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Given the sets of consecutive integers $\{1\}$,$\{2, 3\}$,$\{4,5,6\}$,$\{7,8,9,10\}$,$\; \cdots \;$, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set, determine the sum $S_{15}$ of the elements in the 15th set.
|
1695
| 0.916667 |
Given that the Math Club has 15 students, the Science Club has 10 students, the Art Club has 12 students, and 5 students are enrolled in both the Science and Math Clubs but not in the Art Club, calculate the total number of students participating in these clubs.
|
15 (Math) + 10 (Science) + 12 (Art) - 5 (overlap of Math and Science) = 15 + 10 + 12 - 5 = 32
| 0.5 |
Given $\triangle ABC$, where $AB=6$, $BC=8$, $AC=10$, and $D$ is on $\overline{AC}$ with $BD=6$, find the ratio of $AD:DC$.
|
\frac{18}{7}
| 0.583333 |
Given a rectangular box with dimensions $2 \times b \times c$, where $b$ and $c$ are integers with $b \leq c$, find the number of ordered pairs $(b, c)$ such that the sum of the volume and the surface area of the box equals 120.
|
0
| 0.25 |
Given the expression $2-(-3) \times 2 - 4 - (-5) \times 3 - 6$, evaluate the expression.
|
13
| 0.833333 |
An insect lives on the surface of a regular octahedron with edges of length $\sqrt{2}$. It wishes to travel from the midpoint of one edge to the midpoint of an edge that shares no common faces with the starting edge. What is the length of the shortest such trip?
A) $\frac{\sqrt{2}}{2}$
B) $\frac{3\sqrt{2}}{2}$
C) $\sqrt{2}$
D) $2\sqrt{2}$
E) $2$
|
\sqrt{2}
| 0.333333 |
Given the sales from January through May are documented as $120$, $80$, $70$, $150$, and in May there was a 10% discount on the total sales of $50$, calculate the average sales per month in dollars after applying the discount for May.
|
93
| 0.166667 |
Given that 300 students preferred spaghetti and 120 students preferred tortellini, and that these preferences were evenly distributed across the grade levels, find the ratio of the number of sophomores who preferred spaghetti to the number of sophomores who preferred tortellini.
|
\frac{5}{2}
| 0.916667 |
Given the polynomial expansion of $(1+3x-2x^2)^5$, calculate the coefficient of $x^9$.
|
240
| 0.916667 |
An inverted cone with a base radius of $8 \, \mathrm{cm}$ and a height of $24 \, \mathrm{cm}$ is full of water. This water is first poured into another cone with a base radius of $6 \, \mathrm{cm}$ to form a certain water level. Then all the water from this cone is completely poured into a cylinder with a radius of $6 \, \mathrm{cm}$. Determine the height of the water in the cylinder.
|
\frac{128}{9}
| 0.916667 |
The parabolic arch has a height of $20$ inches and a span of $50$ inches. Find the height, in inches, of the arch at the point $10$ inches from the center.
|
16.8
| 0.916667 |
The High School Ten basketball conference has 10 teams. Each team plays every other conference team twice and also plays 5 games against non-conference opponents. Calculate the total number of games in a season involving the High School Ten teams.
|
140
| 0.916667 |
Find the polynomial whose roots satisfy the condition that the sum is double the product of the roots, with the sum of the roots being $2k$ and the product being $k$, and given that one of the roots is $1$.
|
x^2 - 2x + 1
| 0.833333 |
Keiko walks around a rectangular track with semicircular ends each day. The straight lengths are constant, and the track width is 8 meters. It takes her 48 seconds longer to walk around the outside than around the inside. Given this, calculate Keiko's average speed in meters per second.
|
\frac{\pi}{3}
| 0.5 |
The number of lines that are simultaneously tangent to two circles with radii of 4 cm and 5 cm can be determined based on the different placements of the circles.
|
0, 1, 2, 3, 4
| 0.25 |
Jake and Ellie alternately toss their coins until someone gets heads. Jake's coin lands heads with probability $\frac{1}{4}$, and Ellie's coin lands heads with probability $\frac{1}{3}$. Jake goes first. Find the probability that Jake wins.
|
\frac{1}{2}
| 0.333333 |
In a small reserve, a biologist counted a total of 300 heads comprising of two-legged birds, four-legged mammals, and six-legged insects. The total number of legs counted was 980. Calculate the number of two-legged birds.
|
110
| 0.083333 |
Evaluate the expression $1 + \frac {1}{1 + \frac {1}{1 + \frac{1}{2}}}$.
