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stringlengths 18
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|---|---|---|
Given a box containing five pieces of paper, each labeled with one of the digits $1$, $2$, $4$, $5$, or $6$, calculate the probability that a three-digit number formed by drawing three of these digits (one at a time without replacement) has a sum of digits that is a multiple of $5$.
|
\frac{1}{5}
| 0.916667 |
What is the hundreds digit of $(25! - 20!)$?
|
0
| 0.916667 |
What is the total cost of three purchases for $\textdollar 2.99$, $\textdollar 6.51$, and $\textdollar 10.49$, rounded to the nearest dollar using the rounding rule where exact half amounts ($x.50$) are rounded down instead of up?
|
20
| 0.416667 |
If the product $\dfrac{5}{3}\cdot \dfrac{6}{5}\cdot \dfrac{7}{6}\cdot \dfrac{8}{7}\cdot \ldots\cdot \dfrac{a}{b} = 16$, calculate the sum of $a$ and $b$.
|
95
| 0.583333 |
Consider the 16 integers from $-8$ to $7$, inclusive, arranged to form a 4-by-4 square. What is the value of the common sum of the numbers in each row, the numbers in each column, and the numbers along each of the main diagonals?
|
-2
| 0.416667 |
Given the polynomial $x^3+2x^2-3$ divided by the polynomial $x^2+2$, determine the remainder.
|
-2x - 7
| 0.833333 |
Find the units digit of the decimal expansion of $\left(12 + \sqrt{245}\right)^{17} + \left(12 + \sqrt{245}\right)^{76}$.
|
6
| 0.166667 |
An equilateral triangle has a side length of 2. Inside it, three congruent isosceles triangles are constructed such that their bases collectively cover the entire perimeter of the equilateral triangle equally. The sum of the areas of these three isosceles triangles equals the area of the equilateral triangle. Find the length of one of the congruent sides of one of the isosceles triangles.
|
\frac{2\sqrt{3}}{3}
| 0.833333 |
Emily traveled the first one-fifth of her journey on a country road, the next 30 miles through a city, and then one-third on a highway before completing the final one-sixth on a rural path. Calculate the total length of Emily's journey in miles.
|
100
| 0.916667 |
Given that $800$ students were surveyed and $320$ students preferred spaghetti and $160$ students preferred manicotti, calculate the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti.
|
2
| 0.5 |
If \( a \) and \( b \) are positive numbers such that \( a^b = b^a \) and \( b = 4a \), find the value of \( a \).
|
\sqrt[3]{4}
| 0.916667 |
While Steve and LeRoy are fishing 2 miles from shore, their boat springs a leak, and water comes in at a constant rate of 12 gallons per minute. The boat will sink if it takes in more than 40 gallons of water. Steve starts rowing towards the shore at a constant rate of 3 miles per hour while LeRoy bails water out of the boat. Determine the slowest rate, in gallons per minute, at which LeRoy must bail if they are to reach the shore without sinking.
|
11
| 0.916667 |
Points $P$ and $Q$ are on a circle of radius $7$ and $PQ = 8$. Point $R$ is the midpoint of the minor arc $PQ$. Calculate the length of the line segment $PR$.
|
\sqrt{98 - 14\sqrt{33}}
| 0.083333 |
A box contains 34 red balls, 25 green balls, 18 yellow balls, 21 blue balls, and 13 purple balls. Calculate the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 18 balls of a single color will be drawn.
|
82
| 0.25 |
Maria buys computer disks at a price of 5 for $7 and sells them at a price of 4 for $7. Find the number of computer disks Maria must sell in order to make a profit of $125.
|
358
| 0.916667 |
Two carts are racing down two parallel hills. The first cart travels $6$ inches in the first second and accelerates so that each successive $1$-second time interval, it travels $8$ inches more than during the previous $1$-second interval. It takes $35$ seconds to reach the bottom of the hill. The second cart starts $2$ seconds after the first and travels $7$ inches in the first second, then each successive $1$-second time interval, it travels $9$ inches more than the previous interval. Find the distance, in inches, the second cart travels by the time the first cart reaches the bottom of the hill.
