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If two factors of $3x^3 - hx + k$ are $x+2$ and $x-1$, find the value of $|3h-2k|$. | 15 | 0.916667 |
Two pictures, each 4 feet across, are hung in the center of a wall that is 30 feet wide. The pictures are spaced 1 foot apart. Find the distance from the end of the wall to the nearest edge of the first picture. | 10.5 | 0.166667 |
Given the yearly changes in the population census of a city for three consecutive years are, respectively, a 40% increase, a 15% decrease, and a 15% decrease, calculate the net change over the three years, rounded to the nearest percent. | 1\% | 0.75 |
A circular region has a radius of 4 units. What is the probability that a point chosen at random from within this region is closer to the center than it is to the boundary of the region? | \frac{1}{4} | 0.916667 |
A ticket for a concert costs $x$ dollars, where $x$ is a whole number. A group of middle schoolers buys tickets costing a total of $90$, and a group of high schoolers buys tickets costing a total of $150$. Find the number of values for $x$ that are possible. | 8 | 0.916667 |
The smallest whole number larger than the perimeter of any triangle with sides of lengths $6$ and $21$ and a third side of length $x$ can be found by solving for x in the triangle inequality, $x>21-6$ and $x<6+21$. | 54 | 0.416667 |
Given that the endpoints of a line segment are $(7,31)$ and $(61,405)$, determine how many lattice points are on the line segment, including both endpoints. | 3 | 0.583333 |
Consider the sequence of numbers where each integer \( n \) appears \( n \) times, for \( 1 \leq n \leq 100 \). What is the median of the numbers in this list? | 71 | 0.916667 |
Given that a rectangular piece of metal with dimensions 8 units by 10 units is used to cut out the largest possible circle, which is then used to cut out the largest possible square, calculate the total amount of metal wasted. | 48 | 0.416667 |
Given a rectangular prism with distinct edge lengths, determine the number of unordered pairs of edges that determine a plane. | 42 | 0.083333 |
The taxi fare in New Metro City is $3.00 for the first mile and additional mileage charged at the rate $0.25 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate the number of miles that can be ridden for $15. | 4.6 | 0.666667 |
What is the largest number of solid $3\text{-in} \times 2\text{-in} \times 1\text{-in}$ blocks that can fit in a $4\text{-in} \times 3\text{-in}\times3\text{-in}$ box? | 6 | 0.416667 |
Given Carlos took $65\%$ of a whole pie, then Maria took half of what remained. Calculate the portion of the whole pie that was left. | 17.5\% | 0.833333 |
In the $xy$-plane, find the number of lines whose $x$-intercept is a positive integer and whose $y$-intercept is a positive prime number that pass through the point $(5,4)$. | 1 | 0.75 |
Given that the team earned $72$ points in total, where the points from three-point shots were double the points from two-point shots, and the number of successful free throws was twice the number of successful two-point shots, determine the number of free throws they made. | 18 | 0.916667 |
For a triangle with side lengths $12, 30$, and $x$ to have all its angles acute, determine the number of integers $x$ that satisfy the condition. | 5 | 0.583333 |
A line $x=k$ intersects the graph of $y=\log_2 x$ and the graph of $y=\log_2 (x + 2)$. The distance between the points of intersection is $1$. Given that $k = a + \sqrt{b}$, where $a$ and $b$ are integers, find the value of $a+b$. | 2 | 0.916667 |
Let $ABCDEF$ be an equiangular convex hexagon with a perimeter of 2. Determine the perimeter $s$ of the six-pointed star formed by the pairwise intersections of the extended sides of the hexagon. | 4 | 0.916667 |
Paul owes Paula $60$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, $25$-cent coins, and $50$-cent coins that he can use to pay her. Calculate the difference between the largest and smallest number of coins he can use to pay her. | 10 | 0.916667 |
