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stringlengths 18
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|---|---|---|
Given the factorial $15!$, calculate the probability that a randomly chosen positive integer divisor is an odd number.
|
\frac{1}{12}
| 0.166667 |
Given the functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$, determine the least common positive period $p$ for all such functions.
|
36
| 0.583333 |
Calculate the value of the product: $\left(\frac{1\cdot5}{3\cdot3}\right)\left(\frac{3\cdot7}{5\cdot5}\right)\left(\frac{5\cdot9}{7\cdot7}\right)\cdots\left(\frac{95\cdot99}{97\cdot97}\right)\left(\frac{97\cdot101}{99\cdot99}\right)$.
|
\frac{101}{297}
| 0.583333 |
Given $a$ and $b$ are two distinct non-zero numbers, determine the relationship between $\frac{a^2 + b^2}{2}$, $\frac{a+b}{2}$, and $\sqrt{ab}$.
|
\frac{a^2 + b^2}{2} > \frac{a+b}{2} > \sqrt{ab}
| 0.916667 |
A half-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 aligning the two radii. Determine the volume of the cone formed in cubic inches.
|
9\pi \sqrt{3}
| 0.833333 |
Given that a discount sign indicating “$\frac{1}{4}$ off” was displayed, and the actual cost of the laptops is $\frac{5}{6}$ of the price at which they were sold, calculate the ratio of the cost to the posted price.
|
\frac{5}{8}
| 0.666667 |
Given that Paula has exactly enough money to buy 50 popcorn bags at full price, and there is a promotional deal where one bag is full price and the next two bags are $\frac{1}{4}$ off the regular price, calculate the maximum number of popcorn bags Paula could buy.
|
60
| 0.583333 |
Given the function \( g_1 \) defined on the positive integers by \( g_1(1) = 1 \) and \( g_1(n) = (p_1-1)^{e_1-1} (p_2-1)^{e_2-1} \cdots (p_k-1)^{e_k-1} \), where \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \) is the prime factorization of \( n > 1 \), and for every \( m \geq 2 \), let \( g_m(n) = g_1(g_{m-1}(n)) \), determine the number of integers \( N \) in the range \( 1 \leq N \leq 100 \) for which the sequence \( (g_1(N), g_2(N), g_3(N), \dots) \) is unbounded.
|
0
| 0.166667 |
Triangle PQR has P=(0,0), R=(8,0), and Q in the first quadrant. Additionally, ∠QRP=90° and ∠QPR=45°. Determine the coordinates of the image of Q after P is rotated 120° counterclockwise about P.
|
(-4 - 4\sqrt{3}, 4\sqrt{3} - 4)
| 0.083333 |
Given two polygons $P_1$ and $P_2$, where $P_1$ is an equiangular decagon (10 sides) and each angle of $P_2$ is $kx$ degrees, with $k$ being an integer greater than $1$. If $\frac{kx}{x}$ determines the ratio between the number of sides of $P_2$ and $P_1$, find the smallest integer value of $k$ such that $P_2$ can exist as a valid polygon whose interior angles are also expressed in terms of $x$.
|
2
| 0.666667 |
Given $8^y - 8^{y-1} = 56$, calculate $\left(3y\right)^y$.
|
36
| 0.916667 |
Let $M$ be the greatest four-digit number whose digits have a product of 36. Find the sum of the digits of $M$.
|
15
| 0.75 |
$\frac{8^x}{4^{x+y}}=32$ and $\frac{25^{x+y}}{5^{7y}}=3125$, find the product $xy$.
|
75
| 0.916667 |
Given a rectangular park split into a trapezoidal lawn area and two congruent isosceles right triangle flower beds, the parallel sides of the trapezoidal lawn are 15 meters and 30 meters long. Calculate the fraction of the park's total area that is occupied by the flower beds.
|
\frac{1}{4}
| 0.333333 |
Let M be the greatest five-digit number whose digits have a product of 210. Find the sum of the digits of M.
|
20
| 0.083333 |
Given that a regular octagon and an equilateral triangle have equal areas, determine the ratio of the length of a side of the triangle to the length of a side of the octagon.
|
\sqrt{ \frac{8\sqrt{3}(1+\sqrt{2})}{3} }
| 0.166667 |
Given a regular decagon, a triangle is formed by connecting three randomly chosen vertices of the decagon. Calculate the probability that at least one of the sides of the triangle is also a side of the decagon.