|
4983
| 0.833333 |
A circle with area $A_1$ is inside a larger circle with area $A_1 + A_2$, where the larger circle has a radius of 5. If $A_1, A_2, A_1 + A_2$ are in an arithmetic progression, find the radius of the smaller circle.
|
\frac{5\sqrt{3}}{3}
| 0.75 |
A pair of standard 10-sided dice is rolled once, each die numbered from 1 to 10. The sum of the numbers rolled determines the diameter of a circle. Calculate the probability that the numerical value of the circle's area is less than the numerical value of its circumference.
|
\frac{3}{100}
| 0.833333 |
The average age of 8 people in a room is 28 years. A 20-year-old person leaves the room. Then a 25-year-old person enters the room. Calculate the new average age of the people now in the room.
|
28.625
| 0.916667 |
Find the number of positive factors of 90 that are also multiples of 6.
|
4
| 0.5 |
Given a team with 25 players, where each player must receive a minimum salary of $18,000, and the total salary for all players cannot exceed $900,000, calculate the maximum possible salary, in dollars, for any single player on the team.
|
468,000
| 0.666667 |
Starting from "1," determine the fewest number of keystrokes needed to reach "480" using only the keys [+1] and [x2].
|
11
| 0.583333 |
Given strides, leaps, and bounds are specific units of length. If $3$ strides equals $4$ leaps, $5$ bounds equals $7$ strides, and $2$ bounds equals $9$ meters, then determine the number of leaps in one meter.
|
\frac{56}{135}
| 0.583333 |
If $h(x) = 1 - 2x^2$ and $f(h(x)) = \frac{1-2x^2}{x^2}$ when $x\not=0$, find $f(\frac{3}{4})$.
|
6
| 0.916667 |
Ed and Rose start by taking turns counting aloud, beginning with Rose saying the number $4$. Each subsequent number is obtained by adding $3$ to the last number spoken by the other person. Find the $26^{\text{th}}$ number spoken in this sequence.
|
79
| 0.833333 |
Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of $12$ hours per week helping around the house for $6$ weeks. For the first $5$ weeks, she helps around the house for $10$, $13$, $9$, $14$, and $11$ hours. Find the number of hours that Theresa must work in the final week to earn the tickets.
|
15
| 0.833333 |
Suppose $d$ is a digit. For how many values of $d$ is $2.0d05 > 2.010$?
|
9
| 0.333333 |
Given externally tangent circles with centers at points $A$ and $B$ and radii of lengths $6$ and $4$, respectively, a line externally tangent to both circles intersects ray $AB$ at point $C$. Calculate $BC$.
|
20
| 0.083333 |
Josh has $40$ thin rods, one each of every integer length from $1$ cm through $40$ cm. Given that he places the rods with lengths $5$ cm, $12$ cm, and $20$ cm on a table, how many of the remaining rods can he choose as the fourth rod to form a quadrilateral with positive area?
|
30
| 0.166667 |
A 7 × 7 grid of square blocks is arranged. Determine how many different combinations of 4 blocks can be selected from this set such that no two blocks are in the same row or column.
|
29400
| 0.5 |
Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from exactly half of the other 7 balls.
|
\frac{35}{128}
| 0.75 |
Carlos took $80\%$ of a whole pie, and Maria took one fourth of the remainder. Calculate the portion of the whole pie that was left.
|
15\%
| 0.833333 |
Given that 50% of the students at Central College like dancing, and those who like dancing have a 30% false negative rate, with the remaining 70% accurately expressing their liking, and those who dislike dancing have an 20% false positive rate, with the remaining 80% accurately expressing their dislike, find the fraction of students who say they dislike dancing who actually like dancing.
|
\frac{3}{11}
| 0.75 |
A cylindrical tank with radius $6$ feet and height $12$ feet is lying on its side. The tank is filled with water to a depth of $3$ feet. Calculate the volume of the water in the tank in cubic feet.
