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stringlengths 18
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A dart board is a regular hexagon divided into regions as shown. The center of the board is a smaller regular hexagon, and each of the six surrounding regions is a trapezoid. The side length of the larger hexagon is twice that of the smaller hexagon. If a dart thrown at the board is equally likely to land anywhere on the board, determine the probability that the dart lands within the center hexagon.
|
\frac{1}{4}
| 0.916667 |
Given a $3\times3$ block of calendar dates shown, first, the order of the numbers in the second row is reversed, and then the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?
|
0
| 0.583333 |
A particle moves such that its speed starting from the third mile varies inversely with the square of the integral number of miles already traveled plus 1. Given that the third mile is traversed in 3 hours, determine the time, in hours, required to traverse the nth mile, where n ≥ 3.
|
\frac{n^2}{3}
| 0.416667 |
A triangle has an area of $24$, one side of length $8$, and the median to that side of length $7.2$. Let $\theta$ be the acute angle formed by that side and the median. Calculate $\sin{\theta}$.
|
\frac{5}{6}
| 0.75 |
Calculate the sum of $2367 + 3672 + 6723 + 7236$.
|
19998
| 0.666667 |
Dianna's teacher gave her the task to substitute numbers for $a$, $b$, $c$, $d$, $e$, and $f$ in the expression $a-(b-(c-(d+(e-f))))$ and compute the result. Dianna misinterpreted the parentheses but performed the addition and subtraction accurately, coincidentally arriving at the correct result. Dianna used the numbers $1$, $2$, $3$, $4$, and $5$ respectively for $a$, $b$, $c$, $d$, and $e$. Find the number that Dianna substituted for $f$.
|
2
| 0.083333 |
Given a 24-inch by 15-inch pan of brownies cut into pieces that measure 3 inches by 2 inches, calculate the number of pieces of brownies the pan contains.
|
60
| 0.833333 |
At 3:30 o'clock, calculate the angle formed by the hour and minute hands of a clock.
|
75^\circ
| 0.833333 |
Given that a new kitchen mixer is listed in a store for $\textdollar 129.99$ and an online advertisement offers the same mixer for four easy payments of $\textdollar 29.99$ and a one-time shipping and handling fee of $\textdollar 19.99$, calculate how many cents are saved by purchasing the mixer through the online advertisement instead of in-store.
|
996
| 0.916667 |
A 6 × 6 grid of blocks is used. In how many different ways can 4 blocks be selected such that no two blocks are in the same row or column?
|
5400
| 0.666667 |
The number $b = \frac{r}{s}$, where $r$ and $s$ are relatively prime positive integers, has the property that the sum of all real numbers $y$ satisfying
\[
\lfloor y \rfloor \cdot \{y\} = b \cdot y^2
\]
is $360$, where $\lfloor y \rfloor$ denotes the greatest integer less than or equal to $y$ and $\{y\} = y - \lfloor y \rfloor$ denotes the fractional part of $y$. What is $r + s$?
**A)** 861
**B)** 900
**C)** 960
**D)** 1021
|
861
| 0.083333 |
A laser is placed at the point $(4,6)$. The laser beam travels in a straight line. The requirement is for the beam to hit and bounce off the $y$-axis, then the $x$-axis, again off the $y$-axis, and finally hit the point $(8,6)$. Calculate the total distance the beam will travel along this path.
|
4\sqrt{10}
| 0.083333 |
For how many positive integer values of \(N\) is the expression \(\dfrac{48}{N+3}\) an integer?
|
7
| 0.75 |
Let $u = 3 + 3^q$ and $v = 3 + 3^{-q}$. Express $v$ in terms of $u$.
|
\frac{3u-8}{u-3}
| 0.166667 |
Let $\angle ABC = 40^{\circ}$ and $\angle ABD = 28^{\circ}$. If there is a point $E$ on line segment $BD$ such that $\angle DBE = 10^{\circ}$, determine the smallest possible degree measure for $\angle CBE$.
|
2
| 0.666667 |
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $15!$?
|
10
| 0.75 |
Let \( S_n = \frac{1}{2} (x^n + y^n) \) where \( x = 4 + \sqrt{15} \) and \( y = 4 - \sqrt{15} \), and \( n = 5432 \). Determine the units digit of \( S_{5432} \).
|
1
| 0.583333 |
Jessica has $3p + 2$ nickels, and Samantha has $2p + 6$ nickels. Compute the difference in their money in pennies.
