result by the third number. What is the smallest possible result? | 22 | 0.25 |
Given four spheres of radius $2$, each tangent to three orthogonal planes at the corners of a room, find the radius of the smallest sphere centered at the room's geometric center that contains these four spheres. | 2\sqrt{3} + 2 | 0.833333 |
Given the sequence where each integer \( n \) appears \( n \) times, for \( 1 \leq n \leq 150 \), calculate the median of the numbers in this list. | 106 | 0.916667 |
Given the data set $[8, 22, 36, 36, 44, 45, 45, 48, 56, 62]$, with a median $Q_2 = 44.5$, first quartile $Q_1 = 36$, and third quartile $Q_3 = 48$, determine the number of outliers in the data set, where an outlier is defined as a value that is more than $2$ times the interquartile range below the first quartile ($Q_1$) or more than $2$ times the interquartile range above the third quartile ($Q_3$). | 1 | 0.75 |
In an island kingdom, five fish can be traded for three jars of honey, and a jar of honey can be traded for six cobs of corn. Find the number of cobs of corn one fish is worth. | 3.6 | 0.666667 |
Given John the painter had enough paint initially to paint 50 similarly sized rooms, and after dropping five cans of paint he was left with enough to paint 40 rooms, determine how many cans of paint he used for these 40 rooms. | 20 | 0.916667 |
Jack had a bag of $150$ apples. He sold $20\%$ of them to Jill. Next, he sold $30\%$ of the remaining apples to June. Afterwards, he decides to sell $10\%$ of what's left to Jeff. Finally, Jack donates $5\%$ of the remaining apples to a local school. What is the number of apples that Jack has left after all these transactions? | 72 | 0.75 |
Expand $\left ( 1 - \frac{1}{a} \right )^8$ and find the sum of the last three coefficients. | 21 | 0.666667 |
What is the number of terms in the simplified expansion of $[(2a+5b)^3(2a-5b)^3]^3$? | 10 | 0.5 |
The sum of the angles of the pentagon inscribed in a circle cut off by its sides is 180 degrees. | 180^{\circ} | 0.25 |
A square piece of wood with a side length of 4 inches and uniform density weighs 16 ounces. Calculate the weight, in ounces, of a square piece of the same wood with a side length of 6 inches. | 36 | 0.916667 |
A hemisphere with a radius of 3 rests on the base of a triangular pyramid of height 9, and the hemisphere is tangent to all three faces of the pyramid. Calculate the length of each side of the base of the equilateral triangle. | 6\sqrt{3} | 0.833333 |
Suppose the product $\dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{c}{d} = 16$, find the sum of $c$ and $d$. | 95 | 0.666667 |
Let $x$ be the number of students in Mina's high school. Given that Nina's high school has 5 times as many students as Mina's high school, the number of students in Nina's high school is $5x$. Furthermore, if 200 students were to transfer from Nina's high school to Mina's high school, the total number of students in Mina's high school would be $x+200$ and the total number of students in Nina's high school, after the transfer, would be $5x-200$. The problem states that after the transfer, the two schools would have twice as many students as the smaller school. Therefore, the equation $2(x+200) = 5x-200$ can be formed. Solving for $x$, we find that $x=800. Thus, there are $\boxed{4000}$ students in Nina's high school. | 4000 | 0.916667 |
Consider a number formed by subtracting one from $2^{15}$, i.e., $32,767$. Find the sum of the digits of the greatest prime number that is a divisor of this number. | 7 | 0.583333 |
A worker's salary is reduced by 30%. What percentage increase is needed on the new salary to restore it to the original salary? | 42.86\% | 0.666667 |
Given the digits in the set $\{1, 3, 4, 5, 6, 9\}$, count the number of even integers between 300 and 800 whose digits are all different. | 24 | 0.166667 |
Suppose the estimated $50$ billion dollar cost to construct a space station is shared equally by the $500$ million people in a coalition of countries, calculate the amount each person's share. | \ 100 | 0.916667 |
Find the number of ordered pairs $(b,c)$ of positive integers such that both $x^2+bx+c=0$ and $x^2+cx+b=0$ have exactly one real solution. | 1 | 0.583333 |
Calculate the expression $200(200-5)-(200\cdot200-5)$. | -995 | 0.916667 |