If the sum of the first 15 terms and the sum of the first 85 terms of an arithmetic progression (AP) are 45 and 255, respectively, then find the sum of the first 100 terms. | 300 | 0.833333 |
Calculate the equivalent single discount for successive discounts of $15\%$ and $25\%$ on an item. | 36.25\% | 0.916667 |
A tetrahedron has 4 vertices. You choose three vertices at random to form a triangle. | 1 | 0.083333 |
Given that points $P$ and $Q$ are on a circle of radius $10$ and $PQ=12$, and point $R$ is the midpoint of the minor arc $PQ$, calculate the length of the line segment $PR$. | 2\sqrt{10} | 0.5 |
Given the expression $\frac{x^2-4x+3}{x^2-6x+9} \div \frac{x^2-3x+2}{x^2-4x+4},$ simplify the expression. | \frac{x-2}{x-3} | 0.833333 |
How many positive integer divisors of $221^{10}$ are perfect squares or perfect cubes (or both)? | 48 | 0.333333 |
Find the quadratic equation whose roots sum up to $7$ and the absolute value of whose difference is $9$. | x^2-7x-8=0 | 0.166667 |
Determine the smallest positive integer \(x\) for which \(1800x = M^3\), where \(M\) is an integer. | 15 | 0.916667 |
Determine the number of distinct terms in the expansion of $[(x+4y)^2(x-4y)^2]^3$ when simplified. | 7 | 0.666667 |
Positive integers $a$ and $b$ are such that $a$ is less than $6$ and $b$ is less than $10$. Calculate the smallest possible value for $2 \cdot a - a \cdot b$. | -35 | 0.666667 |
Determine the largest number by which the expression $2n^3 - 2n$ is divisible for all integral values of $n$. | 12 | 0.333333 |
Determine the maximum value of the function $10x - 4x^2$ for any real value of $x$. | \frac{25}{4} | 0.916667 |
Find the minimum value of $\sqrt{x^2+y^2}$ given the constraints $3x+4y=24$ and $x \geq 0$. | \frac{24}{5} | 0.5 |
On a redesigned dartboard, the outer circle radius is increased to $8$ units and the inner circle has a radius of $4$ units. Additionally, two radii divide the board into four congruent sections, each labeled inconsistently with point values as follows: inner sections have values of $3$ and $4$, and outer sections have $2$ and $5$. The probability of a dart hitting a particular region is still proportional to its area. Calculate the probability that when three darts hit this board, the total score is exactly $12$.
A) $\frac{9}{2048}$
B) $\frac{9}{1024}$
C) $\frac{18}{2048}$
D) $\frac{15}{1024}$ | \frac{9}{1024} | 0.416667 |
The mean of three numbers is $20$ more than the least of the numbers and $30$ less than the greatest. The median of the three numbers is $10$. Calculate the sum of these three numbers. | 60 | 0.75 |
Given four integers $p, q, r,$ and $s$, such that their sums taken three at a time are $210, 230, 250,$ and $270$, find the largest of these four integers. | 110 | 0.916667 |
A can complete a piece of work in 12 days. B is 33% less efficient than A. Determine the number of days it takes B to do the same piece of work. | 18 | 0.416667 |
In parallelogram $ABCD$, $BE$ is the height from vertex $B$ to side $AD$, and segment $ED$ is extended from $D$ such that $ED = 8$. The base $BC$ of the parallelogram is $14$. The entire parallelogram has an area of $126$. Determine the area of the shaded region $BEDC$. | 99 | 0.083333 |
Calculate the harmonic mean of 2, 3, and 6.5. | \frac{234}{77} | 0.833333 |
How many unordered pairs of edges in a regular tetrahedron determine a plane? | 12 | 0.25 |
A positive integer n has 72 divisors, and 3n has 96 divisors. What is the greatest integer k such that 3^k divides n? | 2 | 0.75 |
Given the set $\{41, 57, 82, 113, 190, 201\}$, find the number of subsets containing three different numbers such that the sum of the three numbers is odd. | 8 | 0.166667 |
A telephone number is represented as $\text{ABC-DEF-GHIJ}$, with each letter representing a unique digit. The digits in each segment are in decreasing order: $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore, $D, E,$ and $F$ are consecutive digits, not limited to even or odd. $G, H, I,$ and $J$ are also consecutive digits that include both odd and even numbers. Given that $A + B + C = 17$, determine the value of $A$. | 9 | 0.75 |
What is the smallest prime number dividing the sum $4^{15} + 6^{17}$? | 2 | 0.75 |