|
5p - 20
| 0.916667 |
A right circular cone and a sphere have bases with the same radius, denoted as \( r \). If the volume of the cone is one-third that of the sphere, find the ratio of the altitude of the cone to the radius of its base.
|
\frac{4}{3}
| 0.916667 |
How many ordered pairs (a, b) of positive integers satisfy the equation $a \cdot b + 82 = 25 \cdot \text{lcm}(a, b) + 15 \cdot \text{gcd}(a, b)$?
|
0
| 0.25 |
Suppose $n$ standard 6-sided dice are rolled, and the probability of obtaining a sum of 2027 equals the probability of obtaining another sum $S$. Find the smallest possible value of $S$.
|
339
| 0.833333 |
Expand and simplify the expression $(1-x^2)(1+x^4+x^6)$.
|
1 - x^2 + x^4 - x^8
| 0.083333 |
If $x^4$, $x^2+\frac{1}{x^2}$, and $1+\frac{1}{x}+\frac{1}{x^3}$ are multiplied, determine the degree of the resulting polynomial.
|
6
| 0.833333 |
Consider the set of numbers $\{1, 10, 10^2, 10^3, \ldots, 10^{12}\}$. Find the ratio of the largest element of the set to the sum of the other twelve elements of the set rounded to the nearest tenth.
|
9.0
| 0.916667 |
Given that the ratio of AD to AB is 4:3, AB is 36 inches, determine the ratio of the area of the rectangle to the combined area of the semicircles.
|
\frac{16}{3\pi}
| 0.833333 |
Given the area of a rectangle is $18$, plot all the ordered pairs $(w, l)$ of positive integers for which $w$ is the width and $l$ is the length of the rectangle.
|
(1, 18), (2, 9), (3, 6), (6, 3), (9, 2), (18, 1)
| 0.916667 |
The number of revolutions of a wheel, with a fixed center and with an outside diameter of 10 feet, required to cause a point on the rim to go two miles, can be calculated.
|
\frac{1056}{\pi}
| 0.5 |
Given the set $\{-9, -7, -1, 2, 4, 6, 8\}$, find the minimum possible product of three different numbers.
|
-432
| 0.083333 |
Given that Mona's age is a perfect square and at least half of the guesses $16, 25, 27, 32, 36, 40, 42, 49, 64, 81$ are too low, calculate Mona's age.
|
49
| 0.166667 |
Given a rectangular box with dimensions A, B, and C, and surface areas of its faces given as 40, 40, 90, 90, 100, and 100 square units, determine the sum A + B + C.
|
\frac{83}{3}
| 0.75 |
In a circle with center $O$ and radius $r$, a chord $AB$ is drawn with length equal to $2r$. From $O$, a perpendicular to $AB$ meets $AB$ at $M$. From $M$ a perpendicular to $OA$ meets $OA$ at $D$. Determine the area of triangle $MDA$ in square units.
|
0
| 0.25 |
For how many integer values of $n$ does the value of $4800 \cdot \left(\frac{2}{3}\right)^n$ result in an integer?
|
8
| 0.333333 |
Rectangle $ABCD$ has a length of 8 inches and a width of 6 inches. The diagonal $AC$ is divided into four equal segments by points $E$, $F$, and $G$. Determine the area of triangle $BFG$.
|
6
| 0.75 |
Given a rectangular pan of brownies that measures $24$ inches by $15$ inches is cut into pieces that measure $3$ inches by $2$ inches, calculate the number of brownies the pan contains.
|
60
| 0.25 |
Evaluating the expression $\left(\left((3+1)^{-1} \cdot 2\right)^{-1} \cdot 2\right)^{-1} \cdot 2$.
|
\left(\left((3+1)^{-1} \cdot 2\right)^{-1} \cdot 2\right)^{-1} \cdot 2 = \frac{1}{2}
| 0.916667 |
A triangle has sides of length $7$ and $23$. What is the smallest whole number greater than the perimeter of any triangle with these side lengths?
|
60
| 0.25 |
When a certain unfair die is rolled, an even number is $5$ times as likely to appear as an odd number. The die is rolled twice. Calculate the probability that the sum of the numbers rolled is odd.
|
\frac{5}{18}
| 0.25 |
Given a regular dodecagon (12-sided polygon), determine the number of diagonals it has and double that number to account for possible symmetrical line segments inside the polygon that don't necessarily connect vertices.
|
108
| 0.666667 |
Calculate the percentage discount d and the total amount paid if the original price of an item is P dollars and it is sold for Q dollars, where P > Q > 0, and 10 items are purchased.
