problem
stringlengths 18
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|---|---|---|
Given the city's water tower stands 50 meters high and holds 150,000 liters of water, and Maya's miniature version holds 0.2 liters of water, determine the height of Maya's model water tower.
|
0.55
| 0.5 |
Evaluate the expression $2^{-(3k+2)} - 3\cdot2^{-(3k)} + 2^{-3k}$ for any integer $k$.
|
-\frac{7}{4} \cdot 2^{-3k}
| 0.583333 |
Evaluate the limit of $\frac{x^3 - 3x + 2}{x - 1}$ as $x$ approaches $1$.
|
0
| 0.916667 |
Given that the really non-overlapping minor arcs PQ, QR, and RP in circle O are x+85°, 2x+15°, and 3x-32°, respectively, find one interior angle of triangle PQR.
|
57
| 0.25 |
Given the 9 data values $70, 110, x, 60, 50, 220, 100, x, 90$ are such that the mean, median, and mode are all equal to $x$, determine the value of $x$.
|
100
| 0.75 |
Suppose that $\tfrac{3}{4}$ of $16$ apples are worth as much as $6$ pears. How many pears are worth as much as $\tfrac{1}{3}$ of $9$ apples?
|
1.5
| 0.583333 |
Given the equation $\sqrt{9 - 3x} = x\sqrt{9 - 9x}$, determine the number of roots that satisfy this equation.
|
0
| 0.416667 |
Consider two points $(a, b)$ and $(c, d)$ on the parabola whose equation is $y = mx^2 + k$. Determine the distance between these two points in terms of $a, c, m$.
|
|c - a|\sqrt{1 + m^2(c+a)^2}
| 0.75 |
A construction manager calculates that one of his two workers would take 8 hours and the other 12 hours to complete a particular paving job alone. When both workers collaborate, their combined efficiency is reduced by 8 units per hour. The manager decides to assign them both to speed up the job, and it takes them precisely 6 hours to finish. What is the total number of units paved?
|
192
| 0.25 |
Given the function $y = 2(x-a)^2 + 3(x-b)^2$, determine the value of $x$ that minimizes $y$.
|
\frac{2a + 3b}{5}
| 0.583333 |
Tickets to a musical cost $x$ dollars each, where $x$ is a whole number. A group of 8th graders buys tickets costing a total of $120$, a group of 9th graders buys tickets for $180$, and a group of 10th graders buys tickets for $240$. Determine the number of possible values for $x$.
|
12
| 0.916667 |
Given that Alice paid $120, Bob paid $150, and Charlie paid $180, calculate how much Bob should give to Alice so that all three of them share the costs equally.
|
0
| 0.166667 |
Given a $60$-question multiple choice test, students earn $5$ points for a correct answer, $0$ points for an answer left blank, and $-2$ points for an incorrect answer. Owen's total score was $150$. Determine the maximum number of questions that Owen could have answered correctly.
|
38
| 0.833333 |
Five fish can be traded for three loaves of bread, and one loaf of bread can be traded for six bags of rice, and three bags of rice can be traded for two apples. Express the number of apples equivalent to one fish in terms of a fraction.
|
\frac{12}{5}
| 0.916667 |
The cost for the first $\frac{3}{4}$ mile is $3.50, and the remaining cost for additional mileage is $0.25 for each $0.1 mile. If you plan to give the driver a $3 tip, calculate the number of miles you can ride for a total cost of $15.
|
4.15
| 0.75 |
A person drives z miles due east at a speed of 3 minutes per mile, and then returns to the starting point driving due west at 3 miles per minute. Determine the average speed for the entire round trip in miles per hour.
|
36
| 0.5 |
Determine the value of $k$ such that the polynomial $2x^3 - 8x^2 + kx - 10$ is divisible by $x-2$.
|
k = 13
| 0.916667 |
Given the teacher has 30 cookies to distribute among five students so that each student receives at least three cookies, find the number of ways the cookies can be distributed.