A parking lot has 20 spaces in a row. Fifteen cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. Calculate the probability that she is able to park. | \frac{232}{323} | 0.666667 |
Given that one root of $3x^2 + rx + s = 0$, with $r$ and $s$ real numbers, is such that its square is $4-3i$, determine the value of $s$. | 15 | 0.833333 |
The average of the numbers $4, 6, 9, a, b, c, d$ is $20$. Find the average of $a$, $b$, $c$, and $d$. | 30.25 | 0.75 |
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 55? | 3599 | 0.25 |
Given that twelve points are equally spaced around a $3 \times 3$ square, calculate the probability that two randomly chosen points are one unit apart. | \frac{2}{11} | 0.333333 |
David drives from his home to the airport to catch a flight. He drives 40 miles in the first hour, but realizes that he will be 45 minutes late if he continues at this speed. He increases his speed by 20 miles per hour for the rest of the way to the airport and arrives 15 minutes early. Determine the distance from his home to the airport. | 160 | 0.583333 |
Given the speed of sound is approximated as 1100 feet per second, the time between the lightning flash and the thunder is 15 seconds, and one mile is 5280 feet, estimate, to the nearest quarter-mile, how far Charlie Brown was from the flash of lightning. | 3.25 | 0.75 |
Given that $\frac{1}{4}$ of all eighth graders are paired with $\frac{1}{3}$ of all fifth graders for a buddy system, find the fraction of the total number of fifth and eighth graders that have a buddy. | \frac{2}{7} | 0.5 |
Evaluate the expression $2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{3}}}$. | \frac{41}{17} | 0.666667 |
Given three wheels with Wheel One having three even and one odd number, Wheel Two having two even and two odd numbers, and Wheel Three having one even and three odd numbers, calculate the probability that the sum of the results from spinning all three wheels is even. | \frac{1}{2} | 0.666667 |
The region consisting of all points in three-dimensional space within $5$ units of line segment $\overline{CD}$ has a volume of $900 \pi$. What is the length of $\overline{CD}$?
A) 26.67
B) 27.33
C) 28.67
D) 29.33 | D) = 29.33 | 0.833333 |
A chord, which is the perpendicular bisector of a radius of length 10 in a circle, intersects the radius at its midpoint. Calculate the length of the chord. | 10\sqrt{3} | 0.833333 |
If 1 quart of paint is required to paint a statue 8 ft. high, determine the total amount of paint required to paint 360 statues, each 2 ft. high, with each statue needing a double layer of paint for full coverage. | 45 | 0.083333 |
Given two different third degree polynomial functions y=p(x) and y=q(x), each with leading coefficient 1, calculate the maximum number of points of intersection of their graphs. | 2 | 0.083333 |
Six test scores have a mean of $92$, a median of $93$, and a mode of $94$. Find the sum of the two lowest test scores. | 178 | 0.666667 |
Given a number $2310$, find how many of its positive integer factors have more than four factors. | 16 | 0.166667 |
Given that triangle PQR is an isosceles triangle with an area of 100 and contains 20 smallest identical triangles, each with an area of 1, and PQS is composed of 6 of these smallest triangles and is similar to PQR, find the area of trapezoid RQS. | 94 | 0.666667 |
Given that Elmer's new car provides $80\%$ better fuel efficiency than his old car, and the diesel used by the new car is $30\%$ more expensive per liter than the gasoline used by the old car, calculate the percentage Elmer will save on fuel if he uses his new car for a long trip. | 27.78\% | 0.583333 |
Find the smallest integer value of $m$ such that $3x(mx-5)-2x^2+7=0$ has no real roots. | 4 | 0.833333 |
A rectangle in the coordinate plane has vertices at $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. Compute the radius $d$ to the nearest tenth such that the probability the point is within $d$ units from any lattice point is $\tfrac{1}{4}$. | 0.3 | 0.166667 |