|
10Q
| 0.583333 |
If it costs $206.91 to label all the lockers consecutively, where each digit in a locker number costs three cents, determine the number of lockers at Wellington Middle School.
|
2001
| 0.083333 |
Three distinct vertices of a regular tetrahedron are chosen at random. Determine the probability that the plane determined by these three vertices contains points inside the tetrahedron.
|
0
| 0.333333 |
Given an n-digit positive integer, determine how many charming n-digit integers exist, where the digits are an arrangement of the set {1,2,...,n} and its first k digits form an integer that is divisible by k for k = 1,2,...,n, and the entire number ends in digit 7.
|
0
| 0.666667 |
Maria starts working at 7:25 A.M. and works for 9 hours, not including her 1-hour lunch break, and goes for lunch at noon. At what time will Maria's working day end?
|
5:25 \text{ P.M.}
| 0.083333 |
How many whole numbers between 1 and 2000 do not contain the digits 1 or 2?
|
511
| 0.083333 |
Consider a region bounded by the lines \((x-a)^2 = 25\) and \(y^2 = 9x\). Calculate the area of this region for \(a = 5\).
|
40\sqrt{10}
| 0.833333 |
Given a tesseract (4-dimensional hypercube), calculate the sum of the number of edges, vertices, and faces.
|
72
| 0.416667 |
Calculate the average monthly sales from Booster's Club sales data, given that the sales from January to June are $150, 90, X, 180, 210, 240$ respectively, where $X$ represents the sales in March, which were 50% greater than those in February.
|
167.5
| 0.833333 |
Given Aniyah has a $5 \times 7$ index card and if she shortens the length of one side of this card by $2$ inches, the card would have an area of $21$ square inches, determine the area of the card in square inches if she shortens the length of the other side by $2$ inches.
|
25
| 0.166667 |
Twelve points are evenly spaced, with one unit intervals, around a 3x3 square. If two of these twelve points are selected at random, calculate the probability that the two points are one unit apart.
|
\frac{2}{11}
| 0.5 |
Given that Marla has a large white cube with an edge length of 12 feet and enough green paint to cover 432 square feet, find the area of one of the white squares surrounded by green on each face of the cube.
|
72
| 0.416667 |
Given the container $3$ centimeters high, $5$ centimeters wide, and $7$ centimeters long can hold $105$ grams of clay, and a second container with triple the height, double the width, and the same length as the first container can hold $n$ grams of clay, determine the value of $n$.
|
630
| 0.916667 |
The AMC 8 was first given in 1985 and has been administered annually. Jessica turned 15 years old the year she took the tenth AMC 8, in what year was Jessica born?
|
1979
| 0.916667 |
Given that circle O's diameter AB, angle AOE is 60 degrees, and angle FOB is 90 degrees, find the ratio of the area of the smaller sector EOF to the area of the circle.
|
\frac{1}{12}
| 0.666667 |
What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{15} \cdot 5^7}$ as a decimal?
|
15
| 0.166667 |
Given that triangle ABC has a right angle at B, point D is the foot of the altitude from B, with AD = 5 and DC = 7, find the area of triangle ABC.
|
6\sqrt{35}
| 0.166667 |
The taxi fare in Rivertown is $3.00 for the first $\frac{3}{4}$ mile and additional mileage charged at the rate of $0.25 for each additional 0.1 mile. You plan to give the driver a $3 tip. Determine the number of miles that can be ridden for a total of $15.
|
4.35
| 0.75 |
The "High School Elite" basketball league consists of $8$ teams. Each team plays every other team three times and also plays $5$ games against non-conference opponents. Calculate the total number of games in a season involving the "High School Elite" league teams.
|
124
| 0.833333 |
Given a square in the coordinate plane with vertices at $(0,0)$, $(100,0)$, $(100,100)$, and $(0,100)$, determine the value of $d$ (to the nearest tenth) for which the probability that a randomly chosen point within the square is within $d$ units of a lattice point is $\tfrac{1}{4}$.
|
0.3
| 0.166667 |
A half-circle sector of a circle of radius $6$ inches together with its interior is rolled up to form the lateral surface area of a right circular cone by taping together along the two radii. Calculate the volume of the cone in cubic inches.
|
9\pi \sqrt{3}
| 0.916667 |
Find the number of terms with rational coefficients in the expansion of $\left(x\sqrt[4]{2}+y\sqrt[5]{3}\right)^{1250}$.
|
63
| 0.75 |
Evaluate $\left(2004 - \left(2011 - 196\right)\right) + \left(2011 - \left(196 - 2004\right)\right)$.