|
\frac{8}{5}
| 0.75 |
Find the value of $x$ that satisfies the equation $25^{-3} = \frac{5^{60/x}}{5^{36/x} \cdot 25^{21/x}}.$
|
3
| 0.833333 |
Given a positive number $y$, determine the condition that satisfies the inequality $\sqrt{y} < 3y$.
|
y > \frac{1}{9}
| 0.666667 |
What are the times between $7$ and $8$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $120^{\circ}$?
|
7:16 \text{ and 7:56}
| 0.416667 |
Calculate the value of \(x[x\{x(3-x)-3\}+5]+1\).
|
-x^4 + 3x^3 - 3x^2 + 5x + 1
| 0.166667 |
Given a quadrilateral with vertices $P(a,c)$, $Q(c,a)$, $R(-a, -c)$, and $S(-c, -a)$, where $a$, $c$ are integers with $a>c>0$, and with area 24, find the sum of $a$ and $c$.
|
6
| 0.666667 |
Given that a fly is 3 meters from one wall, 7 meters from the other wall, and 10 meters from point P where two walls and the ceiling meet at a right angle, calculate the distance of the fly from the ceiling.
|
\sqrt{42}
| 0.916667 |
If the markings on a number line are equally spaced, what is the number y after taking four steps starting from 0, and reaching 25 after five steps?
|
20
| 0.75 |
Given all three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=2x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis, find the length of $BC$ if the area of the triangle is $128$.
|
8
| 0.666667 |
Given that Beth gets the disease calcuvitis and must take one red pill and one blue pill each day for three weeks, and a red pill costs $3 more than a blue pill, and Beth's pills cost a total of $\textdollar 966$ for the three weeks, calculate the cost of one red pill.
|
24.5
| 0.75 |
Given square ABCD has side length s, calculate the ratio of the area of square EFGH to the area of square ABCD, where E, F, G, and H are the centers respectively of isosceles right-angled triangles with bases AB, BC, CD, and DA, each exterior to the square with the right angle at the base vertex.
|
2
| 0.166667 |
Given that a randomly drawn positive factor of $84$ is selected, calculate the probability that it is less than $8$.
|
\frac{1}{2}
| 0.333333 |
A circle is drawn through vertices $E$ and $H$ and tangent to side $FG$ of square $EFGH$ with side $12$ feet. Calculate the radius of the circle.
|
7.5
| 0.333333 |
Suppose $m$ and $n$ are positive integers such that $45m = n^3$. Additionally, it is given that $n$ is a multiple of 5. Find the minimum possible value of $m + n$.
|
90
| 0.666667 |
Evaluate the square of \(8 - \sqrt{x^2 + 64}\).
|
x^2 + 128 - 16\sqrt{x^2 + 64}
| 0.5 |
Initially, a square is completely painted black. Each time the square is changed, the middle half of each black square turns white. After three changes, what fraction of the original area of the black square remains black.
|
\frac{1}{8}
| 0.583333 |
In an arithmetic progression, any term equals the average of the next two terms. The first term is 12. Find the common difference.
|
0
| 0.583333 |
Given Max makes a larger batch of lemonade using 500 grams of lemon juice, 200 grams of sugar, and 1000 grams of water, and each 100 grams of lemon juice contains 30 calories, while each 100 grams of sugar has 400 calories, determine the total number of calories in 300 grams of this prepared lemonade.
|
168
| 0.666667 |
Given that the sum of all the angles except one of a convex polygon is $2790^{\circ}$, calculate the number of sides of the polygon.
|
18
| 0.916667 |
At a local beach festival, a vendor offers a special deal on flip-flops. If you buy one pair at the regular price of $45, you get the second pair at a 30% discount, and if you buy a third pair, it costs only 40% of the regular price. Maria opts to buy three pairs under this festival offer. Calculate the percentage of the total regular price that Maria saved.
|
30\%
| 0.666667 |
One day a drink kiosk sold 360 cans of soda to 150 customers, with every customer buying at least one can of soda. Determine the maximum possible median number of cans of soda bought per customer on that day.
|
3
| 0.25 |
A cell phone plan costs $25$ dollars each month, plus $0.10$ dollars per text message sent, plus $0.15$ dollars for each minute used over $30$ hours, but not exceeding $31$ hours. Any usage beyond $31$ hours costs $0.20$ dollars per minute. Given that Lucas sent $150$ text messages and talked for $31.5$ hours, calculate the total cost of the plan.
|
55.00
| 0.083333 |
Given the inequalities \(x - y + 3 > x\) and \(x + y - 2 < y\), find the relationship between \(x\) and \(y\).