|
3876
| 0.916667 |
Evaluate the expression $3 + \sqrt{3} + \frac{1}{3 + \sqrt{3}} + \frac{1}{\sqrt{3} - 3}.$
|
3 + \frac{2\sqrt{3}}{3}
| 0.916667 |
On circle $O$, points $E$ and $F$ are on the same side of diameter $\overline{AB}$, $\angle AOE = 40^{\circ}$, and $\angle FOB = 60^{\circ}$. Find the ratio of the area of the smaller sector $EOF$ to the area of the circle.
|
\frac{2}{9}
| 0.166667 |
If $x$ cows give $x+1$ cans of milk in $x+2$ days, determine the number of days it will take $x+4$ cows to give $x+7$ cans of milk.
|
\frac{x(x+2)(x+7)}{(x+1)(x+4)}
| 0.083333 |
Determine the largest number by which the expression $n^4 - n^2$ is divisible for all possible integral values of $n$.
|
12
| 0.416667 |
Determine the distance in feet between the 5th red light and the 23rd red light, where the lights are hung on a string 8 inches apart in the pattern of 3 red lights followed by 4 green lights. Recall that 1 foot is equal to 12 inches.
|
28
| 0.166667 |
What is the value of \[\left(\sum_{k=1}^{50} \log_{10^k} 2^{k^2}\right)\cdot\left(\sum_{k=1}^{10} 3^k \log_{27^k} 125^k\right)?\]
A) $752986 \qquad$
B) $75298600 \qquad$
C) $752986000 \qquad$
D) $7529860 \qquad$
E) $59048$
|
75298600
| 0.333333 |
Elena earns 25 dollars per hour, of which $2\%$ is deducted to pay local taxes and an additional $1.5\%$ for health benefits. Calculate the total amount of cents per hour of Elena's wages that are used to pay these deductions.
|
87.5
| 0.666667 |
Let $x = .123456789101112....497498499$, where the digits are obtained by writing the integers $1$ through $499$ in order. Calculate the $1000$th digit to the right of the decimal point.
|
3
| 0.916667 |
Given $(-\frac{1}{343})^{-5/3} + 1$, simplify the expression.
|
-16806
| 0.833333 |
If Lena has a $3 \times 7$ index card, and shortening the length of one side of this card by $2$ inches results in an area of $15$ square inches, find the area of the card in square inches if she shortens the length of the other side by $2$ inches.
|
7
| 0.583333 |
Given the taxi fare of $3.00 for the first $\frac{1}{2}$ mile and then $0.30 for each additional 0.1 mile, and a $3 tip, calculate the number of miles you can ride for a total of $15.
|
3.5
| 0.916667 |
For how many $n$ in $\{1, 2, 3, ..., 50 \}$ is the tens digit of $n^2$ odd when $n$ ends in 3 or 7?
|
0
| 0.75 |
If Haruto has a $6 \times 8$ index card, and he shortens the length of one side by $2$ inches and the area would be $36$ square inches, determine the area of the card in square inches if he instead shortens the length of the other side by $2$ inches.
|
32
| 0.666667 |
Given a list of $2057$ positive integers with a unique mode occurring exactly $15$ times, find the least number of distinct values that can occur in the list.
|
147
| 0.25 |
Given Ben's current test scores are $95, 85, 75, 65,$ and $90$, find the minimum score he needs for the next test to increase his average score by at least $4$ points.
|
106
| 0.75 |
How many 4-digit numbers greater than 1000 can be formed using all the digits from the number 2013?
|
18
| 0.833333 |
Let $f(x) = |x-3| + |x-5| - |2x-8|$ for $3 \leq x \leq 10$. Find the sum of the largest and smallest values of $f(x)$.
|
2
| 0.916667 |
Determine the number of distinct terms in the simplified form of the expansion \( [(2a+4b)^2(2a-4b)^2]^3 \).
|
7
| 0.75 |
How many $3$-digit positive integers have digits whose product equals $30$ and at least one of the digits is an even number?
|
12
| 0.333333 |
Determine the simplest form of \( T = (x-2)^4 + 4(x-2)^3 + 6(x-2)^2 + 4(x-2) + 1 \).
|
(x-1)^4
| 0.916667 |
The sum of the numerical coefficients in the expansion of the binomial $(a-b)^7$ is what value?
|
0
| 0.916667 |
Let \( S \) be a set of \( 7 \) integers taken from \( \{1, 2, \dots, 15\} \) with the property that if \( a \) and \( b \) are elements of \( S \) with \( a < b \), then \( b \) is not a multiple of \( a \). Find the median value of an element in \( S \).