|
\frac{7}{12}
| 0.083333 |
What is $9\cdot\left(\tfrac{1}{3}+\tfrac{1}{6}+\tfrac{1}{9}+\tfrac{1}{18}\right)^{-1}$?
|
\frac{27}{2}
| 0.916667 |
Nine points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect at a single point inside the circle. Find the number of triangles with all three vertices in the interior of the circle.
|
84
| 0.416667 |
In a three-horse race with horses A, B, and C, where no ties can occur, the odds against A winning are $4:1$, and the odds against B winning are $3:4$. Determine the odds against C winning.
|
\frac{27}{8}
| 0.083333 |
A farmer buys 600 cows. He sells 500 of them for the price he paid for all 600 cows. The remaining 100 cows are sold for 10% more per cow than the price of the 500 cows. Calculate the percentage gain on the entire transaction.
|
22\%
| 0.833333 |
Each vertex of a convex hexagon $ABCDEF$ is to be assigned a color. There are $7$ colors to choose from, and no two adjacent vertices can have the same color, nor can the vertices at the ends of each diagonal. Calculate the total number of different colorings possible.
|
5040
| 0.083333 |
The smallest whole number larger than the perimeter of any triangle with a side of length $6$ and a side of length $19$.
|
50
| 0.333333 |
The angle between the hands of a clock at 3:30 can be calculated.
|
75^\circ
| 0.916667 |
In a set of $36$ square blocks arranged into a $6 \times 6$ square, how many different combinations of $4$ blocks can be selected from that set so that no two are in the same row or column?
|
5400
| 0.916667 |
A point P is situated 15 inches from the center of a circle. A secant from P intersects the circle at points Q and R such that the external segment PQ measures 10 inches and segment QR measures 8 inches. Calculate the radius of the circle.
|
3\sqrt{5}
| 0.833333 |
A cinema has 150 seats arranged in a row. What is the minimum number of seats that must be occupied such that the next person to come must sit adjacent to someone already seated?
|
50
| 0.083333 |
Two different numbers are randomly selected from the set $\{1, 2, 5, 7, 11, 13\}$ and multiplied together. Calculate the probability that the product is an even number.
|
\frac{1}{3}
| 0.916667 |
The number $25!$ has a significantly large number of positive integer divisors. Determine the probability that an odd divisor is chosen at random from these.
|
\frac{1}{23}
| 0.25 |
A square and a regular octagon have equal perimeters. If the area of the square is 16, calculate the area of the octagon.
|
8 + 8\sqrt{2}
| 0.083333 |
A skateboard rolls down a slope, traveling 8 inches in the first second and accelerating so that during each successive 1-second time interval, it travels 9 inches more than during the previous 1-second interval. The skateboard takes 20 seconds to reach the end of the slope. Find the total distance, in inches, that it travels.
|
1870
| 0.333333 |
Nine rectangles each have widths $1, 3, 5, 7, 9, 11, 13, 15, 17$ and lengths $1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2, 9^2$. Calculate the sum of the areas of all nine rectangles.
|
3765
| 0.916667 |
Mina writes down one integer three times and another integer four times. The total sum of these seven numbers is $140$, and one of the numbers is $20$. Determine the value of the other number.
|
20
| 0.75 |
Given that soda is sold in packs of 8, 16, and 32 cans, find the minimum number of packs needed to buy exactly 120 cans of soda.
|
5
| 0.916667 |
A triangular array of $3003$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. Calculate the sum of the digits of $N$.
|
14
| 0.916667 |
Given the original price of a backpack is $120.00 and the sales tax rate is 7%, if Alex calculates the final price by adding 7% sales tax on the original price first and then applying a 15% discount to this total, and if Sam calculates the final price by subtracting 15% of the price first and then adding 7% sales tax on the reduced price and finally adding a fixed packaging fee of $2.50, calculate Alex's total minus Sam's total.
|
-\$2.50
| 0.083333 |
Using three-digit powers of $3$ and $7$ in a cross-number puzzle, determine the unique possible digit for the intersecting square.
|
3
| 0.833333 |
For how many values of $d$ is $2.d05 > 2.005$?