|
x < 2 \text{ and } y < 3
| 0.333333 |
Given that $S_n=1-2+3-4+\cdots +(-1)^{n-1}n$, where $n=1,2,\cdots$, and $T_n = S_n + \lfloor \sqrt{n} \rfloor$, find $T_{19}+T_{21}+T_{40}$.
|
15
| 0.916667 |
Given that Ike and Mike enter a bakery with a total of $50.00 to spend, calculate the number of items, consisting of sandwiches at $5.00 each and cakes at $2.50 each, that they will buy altogether, with the condition that they do not want to buy more than 12 items in total.
|
12
| 0.916667 |
Professor Lee has eleven different language books lined up on a bookshelf: three Arabic, four German, and four Spanish. Calculate the number of ways to arrange the eleven books on the shelf while keeping the Arabic books together.
|
2,177,280
| 0.666667 |
A rectangular solid has side faces with areas of $18\text{ in}^2$, $50\text{ in}^2$, and $45\text{ in}^2$. Calculate the volume of this solid.
|
90\sqrt{5}\text{ in}^3
| 0.083333 |
Evaluate the expression $\sqrt[3]{8 + 3 \sqrt{21}} + \sqrt[3]{8 - 3 \sqrt{21}}$ and square the result.
|
1
| 0.916667 |
Triangle $OPQ$ is defined with $O=(0,0)$, $Q=(3,0)$, and $P$ in the first quadrant. Suppose $\angle PQO = 90^\circ$ and $\angle POQ = 45^\circ$. If $PO$ is rotated $-90^\circ$ (clockwise) around $O$, determine the new coordinates of $P$.
|
(3, -3)
| 0.5 |
The final answer is:
Let the original price of the jacket be $120.00. During a sale, this price is discounted by 25%, then the sales tax of 8% is applied. Two different methods of calculating the bill are presented. The first method involves adding 8% sales tax to the discounted price, whereas the second method involves applying a 25% discount to the original price, followed by adding 8% sales tax to this discounted price. What is the difference between the two totals.
|
0
| 0.916667 |
The sum of the greatest integer less than or equal to \(x\) and the least integer greater than or equal to \(x\) is 7. What is the solution set for \(x\)?
A) $\left\{\frac{7}{2}\right\}$
B) $\{x | 3 \le x \le 4\}$
C) $\{x | 3 \le x < 4\}$
D) $\{x | 3 < x \le 4\}$
E) $\{x | 3 < x < 4\}$
|
\{x | 3 < x < 4\}
| 0.083333 |
Maria bakes a $24$-inch by $30$-inch pan of brownies, and the brownies are cut into pieces that measure $3$ inches by $4$ inches. Calculate the total number of pieces of brownies the pan contains.
|
60
| 0.333333 |
Given that $7 = k\cdot 3^r$ and $49 = k\cdot 9^r$, solve for $r$.
|
\log_3 7
| 0.666667 |
A lemming starts at a corner of a square with side length $8$ meters. It moves $4.8$ meters along a diagonal towards the opposite corner, then makes a $90^{\circ}$ left turn and runs $2.5$ meters. Calculate the average of the shortest distances from the lemming to each side of the square.
|
4
| 0.666667 |
A lemming sits at a corner of a rectangle with side lengths 12 meters and 16 meters. The lemming runs 9.6 meters along a diagonal toward the opposite corner. It stops, makes a $90^{\circ}$ left turn, and runs 3 more meters. Calculate the average of the shortest distances from the lemming to each side of the rectangle.
|
7
| 0.833333 |
Given a square chalkboard divided into squares of equal size, with one diagonal drawn in each square, find the smallest number of total squares that satisfies the condition that the total number of smaller squares along the perimeter of the chalkboard is twice the number of them along one of its diagonals.
|
4
| 0.916667 |
Given that Jo needs to climb an 8-stair flight, find the total number of ways Jo can take 2 or 3 stairs at a time to reach the top.
|
4
| 0.916667 |
Suppose $a$ and $b$ are nonzero real numbers such that $\frac{4a+b}{a-4b}=3$. Evaluate the value of $\frac{a+4b}{4a-b}$.
|
\frac{9}{53}
| 0.916667 |
Given that a positive even multiple of $3$ less than 5000 is a perfect square, how many such numbers exist?
|
11
| 0.583333 |
If $x > 0$ but not too large so that $\frac{x^2}{1+x}$ remains relevant, establish the correct relationship between $\log(1+x)$ and $\frac{x^2}{1+x}$.