|
4008
| 0.75 |
2-(-3)×2-4-(-5)-6-(-7)×2.
|
17
| 0.833333 |
Let $M$ be the greatest five-digit number whose digits have a product of $180$. Find the sum of the digits of $M$.
|
20
| 0.25 |
Two 8-sided dice are rolled once. Let the sum of the numbers rolled be the diameter of a circle. Determine the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference.
|
\frac{3}{64}
| 0.666667 |
Given that a magician's hat contains 4 blue marbles and 3 yellow marbles, marbles are drawn randomly, one at a time without replacement, until all 4 blue marbles are drawn or until all 3 yellow marbles are drawn, calculate the probability that all 4 blue marbles are drawn.
|
\frac{3}{7}
| 0.083333 |
An 18-inch pizza can perfectly accommodate eight equal circles of pepperoni across its diameter, and thirty-six circles of pepperoni are placed on this pizza without overlapping. Calculate the fraction of the pizza's surface that is covered by pepperoni.
|
\frac{9}{16}
| 0.5 |
Choose three different numbers from the set $\{2,3,5,7,11,13\}$. Add any two of these numbers. Multiply their sum by the third number. What is the smallest result that can be obtained from this process?
|
16
| 0.333333 |
Given the expression \[
\left(\frac{x^3+1}{x}\right)\left(\frac{y^3+1}{y}\right)+\left(\frac{x^3-1}{y}\right)\left(\frac{y^3-1}{x}\right), \quad xy \neq 0
\], simplify the expression.
|
2x^2y^2 + \frac{2}{xy}
| 0.583333 |
Given that \( (a_1, a_2, \ldots, a_{15}) \) is a list of the first 15 positive integers such that for each \( 2 \leq i \leq 15 \), either \( a_i + 1 \) or \( a_i-1 \) (or both) appear somewhere before \( a_i \) in the list, and \( a_1 \) must be either 1 or 2, and \( a_{15} \) must be either 14 or 15, determine the number of such lists.
|
4 \times 4096 = 16384
| 0.166667 |
Five friends did yard work over the weekend, earning amounts of $10, $30, $50, $40, and $70. Calculate the amount the friend who earned $70 will give to the others when they equally share their total earnings.
|
30
| 0.916667 |
Calculate the sum: $\dfrac{2}{100} + \dfrac{5}{1000} + \dfrac{8}{10000} + \dfrac{6}{100000}$.
|
0.02586
| 0.666667 |
The ferry boat begins transporting tourists to an island every hour starting at 9 AM until its last trip, which starts at 4 PM. On the first trip at 9 AM, there were 120 tourists, and on each successive trip, there were 2 fewer tourists than on the previous trip. Determine the total number of tourists the ferry transported to the island that day.
|
904
| 0.916667 |
Given the function g(n) = log<sub>27</sub>n if log<sub>27</sub>n is rational, and 0 otherwise, find the value of the sum from n=1 to 7290 of g(n).
|
12
| 0.083333 |
Given that $40\%$ of all students initially answered "Yes" and $60\%$ answered "No", and at the end of the year $80\%$ answered "Yes" and $20\%$ answered "No", determine the minimum possible percentage of students that changed their mind during the year.
|
40\%
| 0.833333 |
Given that John's flight departed from Chicago at 3:42 PM and arrived in Denver at 6:57 PM, both cities being in the same time zone, compute the sum of the hours and minutes that his flight took.
|
18
| 0.833333 |
Determine the number of points C in a plane that exist such that the perimeter of triangle ABC is 36 units and the area of triangle ABC is 48 square units.
|
4
| 0.166667 |
Given the arithmetic series starting with the term $k^2 - 1$ and each subsequent term increasing by $1$, calculate the sum of the first $2k$ terms of this series.
|
2k^3 + 2k^2 - 3k
| 0.5 |
Given that Liam has three older siblings who are triplets and the product of their four ages is 216, calculate the sum of their four ages.
|
19
| 0.166667 |
Given the set $\{-10, -7, -3, 0, 2, 4, 8, 9\}$, calculate the minimum possible product of three different numbers.
|
-720
| 0.166667 |
Given $a_1,a_2,\dots,a_{2023}$ be a strictly increasing sequence of positive integers such that $a_1+a_2+\cdots+a_{2023}=2023^{2023}$. Determine the remainder when $a_1^3 - a_2^3 + a_3^3 - a_4^3 + \cdots - a_{2022}^3 + a_{2023}^3$ is divided by $6$.