|
9
| 0.75 |
Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $\beta$. Given that one root is triple the other, determine the relationship between $a$, $b$, and $c$.
|
3b^2 = 16ac
| 0.916667 |
Let \( T = (x-2)^5 + 5(x-2)^4 + 10(x-2)^3 + 10(x-2)^2 + 5(x-2) + 1 \), then simplify T into the form \( (x - a)^5 \).
|
(x-1)^5
| 0.916667 |
Given the equation \( 4 + 2\cos\theta - 3\sin2\theta = 0 \), determine the number of values of \( \theta \) in the interval \( 0 < \theta \leq 2\pi \) that satisfy the equation.
|
2
| 0.25 |
Given a sequence of 0s and 1s of length 21 that begins with a 0, ends with a 0, contains no two consecutive 0s, and contains no three consecutive 1s, determine the number of such sequences.
|
114
| 0.166667 |
A square has a computed area of $1.4456$ square feet rounded to the nearest ten-thousandth of a square foot after its original area has been increased by $0.0001$ square feet. Find the number of significant digits in the measurement of the side length of the original square.
|
5
| 0.416667 |
Given that $20\%$ of the students are juniors and $80\%$ are seniors with an overall average score of $86$, and the average score of the seniors is $85$, calculate the score received by each of the juniors.
|
90
| 0.75 |
Given that John scored 85% on 28 tests out of 40 tests, in order to achieve an 85% score on 60 tests, determine the minimum number of remaining tests he can afford to score below 85%.
|
0
| 0.5 |
If Menkara 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 instead, she doubles the length of the other side.
|
70
| 0.083333 |
A child has $3$ red, $3$ blue, and $4$ green cubes and wants to build a tower $9$ cubes high. How many different towers can the child build if exactly one cube is left out?
|
4,200
| 0.416667 |
Points M and N are the midpoints of sides PC and PD of triangle PCD. As P moves along a line that is parallel to side CD, determine the number of the four quantities listed that change: the length of the segment MN, the perimeter of triangle PCD, the area of triangle PCD, the area of trapezoid CDNM.
|
1
| 0.166667 |
What is the remainder when $1^2 + 2^2 + 3^2 + \cdots + 25^2$ is divided by 6?
|
5
| 0.333333 |
Find the smallest whole number that is larger than the sum $3\dfrac{1}{3}+5\dfrac{1}{4}+7\dfrac{1}{6}+9\dfrac{1}{8}$.
|
25
| 0.416667 |
Given Jo and Blair take turns counting, with each number said by each person being two more than the last number said by the other person, determine the 30th number said, starting from 2 and including 20, and then from 23 incrementing by 3 instead of 2.
|
80
| 0.333333 |
Given a circle of radius $3$ units, find the area of the region consisting of all line segments of length $6$ units that are tangent to the circle at their midpoints.
|
9\pi
| 0.083333 |
Given $200(200 + 5) - (200 \cdot 200 + 5)$, calculate the expression.
|
995
| 0.916667 |
Lila has a rectangular notecard measuring $5 \times 7$ inches. She decides to shorten one side of the notecard by $2$ inches, resulting in a new area of $21$ square inches. Determine the new area of the notecard if Lila instead shortens the length of the other side by $2$ inches.
|
25
| 0.5 |
What number should be removed from the set $\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$ so that the average of the remaining numbers is $7.1$?
|
6
| 0.916667 |
The population doubles every 20 years starting from an initial population of 500 in the year 2023. Determine the year when the population is closest to 8,000.
|
2103
| 0.916667 |
Given the graph of $y = mx + 3$ passes through no lattice point with $0 < x \le 50$ for all $m$ such that $\frac{1}{3} < m < b$, find the maximum possible value of $b$.
|
\frac{17}{50}
| 0.083333 |
Ten points are spaced evenly along the perimeter of a rectangle measuring $3 \times 2$ units. What is the probability that the two points are one unit apart?
|
\frac{2}{9}
| 0.833333 |
A circle of radius 3 is centered at point $A$. An equilateral triangle with side length 4 has one vertex at $A$. Calculate the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle.
|
9\pi - 4\sqrt{3}
| 0.583333 |
Determine the number of pairs of regular polygons with sides of unit length that have a ratio of their interior angles of $5:3$.