|
\log (1+x) > \frac{x^2}{1+x}
| 0.5 |
If $\omega = \alpha t + \omega_0$ and $\theta = \frac{1}{2}\alpha t^2 + \omega_0t$, solve for $t$ in terms of $\theta$, $\omega$, and $\omega_0$.
|
\frac{2\theta}{\omega + \omega_0}
| 0.75 |
Let $S$ be the sum of all positive real numbers $x$ for which $x^{3^{\sqrt{3}}} = \sqrt{3}^{3^x}$. Determine the value of $S$.
|
S = \sqrt{3}
| 0.5 |
What is the product of all positive odd integers less than $20000$?
|
\dfrac{20000!}{2^{10000} \cdot 10000!}
| 0.416667 |
What is the probability that a randomly drawn positive factor of $36$ is less than $6$?
|
\frac{4}{9}
| 0.583333 |
Marie performs four equally time-consuming tasks consecutively without breaks. She begins the first task at 7:00 AM and completes the second task at 9:20 AM. Determine the time at which she finishes the fourth task.
|
11:40 \text{ AM}
| 0.666667 |
Given that every 8-digit whole number is a potential telephone number except those that begin with $0$ or $\geq 6$, calculate the fraction of telephone numbers that begin with $5$ and end with $2$.
|
\frac{1}{50}
| 0.666667 |
Given the sum of the interior angles of a convex polygon is $2240^\circ$, calculate the measure of the uncounted angle.
|
100
| 0.083333 |
Given that the sum of the interior angles of a convex polygon is 1958°, determine the degree measure of the omitted angle.
|
22
| 0.75 |
Given that Ben spent some amount of money and David spent $0.5 less for each dollar Ben spent, and Ben paid $16.00 more than David, determine the total amount they spent together in the bagel store.
|
48.00
| 0.833333 |
How many positive factors does $27,648$ have?
|
44
| 0.25 |
Triangle $OPQ$ has $O=(0,0)$, $Q=(6,0)$, and $P$ in the first quadrant. Additionally, $\angle PQO=90^\circ$ and $\angle POQ=45^\circ$. Suppose that $OP$ is rotated $90^\circ$ counterclockwise about $O$. Find the coordinates of the image of $P$.
|
(-6, 6)
| 0.416667 |
A cuboid has its side lengths as 8 units, 6 units, and 4 units respectively. Calculate the number of edges that are not parallel to a given edge.
|
8
| 0.583333 |
If the sum of the first 20 terms and the sum of the first 50 terms of a given arithmetic progression are 200 and 150, respectively, find the sum of the first 70 terms of the sequence.
|
-\frac{350}{3}
| 0.75 |
Calculate the product $(1 + x^3)(1 - 2x + x^4)$.
|
1 - 2x + x^3 - x^4 + x^7
| 0.166667 |
Given the taxi fare starts at $3.00 for the first $\frac{3}{4}$ mile, and each additional 0.1 mile is charged at $0.25, and a $3 tip is intended, determine the number of miles that can be ridden for a total cost of $15.
|
4.35
| 0.583333 |
Given $2^9+1$ and $2^{17}+1$, inclusive, determine the number of perfect cubes between these two values.
|
42
| 0.666667 |
Given a list of nine numbers, the average of the first five numbers is 7, and the average of the last five numbers is 10. If the average of all nine numbers is $8\frac{2}{9}$, determine the number common to both sets of five numbers.
|
11
| 0.916667 |
Given the population of Newlandia in the year 2050 is 350,000,000, the area of Newlandia is 4,500,000 square miles, and there are $(5280)^2$ square feet in one square mile, calculate the average number of square feet per person in Newlandia.
|
358,437
| 0.583333 |
Let $\frac{27x - 19}{x^2 - 5x + 6} = \frac{M_1}{x - 2} + \frac{M_2}{x - 3}$ be an identity in $x$. Find the value of $M_1M_2$.
|
-2170
| 0.916667 |
The circle having $(2,2)$ and $(10,8)$ as the endpoints of a diameter intersects the $x$-axis at a second point. Calculate the $x$-coordinate of this point.
|
6
| 0.083333 |
A box contains 3 red marbles, 3 green marbles, and 3 yellow marbles. Carol takes 3 marbles from the box at random; then Claudia takes another 3 marbles at random; lastly, Cheryl takes the remaining 3 marbles, calculate the probability that Cheryl gets 3 marbles of the same color.
|
\frac{1}{28}
| 0.333333 |
Given the number $1036$ represented in factorial base numeration as $1036 = b_1 + b_2 \times 2! + b_3 \times 3! + \ldots + b_n \times n!$, where each $b_k$ satisfies $0 \leq b_k \leq k$, find the value of $b_4$.