|
144\pi - 108\sqrt{3}
| 0.666667 |
Chelsea is 60 points ahead halfway through a 120-shot archery competition, where each bullseye scores 10 points and other possible scores are 7, 3, 1, and 0. Chelsea consistently scores at least 3 points per shot. Calculate the minimum number of consecutive bullseyes Chelsea needs over her next $n$ shots to secure victory, assuming her competitor can score a maximum in each shot hereafter.
|
52
| 0.083333 |
Given that $\angle XYZ = 40^\circ$ and $\angle XYW = 15^\circ$, determine the smallest possible degree measure for $\angle WYZ$.
|
25
| 0.916667 |
Twenty-five percent less than 80 is twenty-five percent more than what number?
|
48
| 0.916667 |
Given that the last initial of Mr. and Mrs. Alpha's baby's monogram is 'A', determine the number of possible monograms in alphabetical order with no letter repeated.
|
300
| 0.333333 |
Olivia drove the first quarter of her trip on a mountain road, the next $30$ miles on a paved road, and the remaining one-sixth on a desert road. What is the total length of Olivia's entire trip?
|
\frac{360}{7}
| 0.916667 |
If $2(5x + 3\pi)=Q$, evaluate the expression $4(10x + 6\pi + 2)$.
|
4Q + 8
| 0.5 |
The Eagles won 3 out of the 5 games they initially played against the Hawks. If the Hawks won all subsequent games that they played against the Eagles, determine the minimum number of additional games they must win to reach a winning percentage of at least 90%.
|
25
| 0.666667 |
Evaluate the expression $(3(3(3(3(3 + 2) + 2) + 2) + 2) + 2)$.
|
485
| 0.666667 |
Let $\omega = -\frac{1}{2} + \frac{i\sqrt{3}}{2}$ and $\omega^2 = -\frac{1}{2} - \frac{i\sqrt{3}}{2}$. Define $T$ as the set of all points in the complex plane of the form $a + b\omega + c\omega^2 + d$, where $0 \leq a, b, c \leq 1$ and $d \in \{0, 1\}$. Find the area of $T$.
|
3\sqrt{3}
| 0.166667 |
Given the expression $\sqrt{36\sqrt{12\sqrt{9}}}$, evaluate the value of this expression.
|
6\sqrt{6}
| 0.916667 |
Given that $\operatorname{log}_{8}(p) = \operatorname{log}_{12}(q) = \operatorname{log}_{18}(p-q)$, calculate the value of $\frac{q}{p}$.
|
\frac{\sqrt{5} - 1}{2}
| 0.333333 |
Segments AD=12, BE=8, CF=20 are drawn from vertices of triangle ABC, each perpendicular to a straight line RS, not intersecting the triangle. Points D, E, F are the intersection points of RS with the perpendiculars. If x is the length of the perpendicular segment GH drawn to RS from the centroid G of the triangle, calculate the value of x.
|
\frac{40}{3}
| 0.833333 |
Calculate the probability of a contestant winning the quiz given that they answer at least 3 out of 4 questions correctly, assuming random guesses for each question with 3 possible answer options.
|
\frac{1}{9}
| 0.916667 |
A student must schedule 3 mathematics courses — algebra, geometry, and number theory — in a 7-period day, such that at most one pair of mathematics courses can be taken in consecutive periods. Calculate the total number of ways this can be done.
|
180
| 0.083333 |
Three fair tetrahedral dice (four sides each) are tossed at random. Calculate the probability that the three numbers turned up can be arranged to form an arithmetic progression with a common difference of one.
|
\frac{3}{16}
| 0.833333 |
Ray's car averages 50 miles per gallon of gasoline, and Tom's car averages 15 miles per gallon of gasoline. If they both drive the same number of miles, calculate their cars' combined rate of miles per gallon of gasoline.
|
\frac{300}{13}
| 0.916667 |
Let $n = \frac{3xy}{x-y}$. Solve for $y$ in terms of $x$ and $n$.
|
\frac{nx}{3x + n}
| 0.666667 |
What is the minimum test score Josanna would need on her next test to increase her test average by $10$ points given that she received scores of $75, 85, 65, 95, 70$ on her previous tests?