Given $i$ be the imaginary unit such that $i^2 = -1$ and $a$ is a non-zero real number, calculate $(ai-i^{-1})^{-1}$. | -\frac{i}{a+1} | 0.416667 |
Four A's, four B's, four C's, and four D's are placed in the sixteen spaces of a 4x4 grid so that each row and column contains one of each letter. If A is placed in the upper left corner, determine the number of possible arrangements. | 144 | 0.083333 |
If Alice is cycling east at a speed of 10 miles per hour and Sam is also cycling east but at a faster speed of 16 miles per hour, with Sam being 3 miles west of Alice, calculate the time in minutes it will take for Sam to catch up to Alice. | 30 | 0.916667 |
Let $ABCD$ be a square, and let $E, F, G, H$ be the midpoints of the hypotenuse of right-angled triangles with one leg as $\overline{AB}, \overline{BC}, \overline{CD}, \overline{DA}$ respectively, each exterior to the square. Given that each triangle has legs equal to the side of the square, find the ratio of the area of square $EFGH$ to the area of square $ABCD$. | 2 | 0.166667 |
Let a sequence $\{u_n\}$ be defined by $u_1=4$ and the recurrence relation $u_{n+1}-u_n=2n^2 - 2n + 1, n=1,2,3,\dots$. If $u_n$ is expressed as a polynomial in $n$, find the algebraic sum of its coefficients. | 4 | 0.916667 |
A solid cube of side length $4$ has a solid cube of side length $2$ removed from each corner. How many edges does the remaining solid have? | 36 | 0.333333 |
Jessica has 30 coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $1.20 more. Calculate the total value of her coins. | \$1.65 | 0.75 |
A rectangular board of 10 rows is numbered in a snaking pattern, starting in the upper left corner. In the first row, the squares are numbered from 1 to 10, in the second row from 20 to 11 (backwards), and so on alternating directions each row. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips three squares and shades square 10, continuing in this manner. Determine the number of the shaded square that is the first to ensure at least one shaded square in each row. | 91 | 0.083333 |
Given two numbers such that their difference, their sum, and their product are to one another as $1:8:40$, calculate the product of these two numbers. | \frac{6400}{63} | 0.666667 |
When all four valves are open, the tank fills in 1.2 hours, and with valves A, B, and D open, it takes 2 hours. With valves A, C, and D open it takes 1.5 hours. Given the rate at which valve D releases water into the tank is half the rate of valve C, determine how many hours it will take to fill the tank with only valves A, B, and C open. | 1.5 | 0.583333 |
The Tigers beat the Sharks 3 out of the 4 times they initially played. They then played N more games, all won by the Sharks. Afterwards, the Sharks ended up winning at least 90% of all the games played. Determine the minimum possible value for N. | 26 | 0.916667 |
In the expansion of $\left(a - \frac{1}{\sqrt{a}}\right)^8$, determine the coefficient of the $a^{-\frac{1}{2}}$ term. | 0 | 0.833333 |
Given the first four terms of an arithmetic sequence $a, y, b, 3y$, determine the ratio of $a$ to $b$. | 0 | 0.916667 |
Given that $63$ students are wearing red shirts and $69$ students are wearing green shirts, and the $132$ students are assigned into $66$ pairs, with $26$ pairs consisting of two red shirts, determine the number of pairs that consist entirely of students wearing green shirts. | 29 | 0.333333 |
Given that $x$ is a real number, evaluate the expression $x[x\{x(3-x)-5\}+12]+2$. | -x^4 + 3x^3 - 5x^2 + 12x + 2 | 0.416667 |
Given Liam has written one integer three times and another integer four times. The sum of these seven numbers is 131, and one of the numbers is 17, determine the value of the other number. | 21 | 0.166667 |
Given a set of tiles numbered from 1 through 150, find the smallest number of operations required to reduce the number of tiles in the set to less than ten. | 4 | 0.166667 |
If a package weighs $W$ ounces, calculate the total mailing cost in cents, given that the service charges five cents for every ounce or portion thereof and there is a fixed handling fee of three cents per package. | 5\lceil W \rceil + 3 | 0.25 |
Given a positive real number is called doubly special if its decimal representation consists entirely of the digits $0$ and $5$, find the smallest integer $n$ such that $1$ can be written as a sum of $n$ doubly special numbers. | 2 | 0.916667 |