|
3
| 0.75 |
Elmer's latest car offers $60\%$ better fuel efficiency, measured in kilometers per liter, than his old car. Additionally, the new car uses diesel fuel, which is $25\%$ more expensive per liter than the gasoline used by his old car. Calculate the percent by which Elmer will save money if he opts to use his new car instead of his old car for a long trip.
|
21.875\%
| 0.833333 |
Given $221^8$, calculate the number of positive integer divisors that are perfect squares or perfect cubes or both.
|
30
| 0.25 |
Given the product of all even negative integers strictly greater than $-2020$, after adding 10 to the product, determine the sign and units digit of the result.
|
0
| 0.25 |
Let $g(t) = \frac{2t}{1+t}$, where $t \not= -1$. Express $w$ as a function of $z$, where $z = g(w)$.
|
\frac{-z}{z-2}
| 0.083333 |
Given that a baby is born every 6 hours and a death occurs every 2 days in the nation of North Southland, calculate the average annual increase in population to the nearest fifty.
|
1300
| 0.666667 |
Calculate the sum of the digits of the greatest prime number that is a divisor of $59,048$.
|
7
| 0.75 |
Evaluate the expression $\frac{2^{2010} \cdot 3^{2012}}{6^{2011}}$.
|
\frac{3}{2}
| 0.916667 |
A sequence of squares is constructed with each square having an edge that is two tiles longer than the edge of the previous square. The sequence starts from a square with a side length of 1 tile. Calculate the difference in the number of tiles required for the eleventh square and the tenth square.
|
80
| 0.916667 |
Find the second-next perfect square after a perfect square number $x$.
|
x + 4\sqrt{x} + 4
| 0.416667 |
It takes Clea 75 seconds to walk down an escalator when it is not moving, and 30 seconds when it is moving. Calculate the time it would take Clea to ride the escalator down when she is not walking.
|
50
| 0.916667 |
Jim traveled to the store at an average speed of 15 kilometers per hour. After traveling two-thirds of the total distance, the skateboard broke, and he walked the remaining distance at 4 kilometers per hour. It took him a total of 56 minutes to complete the trip. Calculate the total distance Jim traveled in kilometers, rounded to the nearest half kilometer.
|
7.5
| 0.916667 |
Given that Rachel has $3030$ coins composed of pennies and nickels, and that she must have at least $10$ times as many pennies as nickels but no less than $3$ nickels, determine the difference in cents between the maximum and minimum monetary amounts Rachel can have.
|
1088
| 0.916667 |
A woman buys a property for $12,000 and plans to earn a $6\%$ return on her investment through renting it out. She sets aside $10\%$ of the monthly rent for repairs and maintenance and pays $400 annually in taxes. Calculate the necessary monthly rent.
|
103.70
| 0.5 |
Given that a reader is guessing at random, calculate the probability that a reader will match all four celebrity photos correctly.
|
\frac{1}{24}
| 0.916667 |
$\log_5{(10x-3)}-\log_5{(3x+4)$ approaches what value as $x$ grows beyond all bounds.
|
\log_5 \left(\frac{10}{3}\right)
| 0.416667 |
Consider a $5\times 5$ block of calendar dates as shown below. First, the order of the numbers in the second, third, and fifth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?
|
4
| 0.416667 |
The yearly changes in the population census of a city for five consecutive years are, respectively, 20% increase, 10% increase, 30% decrease, 20% decrease, and 10% increase. Calculate the net change over these five years, to the nearest percent.
|
-19\%
| 0.916667 |
Point $E$ is the midpoint of side $\overline{BC}$ in square $ABCD$, and $\overline{AE}$ meets diagonal $\overline{BD}$ at $F$. The area of quadrilateral $AFED$ is $25$. Given this information, calculate the area of square $ABCD$.
|
50
| 0.666667 |
Calculate the sum of the sequence $1-2-3+4+5-6-7+8+9-10-11+12+\cdots+2021-2022-2023+2024+2025-2026-2027+2028$.
|
0
| 0.583333 |
In an All-District track meet, $320$ sprinters enter a $100-$meter dash competition. The track has 8 lanes, so only $8$ sprinters can compete at a time. At the end of each race, the seven non-winners are eliminated, and the winner will compete again in a later race. Calculate the number of races needed to determine the champion sprinter.
|
46
| 0.666667 |
A rectangular board consists of an alternating pattern of light and dark squares similar to a chessboard. This board is $8$ rows high and $7$ columns wide. Determine how many more dark squares there are than light squares.
|
0
| 0.666667 |
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