|
138
| 0.166667 |
A painting $4$ feet wide is hung in the center of a wall that is $26$ feet wide. Calculate the distance from the end of the wall to the nearest edge of the painting.
|
11
| 0.833333 |
Given $T' = \{(0,0),(5,0),(0,4)\}$ be a triangle in the coordinate plane, determine the number of sequences of three transformations—comprising rotations by $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, and reflections across the lines $y = x$ and $y = -x$—that will map $T'$ onto itself.
|
12
| 0.166667 |
Given a wooden cube $n$ units on a side is painted blue on all six faces and then cut into $n^3$ unit cubes, exactly one-third of the total number of faces of the unit cubes are blue. What is $n$?
|
3
| 0.916667 |
Given two positive numbers $p$ and $q$ that are in the ratio $3:5$, and the sum of the squares of $p$ and $q$ equals $2k$, calculate the smaller of $p$ and $q$.
|
3\sqrt{\frac{k}{17}}
| 0.666667 |
Given that the number $3103$ has four digits, determine the number of different four-digit numbers that can be formed by rearranging these digits.
|
9
| 0.833333 |
Mia drove the first one-fourth of her journey on a highway, the next 25 miles through a town, and the final one-sixth on a country road. Calculate the total length of Mia's trip in miles.
|
\frac{300}{7}
| 0.916667 |
Given distinct points $G$, $H$, $I$, and $J$ lie on a line, with $GH = HI = \frac{1}{2}$, and $IJ = 1$, and points $K$ and $L$ lie on a second line, parallel to the first, with $KL = 2$, determine the number of possible values for the area of a triangle formed by selecting any three of the six points.
|
4
| 0.083333 |
In an arithmetic sequence, the first term is $3$, the last term is $34$, and the sum of all the terms is $222$. Find the common difference.
|
\frac{31}{11}
| 0.916667 |
Let $x = 0.\overline{527}$, and express $x$ in its simplest fractional form, and then calculate the sum of the numerator and denominator.
|
1526
| 0.916667 |
Points $A(13, 11)$ and $B(5, -1)$ are vertices of $\triangle ABC$ with $AB=AC$. The altitude from $A$ meets the opposite side at $D(2, 7)$. Determine the coordinates of point $C$.
|
(-1, 15)
| 0.583333 |
The numbers $-3, 5, 7, 10, 15$ are rearranged according to the following rules:
1. The largest number is not in the last place, but it is within the last three places.
2. The smallest number is not in the first place, but it is within the first three places.
3. The median number is neither in the first nor in the last place. Determine the sum of the second and fourth numbers.
|
12
| 0.25 |
A digital watch displays hours and minutes in a 24-hour format. What is the largest possible sum of the digits in the display?
|
24
| 0.083333 |
Given a circle with radius 10 units, and two semicircles within it, one with radius 8 units and another with radius 6 units, both passing through the center of the circle, determine the ratio of the combined areas of these two semicircles to the area of the larger circle.
|
\frac{1}{2}
| 0.916667 |
Given the progression \(8^{\frac{2}{11}}, 8^{\frac{3}{11}}, 8^{\frac{4}{11}}, \dots, 8^{\frac{(n+1)}{11}}\), calculate the least positive integer \(n\) such that the product of the first \(n\) terms of the progression exceeds 1,000,000.
|
11
| 0.5 |
For how many positive integer values of $N$ does the expression $\dfrac{48}{N+3}$ result in an integer?
|
7
| 0.916667 |
Given the inequality $(x-1)^2 < 12 - x$, determine the range of values for $x$ that satisfy this inequality.
|
\left(\frac{1 - 3\sqrt{5}}{2}, \frac{1 + 3\sqrt{5}}{2}\right)
| 0.75 |
What is the least possible value of $(xy + 2)^2 + (x - y)^2$ for real numbers $x$ and $y$?