|
11
| 0.25 |
Determine the least positive period $q$ of the functions $g$ such that $g(x+2) + g(x-2) = g(x)$ for all real $x$.
|
12
| 0.5 |
Chloe selects a real number uniformly at random from the interval $[0, 2020]$. Independently, Laurent selects a real number uniformly at random from the interval $[0, 4040]$. Calculate the probability that Laurent's number is greater than Chloe's number.
|
\frac{3}{4}
| 0.166667 |
Given the equation $x^4 + y^2 = 4y$, find the number of ordered pairs of integers $(x, y)$ that satisfy this equation.
|
2
| 0.666667 |
What is the value of $3456 + 4563 + 5634 + 6345$?
|
19998
| 0.333333 |
Given positive integers $a$ and $b$ are members of a set where $a \in \{2, 3, 5, 7\}$ and $b \in \{2, 4, 6, 8\}$, and the sum of $a$ and $b$ must be even, determine the smallest possible value for the expression $2 \cdot a - a \cdot b$.
|
-12
| 0.166667 |
Simplify the expression $\sqrt{1+ \left (\frac{x^6-1}{3x^3} \right )^2}$.
|
\frac{\sqrt{x^{12} + 7x^6 + 1}}{3x^3}
| 0.666667 |
What is the smallest prime number dividing the sum $2^{14} + 7^{12}$?
|
5
| 0.833333 |
Consider a sequence defined by $y_{k+1} = y_k + k$ for $k=1, 2, \dots, n-1$ and $y_1=2$, find the value of $y_1 + y_2 + \dots + y_n$.
|
2n + \frac{(n-1)n(n+1)}{6}
| 0.083333 |
Find the area bounded by the x-axis, the line x = 10, and the curve defined by y = x^2 when 0 ≤ x ≤ 3, and y = 3x - 6 when 3 ≤ x ≤ 10.
|
103.5
| 0.916667 |
The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In three years, Jack will be three times as old as Bill will be then. Calculate the difference in their current ages.
|
36
| 0.75 |
Given two integers have a sum of 34, and when two more integers are added to the first two integers, the sum is 51, and finally, when two more integers are added to the sum of the previous four integers, the sum is 72, determine the minimum number of odd integers among the 6 integers.
|
2
| 0.666667 |
The number of revolutions of a wheel, with fixed center and with an outside diameter of $8$ feet, required to cause a point on the rim to go one mile is to be calculated.
|
\frac{660}{\pi}
| 0.916667 |
The point $P$ moves circumferentially around the square $ABCD$. Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ respectively. Considering this motion, how many of the four quantities listed change: the length of the diagonal $AC$, the perimeter of square $ABCD$, the area of square $ABCD$, the area of quadrilateral $MBCN$?
|
1
| 0.5 |
Two concentric circles have radii $2$ and $4$. Find the probability that the chord joining two points chosen independently and uniformly at random on the outer circle does not intersect the inner circle.
|
\frac{2}{3}
| 0.083333 |
Square $ABCD$ has side length $10$ and $\angle B = 90$°. Region $R$ consists of all points inside the square that are closer to vertex $B$ than any of the other three vertices. Find the area of $R$.
|
25
| 0.333333 |
At the end of $1994$, Walter was one-third as old as his mother. The sum of the years in which they were born was $3900$. Calculate Walter's age at the end of $2004$.
|
32
| 0.583333 |
How many different four-digit numbers can be formed by rearranging the four digits in $2023$?
|
9
| 0.833333 |
Consider a positive real number to be very special if it has a decimal representation that consists entirely of digits $0$ and $5$. For instance, $\frac{500}{99}= 5.\overline{05}= 5.050505\cdots$ and $55.005$ are very special numbers. Find the smallest $n$ such that $1$ can be written as a sum of $n$ very special numbers using only up to three decimal places in each number.
|
2
| 0.333333 |
A rectangular prism has dimensions $A$, $B$, and $C$ with face areas $30$, $30$, $40$, $40$, $60$, and $60$ square units, respectively. Find $A + B + C$.