Five friends — Sarah, Lily, Emma, Nora, and Kate — performed in a theater as quartets, with one friend sitting out each time. Nora performed in 10 performances, which was the most among all, and Sarah performed in 6 performances, which was the fewest among all. Calculate the total number of performances. | 10 | 0.416667 |
Let $x = .123456789101112...495496497498499$, where the digits are obtained by writing the integers from $1$ through $499$ in order. Determine the $1234$th digit to the right of the decimal point. | 4 | 0.916667 |
Determine the sum of all possible x-coordinates where the lines given by the equations $y=2x+4$ and $y=4x+c$ intersect the x-axis. | -2 | 0.583333 |
A sphere with center $O$ has radius $9$. A right triangle with sides of length $12, 35, 37$ is situated in space so that each of its sides is tangent to the sphere. Calculate the distance between $O$ and the plane determined by the triangle. | 2\sqrt{14} | 0.166667 |
Four balls numbered $1, 2, 3$, and $4$ are placed in an urn. One ball is drawn, its number noted, and then returned to the urn. This process is repeated three times. If the sum of the numbers noted is $9$, determine the probability that the ball numbered $3$ was drawn all three times. | \frac{1}{10} | 0.666667 |
A positive integer $n$ has 72 divisors and $5n$ has 90 divisors. What is the greatest integer $k$ such that $5^k$ divides $n$? | 3 | 0.583333 |
Given Chloe chooses a real number uniformly at random from the interval $[ 0,3000 ]$ and Laurent chooses a real number uniformly at random from the interval $[ 0,6000 ]$, find the probability that Laurent's number is greater than Chloe's number. | \frac{3}{4} | 0.25 |
Determine the number of positive factors of 147,456. | 45 | 0.083333 |
There are twenty-four $4$-digit numbers that use each of the four digits $1$, $3$, $6$, and $8$ exactly once. List the numbers in numerical order from smallest to largest and find the number in the $15^{\text{th}}$ position in the list. | 6318 | 0.666667 |
In an exam, $15\%$ of the students scored $60$ points, $20\%$ scored $75$ points, $25\%$ scored $85$ points, $25\%$ scored $90$ points, and the rest got $100$ points. Calculate the difference between the mean and the median score on this exam. | 2.25 | 0.583333 |
Let \( S = \omega^n + \omega^{-n} \), where \( \omega = e^{2\pi i / 5} \) is a complex fifth root of unity, and \( n \) is an integer. Determine the total number of possible distinct values for \( S \). | 3 | 0.583333 |
Numbers between $200$ and $500$ that are divisible by $5$ contain the digit $3$. How many such whole numbers exist? | 24 | 0.25 |
How many positive factors of 48 are also multiples of 8? | 4 | 0.75 |
In the coordinate plane, a point $(x, y)$ needs to be the same distance from the x-axis, the y-axis, and the line $x + y = 4$. What is the x-coordinate of such a point?
A) $\sqrt{2}$
B) $2+\sqrt{2}$
C) $2$
D) $1$
E) Not uniquely determined | 2 | 0.583333 |
Given $15!$ has multiple positive integer divisors. If one of these divisors is chosen at random, calculate the probability that it is odd. | \frac{1}{12} | 0.166667 |
Let $U$ be the set of the first $2010$ positive multiples of $5$, and let $V$ be the set of the first $2010$ positive multiples of $7$. Calculate how many elements are common to both sets $U$ and $V$. | 287 | 0.416667 |
The average age of the members of the Jansen family is $25$, the father is $50$ years old, and the grandfather is $70$ years old. Given that the average age of the mother, grandfather, and children is $20$, calculate the number of children in the family. | 3 | 0.416667 |
Given a decorative window that includes a rectangle with semicircles on either end and an equilateral triangle centered on top of one semicircle, where the ratio of the length of the rectangle's length $AD$ to the width $AB$ is $3:2$ and $AB$ is 20 inches, determine the ratio of the area of the rectangle to the combined area of the semicircles and the triangle. | \frac{6}{\pi + \sqrt{3}} | 0.916667 |
Given that the speed of sound in the current weather conditions is 1100 feet per second, calculate to the nearest quarter-mile the distance Daisy was from the lightning flash 12 seconds after hearing the thunder. | 2.5 | 0.833333 |
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