|
4
| 0.75 |
Given that the probability of a boy being chosen is $\frac{3}{4}$ of the probability that a girl is chosen, find the ratio of the number of boys to the total number of boys and girls in the class.
|
\frac{3}{7}
| 0.916667 |
On hypotenuse $AB$ of a right triangle $ABC$, a second right triangle $ABD$ is constructed with hypotenuse $AB$. If $\overline{BC} = 3$, $\overline{AC} = a$, and $\overline{AD} = 1$, then calculate $\overline{BD}$.
|
$\sqrt{a^2 + 8}$
| 0.916667 |
Given that there are three times as many third graders as fourth graders, twice as many fourth graders as fifth graders, and the average daily running times for third, fourth, and fifth graders are 14, 18, and 8 minutes, respectively, calculate the average number of minutes run per day by these students.
|
\frac{128}{9}
| 0.75 |
Given that Thomas the painter initially had enough paint for 50 identically sized rooms, and 4 cans of paint were lost, determining how many cans of paint he used for 42 rooms.
|
21
| 0.333333 |
Given $S = i^n + i^{-2n}$, where $i = \sqrt{-1}$ and $n$ is an integer, determine the total number of possible distinct values for $S$.
|
4
| 0.916667 |
Determine the times between $7$ and $8$ o'clock, to the nearest minute, when the hands of a clock will form an angle of $120^{\circ}$.
|
7:16
| 0.583333 |
A retailer purchases a product at $50 less 20%. He then intends to sell the product at a profit of 25% relative to his cost after applying a 10% discount on his marked price. What should the marked price be, in dollars?
|
55.56
| 0.25 |
Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. Determine the number of two-digit values of $x$ for which $\clubsuit(\clubsuit(x))=4$.
|
10
| 0.583333 |
Consider the year 2020, which is not a palindrome. What is the product of the digits of the next year after 2020 that reads the same forwards and backwards?
|
4
| 0.333333 |
Given the sum of the first 15 terms of an arithmetic progression is four times the sum of the first 8 terms, find the ratio of the first term to the common difference.
|
-\frac{7}{17}
| 0.833333 |
Simplify the expression: $\left(\frac{(x+1)^{3}(x^{2}-x+1)^{3}}{(x^{3}+1)^{3}}\right)^{3}\cdot\left(\frac{(x-1)^{3}(x^{2}+x+1)^{3}}{(x^{3}-1)^{3}}\right)^{3}$.
|
1
| 0.833333 |
Given that point $E$ lies on $\overline{AB}$ and line $\overline{ED}$ bisects $\angle ADC$, determine the ratio of the area of $\triangle AED$ to the area of square $ABCD$.
|
\frac{1}{2}
| 0.083333 |
Given that $a$ cows give $b$ gallons of milk in $c$ days with a standard efficiency factor of 100%, calculate the number of gallons of milk $d$ cows will give in $e$ days when the efficiency declines by 10%.
|
\frac{0.9bde}{ac}
| 0.75 |
The maximum number of possible points of intersection of a circle and a rectangle.
|
8
| 0.75 |
Given six test scores have a mean of $85$, a median of $86$, and a mode of $88$. Determine the sum of the two lowest test scores.
|
162
| 0.833333 |
In a triangle, one side measures 16 inches and the angle opposite this side is 45°. Calculate the diameter of the circumscribed circle surrounding this triangle.
|
16\sqrt{2}\text{ inches}
| 0.916667 |
Determine the sum of the prime factors of the number $2550$.
|
27
| 0.916667 |
Given the sequence $1, -2, 3, -4, 5, -6, \ldots, $ whose nth term is $(-1)^{n+1}\cdot n$, determine the average of the first $201$ terms of this sequence.
|
\frac{101}{201}
| 0.833333 |
Let $S=\{(x,y) : x\in \{0,1,2,3\}, y\in \{0,1,2,3\},\text{ and } (x,y)\ne (2,2)\}$. Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle{ABC}$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $B$, find the product of the values of $\tan(\angle{ACB})$ for every $t$ in $T$.