|
9\sqrt{5}
| 0.083333 |
How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ so that the mean of the remaining numbers is $5$?
|
4
| 0.916667 |
Simplify the expression $(-\frac{1}{343})^{-2/3}$.
|
49
| 0.916667 |
In a sequence where integer $n$ appears $n$ times for $1 \leq n \leq 300$, find the median of this list.
|
212
| 0.333333 |
A right circular cone and a sphere share the same radius, $r$. The volume of the cone is one-third that of the sphere. Determine the ratio of the cone's altitude, $h$, to the sphere's radius.
|
\frac{4}{3}
| 0.916667 |
The smallest positive integer n such that $\sqrt{n} - \sqrt{n-1} < 0.005$
|
10001
| 0.666667 |
Three distinct numbers are randomly selected from the set $\{-3, -2, 0, 1, 2, 5, 6\}$. Calculate the probability that the product of these three numbers is positive.
|
\frac{8}{35}
| 0.5 |
Ace runs with constant speed and Flash runs \( x^2 \) times as fast, where \( x>1 \). Flash gives Ace a head start of \( y^2 \) yards, and, at a given signal, they start off in the same direction. Determine the number of yards Flash must run to catch Ace.
|
\frac{x^2y^2}{x^2 - 1}
| 0.833333 |
Given that Anne, Cindy, and Ben repeatedly take turns tossing a die in the order Anne, Cindy, Ben, find the probability that Cindy will be the first one to toss a five.
|
\frac{30}{91}
| 0.083333 |
Given that a CNC machine starts its daily operations at $9:00 \text{ AM}$ and completes one-fourth of the day's planned tasks by $12:30 \text{ PM}$, calculate the time at which the CNC machine will finish all tasks.
|
11:00 \text{ PM}
| 0.833333 |
Let $n$ be the number of ways $15$ dollars can be changed into nickels and half-dollars, with at least one of each coin being used. Determine the value of $n$.
|
29
| 0.916667 |
Given the numbers 144 and 756, determine the ratio of the least common multiple to the greatest common factor.
|
84
| 0.833333 |
How many pairs $(m,n)$ of integers satisfy the equation $m + n = mn - 3$?
|
6
| 0.833333 |
In a city of 400 adults, every adult owns a car, a motorcycle, or both, and some also own bicycles. If 350 adults own cars, 60 adults own motorcycles, and 30 adults own bicycles, calculate the number of car owners who do not own a motorcycle but may or may not own a bicycle.
|
340
| 0.75 |
Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x))=4$?
|
10
| 0.666667 |
Sarah's flight took off from Denver at 11:07 AM and landed in Chicago at 2:45 PM. Both cities are in the same time zone. If her flight took $h$ hours and $m$ minutes, with $0 < m < 60$, find the sum $h + m$.
|
41
| 0.833333 |
Given the product of 0.125 and 3.84, calculate the correct result if the decimal points were included.
|
0.48
| 0.833333 |
Given a circle is inscribed in a square, with the circle tangent to two adjacent sides of the square at points $B$ and $C$, determine the fraction of the area of the square that lies outside the circle.
|
1 - \frac{\pi}{4}
| 0.916667 |
In a particular year, the price of a commodity increased by 15% in January, decreased by 25% in February, increased by 30% in March, and fell by \(x\%\) in April. At the end of April, the price of the commodity was the same as it was at the beginning of January. Determine \(x\) to the nearest integer.
|
11
| 0.916667 |
A right triangle $ABC$ with sides $AB = 12$ cm and $BC = 12$ cm is repeatedly subdivided by joining the midpoints of each triangle’s sides, and the bottom-left triangle is shaded in each iteration. Calculate the total shaded area after 100 iterations.
|
24 \text{ cm}^2
| 0.833333 |
Given that $\log_b 256$ is a positive integer, calculate the number of positive integers $b$.
|
4
| 0.833333 |
Given a positive integer $n$ that has $72$ divisors and $5n$ has $96$ divisors, find the greatest integer $k$ such that $5^k$ divides $n$.
|
2
| 0.833333 |
Given a baseball league consisting of two five-team divisions, where each team plays every other team in its division N games and each team plays every team in the other division M games with N > 2M and M > 4, and each team plays an 82 game schedule, determine the number of games a team plays within its own division.