|
1
| 0.166667 |
Given that 4500 Euros equals 3900 pounds, calculate how many Euros the tourist will receive for exchanging 3000 pounds.
|
3461.54
| 0.833333 |
In a rhombus ABCD, the longer diagonal is three times the length of the shorter diagonal. If the area of the rhombus is $L$ square units, express the side length of the rhombus in terms of $L$.
|
\sqrt{\frac{5L}{3}}
| 0.5 |
Points $B$, $C$, and $E$ lie on $\overline{AD}$. The length of $\overline{AB}$ is $3$ times the length of $\overline{BD}$, and the length of $\overline{AC}$ is $7$ times the length of $\overline{CD}$. Point $E$ is situated such that $\overline{DE}$ is twice the length of $\overline{CE}$. Express the length of $\overline{BC}$ as a fraction of $\overline{AD}$.
|
\frac{1}{8}
| 0.583333 |
If $i^2=-1$, then find the value of $(3i-3i^{-1})^{-1}$.
|
-\frac{i}{6}
| 0.916667 |
Given the square defined by the vertices (0,0), (0,5), (5,5), and (5,0), an initially positioned frog at point (2, 2) makes random jumps of length 1 parallel to the coordinate axes. Calculate the probability that the sequence of jumps ends on a vertical side of the square.
|
\frac{1}{2}
| 0.416667 |
In square $ABCD$, point $E$ is on $AB$ such that $AE = 3EB$, and point $F$ is on $CD$ such that $DF = 3FC$. Determine the ratio of the area of triangle $BFC$ to the area of square $ABCD$.
|
\frac{1}{8}
| 0.583333 |
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 70?
|
5183
| 0.916667 |
A rectangular floor that is 15 feet wide and 20 feet long is tiled with one-foot square tiles. A bug starts at the midpoint of one of the shorter sides and walks in a straight line to the opposite side's midpoint. Calculate the number of tiles the bug visits, including the first and the last tile.
|
20
| 0.166667 |
Given Ella rode her bicycle for 4 days, with distances of 3 miles each day, at speeds of 6 miles per hour on Monday, 4 miles per hour on Tuesday, 5 miles per hour on Thursday, and 3 miles per hour on Friday. If Ella had always cycled at 5 miles per hour, determine the difference in time spent bicycling.
|
27
| 0.666667 |
Given Angie, Bridget, Carlos, Diego, and Eliza are seated at random around a circular table with each person occupying one side, find the probability that Angie and Carlos are seated opposite each other, and Bridget is not sitting next to Carlos.
|
\frac{1}{6}
| 0.25 |
What is the value of the expression $\sqrt{25\sqrt{16\sqrt{9}}}$?
|
10\sqrt[4]{3}
| 0.416667 |
Given that Chloe's estimate of 40,000 is within $5\%$ of the actual attendance in Chicago, Derek's estimate of 55,000 is within $15\%$ of the actual attendance in Denver, and Emma's prediction of 75,000 is within $10\%$ of the actual count in Miami, calculate the largest possible difference between the numbers attending any two of these three events to the nearest 1,000.
|
45000
| 0.833333 |
Given Suzanna rides her bike at a constant rate as shown by a graph, if she rides for 40 minutes at the same speed, how many miles would she have ridden?
|
8
| 0.583333 |
A supermarket has 150 crates of oranges. Each crate contains at least 125 oranges and at most 149 oranges. What is the largest integer n such that there must be at least n crates containing the same number of oranges?
|
6
| 0.666667 |
When Cheenu was a boy, he could run 20 miles in 4 hours. As a middle-aged man, he could bicycle 30 miles in 2 hours. Now, as an old man, he can walk 8 miles in 4 hours. Determine the difference in time it takes for him to walk a mile now compared to when he bicycled a mile as a middle-aged man.
|
26
| 0.5 |
The three numbers taken in pairs have sums of 18, 23, and 27. Find the middle number.
|
11
| 0.25 |
Given a wheel with a fixed center and an outside diameter of $8$ feet, determine the number of revolutions required for a point on the rim to travel half a mile.
|
\frac{330}{\pi}
| 0.833333 |
Calculate the value of $2 - 3(-4) - 7 + 2(-5) - 9 + 6(-2)$.