|
1
| 0.5 |
Given two cubic polynomials $P(x)$ and $Q(x)$ where $P(x) = x^3 + 3x^2 - 3x - 9$, construct $Q(x)$ by replacing each nonzero coefficient of $P(x)$ with their arithmetic mean. Determine the value of $Q(1)$.
|
-8
| 0.25 |
A school organizes a talent show with 120 students participating, each capable of singing, dancing, or acting. No student excels in all three. It's known that 50 students cannot sing, 75 students cannot dance, and 35 students cannot act. Determine how many students have exactly two of these talents.
|
80
| 0.916667 |
John covers an initial segment of 40 miles in 4 hours, then takes a 1-hour break, and finally covers another 15 miles in 3 hours. Calculate John's average speed in miles per hour for the entire trip.
|
6.875
| 0.916667 |
Simplify $\left(\sqrt[3]{8} + \sqrt{4 \frac{1}{4}}\right)^2$.
|
\frac{33 + 8\sqrt{17}}{4}
| 0.5 |
Given $\log_{14} \Big(\log_4 (\log_3 y) \Big) = 1$, solve for $y^{-1/3}$.
|
3^{-4^{14}/3}
| 0.916667 |
Given the number $2025$, calculate the number of different four-digit numbers that can be formed by rearranging the four digits.
|
9
| 0.833333 |
Given $x^3y = k$ for a positive constant $k$, find the percentage decrease in $y$ when $x$ increases by $20\%$.
|
42.13\%
| 0.416667 |
Given John owes Jenny $60$ cents and has a pocket full of $5$-cent coins, $20$-cent coins, and $50$-cent coins, calculate the difference between the largest and the smallest number of coins he can use to pay her exactly.
|
9
| 0.833333 |
For \(p=1, 2, \ldots, 10\), let \(S_p\) be the sum of the first 50 terms of the arithmetic progression whose first term is \(p\) and whose common difference is \(3p\). Find the value of \(S_1 + S_2 + \cdots + S_{10}\).
|
204875
| 0.25 |
The sum of the final two numbers after 4 is added to each number and then each of the resulting numbers is tripled is given by what expression in terms of \( S \)?
|
3S + 24
| 0.416667 |
Two subsets of the set S = {a, b, c, d, e, f} are to be chosen so that their union is S and their intersection contains exactly three elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?
|
80
| 0.583333 |
The expression $a^4 - a^{-4} = (a^2 - a^{-2})(a^2 + a^{-2}) = (a^2 - a^{-2})\left( (a^2)^2 + (a^{-2})^2 + 2a^2a^{-2}\right)$. Further simplify this expression to factor it into the product of three simpler expressions.
|
(a - a^{-1})(a + a^{-1})(a^2 + a^{-2})
| 0.083333 |
If $x$ and $y$ are distinct nonzero real numbers such that $x + \frac{3}{x} = y + \frac{3}{y}$ and the sum of $x$ and $y$ is 4, find the product $xy$.
|
3
| 0.916667 |
Star writes down the whole numbers $1$ through $40$. Emilio copies Star's numbers, but he replaces each occurrence of the digit $3$ by the digit $2$. Calculate the difference between Star's sum of the numbers and Emilio's sum of the numbers.
|
104
| 0.333333 |
When the mean, median, and mode of the list
\[4, 9, x, 4, 9, 4, 11, x\]
are arranged in increasing order, they form a non-constant geometric progression. What is the sum of all possible values of $x$?
A) 3
B) 6.83
C) 15.5
D) 17
E) 20
|
15.5
| 0.833333 |
Calculate the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30} \cdot 5^6 \cdot 3}$.
|
30
| 0.166667 |
Positive integers $a$ and $b$ are such that $a < 6$ and $b < 9$. Find the smallest possible value for $3a - 2ab$.
|
-65
| 0.833333 |
When $n$ standard 6-sided dice are rolled, find the smallest possible value of $S$ such that the probability of obtaining a sum of 2000 is greater than zero and is the same as the probability of obtaining a sum of $S$.
|
338
| 0.833333 |
Teams $X$ and $Y$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $X$ has won $\tfrac{3}{4}$ of its games and team $Y$ has won $\tfrac{2}{3}$ of its games. Additionally, team $Y$ has won $5$ more games and lost $5$ more games than team $X$. Determine the number of games that team $X$ has played.
|
20
| 0.916667 |
Determine the number of points $C$ such that the perimeter of $\triangle ABC$ is $60$ units, and the area of $\triangle ABC$ is $120$ square units.
|
4
| 0.083333 |
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