|
52
| 0.833333 |
What is the greatest number of consecutive integers whose sum is $36$?
|
72
| 0.25 |
Evaluate the expression $\sqrt{\log_4{8} + \log_8{16}} + \sqrt{\log_2{8}}$.
|
\sqrt{\frac{17}{6}} + \sqrt{3}
| 0.666667 |
What is the largest number of solid \(1\text{-in} \times 1\text{-in} \times 2\text{-in}\) blocks that can fit in a \(4\text{-in} \times 3\text{-in} \times 2\text{-in}\) box?
|
12
| 0.5 |
If Amara has a $5 \times 7$ index card, and if she shortens one side of this card by $2$ inches, the card would have an area of $21$ square inches, determine the area of the card if instead she reduces the length of the other side by $2$ inches.
|
25
| 0.666667 |
Two cyclists are $2k$ miles apart. When traveling in the same direction, they meet in $3r$ hours, and when traveling in opposite directions, they pass each other in $2t$ hours. Given that the speed of the faster cyclist is twice the speed of the slower cyclist when they travel towards each other, find the ratio of the speed of the faster cyclist to that of the slower cyclist when they travel in the same direction.
|
2
| 0.916667 |
From the set $\{ -3, 0, 0, 4, 7, 8\}$, find the probability that the product of two randomly selected numbers is $0$.
|
\frac{3}{5}
| 0.166667 |
Given that Ahn chooses a two-digit integer, subtracts twice the integer from 300, and triples the result, find the largest number Ahn can get.
|
840
| 0.916667 |
Determine the smallest number of students that could be participating from these grades, given that the ratio of $10^\text{th}$-graders to $8^\text{th}$-graders is $3:2$, and the ratio of $10^\text{th}$-graders to $9^\text{th}$-graders is $5:3$.
|
34
| 0.833333 |
Solve the equation $x - \frac{8}{x-2} = 5 - \frac{8}{x-2}$, where $x$ is an integer.
|
5
| 0.916667 |
Given Alice has $30$ apples and each person must receive at least $3$ apples, calculate the number of ways she can distribute them equally among herself, Becky, and Chris.
|
253
| 0.666667 |
Seven points on a circle are given. Four of the chords joining pairs of the seven points are selected at random. Calculate the probability that the four chords form a convex quadrilateral.
|
\frac{1}{171}
| 0.333333 |
Construct a square and attach an equilateral triangle to one of its sides. On a non-adjacent side of the triangle, construct a pentagon. On a non-adjacent side of the pentagon, construct a hexagon. Continue this pattern until you construct a heptagon. Calculate the total number of visible sides.
|
17
| 0.166667 |
Find the coefficient of $x^9$ in the expansion of $\left(\frac{x^3}{3}-\frac{3}{x^2}\right)^9$.
|
0
| 0.916667 |
Given that a password is composed of five digits (0 to 9) and no password may begin with the sequence $9,1,1,1$, calculate the number of valid passwords that are possible.
|
99990
| 0.916667 |
Determine the number of different total scores the basketball player could have achieved by making 8 baskets, each worth either 2, 3, or 4 points.
|
17
| 0.916667 |
Three pitchers with capacities of $500$ mL, $700$ mL, and $800$ mL have orange juice. The first pitcher is $1/5$ full, the second is $3/7$ full, and the third is $1/4$ full. After topping each pitcher off with water, all the contents are poured into a large container. What fraction of the mixture in the large container is orange juice?
|
\frac{3}{10}
| 0.583333 |
Given $145^9$, determine the total number of positive integer divisors that are perfect squares or perfect cubes, or both.
|
37
| 0.916667 |
In rectangle ABCD, AB = 3 ⋅ AD. Points P and Q are on AB such that AP = PQ = QB. Let E be the midpoint of CD. If lines PE and QE are drawn to meet CD at points R and S respectively, find the ratio of the area of ∆PQS to the area of rectangle ABCD.
|
\frac{1}{6}
| 0.333333 |
Let \( T = (x+2)^4 - 4(x+2)^3 + 6(x+2)^2 - 4(x+2) + 1 \). Simplify \( T \).
|
(x+1)^4
| 0.916667 |
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