|
-24
| 0.75 |
A point is chosen at random within a rectangle in the coordinate plane whose vertices are (0, 0), (4040, 0), (4040, 2020), and (0, 2020). The probability that the point is within $d$ units of a lattice point is $\frac{1}{4}$. What is $d$ to the nearest tenth?
|
0.3
| 0.416667 |
The length AB of the line segment where the region in three-dimensional space consisting of all points within 4 units from AB has a total volume of 320π.
|
\frac{44}{3}
| 0.5 |
Given the integer $2020$, find the sum of its prime factors.
|
108
| 0.083333 |
Let $m = 2011^2 + 2^{2011}$. Determine the units digit of $m^3 + 3^m$.
|
2
| 0.916667 |
Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. Determine the probability that the sum of the die rolls is odd.
|
\frac{7}{16}
| 0.583333 |
Given that $15\%$ of the students scored $60$ points, $25\%$ scored $75$ points, $35\%$ scored $85$ points, $20\%$ scored $95$ points, and the rest scored $110$ points, calculate the difference between the mean and the median score of the students' scores on this test.
|
3
| 0.083333 |
Given the numbers $4, 5, 3, 3, 6, 7, 3, 5$, calculate the sum of the mean, median, and mode.
|
12
| 0.333333 |
Given positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_3{x} = \log_y{81}$ and $xy = 243$, calculate $(\log_3{\tfrac{x}{y}})^2$.
|
9
| 0.916667 |
Given a cube with side length $n$ painted red on all six faces, cut into $n^3$ smaller cubes of unit side length, and having exactly one-third of the total number of faces of these small cubes be red, determine the value of $n$.
|
3
| 0.583333 |
Calculate the geometric mean of the expressions $\frac{x + a}{x}$ and $\frac{x - a}{x}$.
|
\sqrt{1 - \frac{a^2}{x^2}}
| 0.916667 |
The runners run at speeds of 3.2, 4.0, 4.8, 5.6, and 6.4 meters per second. They stop once they are all together again somewhere on the circular 400-meter track. Determine the time in seconds the runners run.
|
500
| 0.083333 |
Let $a$, $b$, $c$, and $d$ be positive integers with $a < 3b$, $b < 2c$, and $c < 5d$. If $d < 150$, calculate the largest possible value for $a$.
|
4460
| 0.916667 |
The product of two 999-digit numbers $400400400...\text{(300 times)}...400$ and $606606606...\text{(300 times)}...606$ has tens digit \( A \) and units digit \( B \). What is the sum of \( A \) and \( B \)?
|
0
| 0.75 |
A regular tetrahedron has four vertices. Three vertices are chosen at random. Determine the probability that the plane determined by these three vertices contains points inside the tetrahedron.
|
0
| 0.833333 |
Two congruent squares, $ABCD$ and $EFGH$, each have a side length of $20$. They overlap to form the $20$ by $35$ rectangle $AEHD$. Calculate the percentage of the area of rectangle $AEHD$ that is shaded and also find the unshaded area.
|
600
| 0.166667 |
Given that Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side, determine the probability that Bridget and Carlos are seated next to each other.
|
\frac{2}{3}
| 0.25 |
Two numbers are such that their difference, their sum, and twice their product are to one another as $2:8:30$. Find the product of the two numbers.
|
15
| 0.833333 |
Karl's rectangular vegetable garden is 22 feet by 50 feet, while Makenna's garden, initially 30 feet by 46 feet, has a 1-foot wide walking path taking up space around the entire inside perimeter, effectively reducing the planting area. Calculate the difference in area between the two gardens.
|
132
| 0.833333 |
Given the expression $(16^{\log_2 2023})^{\frac{1}{4}}$, evaluate the value of the given expression.
|
2023
| 0.916667 |
Jill's grandmother takes three quarters of a pill every three days to manage her condition, and a medication supply contains 60 pills. Determine the approximate time in months that this supply will last.
|
8
| 0.916667 |
Find the number halfway between $\frac{1}{6}$, $\frac{1}{7}$, and $\frac{1}{8}$.
|
\frac{73}{504}
| 0.5 |
Six test scores have a mean of 85, a median of 88, and a mode of 90. The highest score exceeds the second highest by 5 points. Find the sum of the three highest scores.
|
275
| 0.333333 |
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