|
1
| 0.833333 |
A quadrilateral has vertices $P(a+1,b-1)$, $Q(b+1,a-1)$, $R(-a-1, -b+1)$, and $S(-b-1, -a+1)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $24$. Calculate the value of $a+b$.
|
6
| 0.333333 |
Given Paula initially planned to paint 50 rooms with a total amount of paint that is just enough to cover 40 rooms after losing 5 cans, calculate the number of cans of paint she used for these 40 rooms.
|
20
| 0.666667 |
A palindrome between $1000$ and $10,000$ is chosen at random. Calculate the probability that it is divisible by $11$.
|
1
| 0.833333 |
Given that point B is the vertex of three angles $\angle \text{ABC}$, $\angle \text{ABD}$, and $\angle \text{CBD}$, where $\angle \text{CBD}$ is a right angle and the total sum of angles around point B is $180^\circ$, and $\angle \text{ABD} = 30^\circ$, find the measure of $\angle \text{ABC}$.
|
60^\circ
| 0.916667 |
Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar 1 the ratio of blue to green marbles is 7:3, and the ratio of blue to green marbles in Jar 2 is 9:1. There are 80 green marbles in all. Calculate the difference in the number of blue marbles in Jar 1 and Jar 2.
|
40
| 0.916667 |
Five friends earned $18, $23, $28, $35, and $45. If they split their earnings equally among themselves, how much will the friend who earned $45 need to give to the others?
|
15.2
| 0.833333 |
Given an unfair die that has a probability 4 times as likely to result in an odd number as an even number, three such dice are rolled. Find the probability that the sum of the numbers rolled is odd.
|
\frac{76}{125}
| 0.916667 |
Given that the roots of the polynomial $Q(x) = x^3 + px^2 + qx + r$ are $\cos \frac{\pi}{7}, \cos \frac{3\pi}{7},$ and $\cos \frac{5\pi}{7}$, where angles are in radians, determine the value of $pqr$.
|
1. Let $z = e^{\frac{\pi i}{7}}$. The roots of $Q(x) = x^3 + px^2 + qx + r$ are $\cos \frac{\pi}{7}, \cos \frac{3\pi}{7},$ and $\cos \frac{5\pi}{7}$. Using Euler's formula, this can be expressed as:
\[
\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}.
\]
2. The sum of all the real parts (cosines) of $z^k$, where $k=1$ to $6$, noting that $\cos \theta = \cos (2\pi - \theta)$ and $\sum_{k=1}^{6} z^k = -1$, simplifies to:
\[
\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7} = -\frac{1}{2}.
\]
3. Using trigonometric identities for sum of cosines and polynomial coefficients, the polynomial becomes:
\[
x^3 + \frac{1}{2}x^2 - \frac{1}{2}x - \frac{1}{8} = 0.
\]
Here, $p = \frac{1}{2}$, $q = -\frac{1}{2}$, and $r = -\frac{1}{8}$.
4. The product of the coefficients $p$, $q$, and $r$ is given by $pqr = \left(\frac{1}{2}\right)\left(-\frac{1}{2}\right)\left(-\frac{1}{8}\right) = \frac{1}{32}.$
| 0.166667 |
If \(\log_4(\log_5(\log_6 x))=\log_5(\log_6(\log_4 y))=\log_6(\log_4(\log_5 z))=0\), calculate the sum of \(x\), \(y\), and \(z\).
|
12497
| 0.583333 |
Suppose that $\tfrac{3}{4}$ of $14$ apples are worth as much as $9$ lemons. How many lemons are worth as much as $\tfrac{5}{7}$ of $7$ apples?
|
\frac{30}{7}
| 0.916667 |
In a pentagon $ABCDE$, $\angle A = 100^\circ$, $\angle D = 120^\circ$, and $\angle E = 80^\circ$. If $\angle ABC = 140^\circ$, find the measure of $\angle BCD$.
|
100^\circ
| 0.666667 |
Let $F=\frac{8x^2+20x+5m}{8}$ be the square of an expression which is linear in $x$. Find the range in which $m$ lies for this condition to hold.
|
2.5
| 0.083333 |
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