A pair of standard $8$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. Calculate 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} | 0.5 |
Given two circles of radii $4$ and $5$, and distance $d$ between their centers, determine how many different values of $k$ are possible when drawing all possible lines simultaneously tangent to both circles. | 5 | 0.25 |
A rope is cut at a random point along its length of 1 meter. What is the probability that the longer piece is at least three times as large as the shorter piece? | \frac{1}{2} | 0.833333 |
Given Jasmine has two types of bottles, one that can hold 45 milliliters and another that can hold 675 milliliters, and a vase that can hold 95 milliliters, determine the total number of small bottles she must buy to fill the large bottle as much as possible and the vase. | 18 | 0.333333 |
Given a two-digit positive integer $N$, find how many such integers have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ equals twice a perfect square. | 0 | 0.416667 |
Determine how many ordered pairs of integers \( (x, y) \) satisfy the equation \( x^{4} + y^2 = 4y. \) | 2 | 0.666667 |
Given the progression $2^{\dfrac{1}{7}}, 2^{\dfrac{2}{7}}, 2^{\dfrac{3}{7}}, 2^{\dfrac{4}{7}}, \dots , 2^{\dfrac{n}{7}}$, find the least positive integer $n$ such that the product of the first $n$ terms of the progression exceeds $1,024$. | 12 | 0.916667 |
Given that Mary is $30\%$ older than Sally, and Sally is $25\%$ younger than Danielle, and the sum of their ages is $30$ years, calculate Mary's age on her next birthday. | 11 | 0.166667 |
Given triangle $\triangle DEF$ has vertices $D = (4,0)$, $E = (0,4)$, and $F$ is on the line $x + y = 9$, determine the maximum area of $\triangle DEF$. | 10 | 0.833333 |
A circle with radius 7 is inscribed in a rectangle and has a ratio of length to width of 3:1. Calculate the area of the rectangle. | 588 | 0.916667 |
Find the coefficient of \(x^8\) in the expansion of \(\left(\frac{x^3}{3} - \frac{3}{x^2}\right)^9\). | 0 | 0.916667 |
Find $f(2x)$ for $x \neq 1$ and $x \neq 2$, where $f(x) = \frac{2x+3}{x-2}$. | \frac{4x+3}{2(x-1)} | 0.333333 |
Four years ago, Sam was four times as old as his brother Tim. Six years before that, Sam was five times as old as Tim. Find the number of years after which the ratio of their ages will be 3 : 1. | 8 | 0.083333 |
How many distinguishable rearrangements of the letters in "BALANCE" have all the vowels at the end. | 72 | 0.916667 |
Given points $A(-3, -4)$ and $B(6, 3)$ in the xy-plane; point $C(1, m)$ is taken so that $AC + CB$ is a minimum. Find the value of $m$. | -\frac{8}{9} | 0.083333 |
A regular hexagon $ABCDEF$ with side length 2 has two circles positioned outside it. The first circle is tangent to $\overline{AB}$ and the second circle is tangent to $\overline{CD}$. Both circles are also tangent to lines $BC$ and $FA$. Find the ratio of the area of the second circle to that of the first circle. | 1 | 0.333333 |
The total in-store price for a blender is $\$75.99$. A television commercial advertises the same blender for four easy payments of $\$17.99$ and a one-time shipping fee of $\$6.50$ coupled with a handling charge of $\$2.50$. Calculate the number of cents saved by purchasing the blender from the television advertiser. | 497 | 0.583333 |
A box 2 centimeters high, 4 centimeters wide, and 6 centimeters long can hold 48 grams of clay. A second box with three times the height, twice the width, and one and a half times the length as the first box can hold n grams of clay. Determine the value of n. | 432 | 0.916667 |
Given that the product of 900 and $x$ is a square and the product of 900 and $y$ is a fifth power, determine the sum of $x$ and $y$. | 27001 | 0.833333 |
A school has $120$ students and $6$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $40, 40, 20, 10, 5, 5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. Find the value of $t-s$. | -11.25 | 0.583333 |
A beam of light originates from point $A$ on a plane and is reflected several times between lines $AD$ and $CD$ before finally striking a point $B$ on line $CD$ at a $15^\circ$ angle and then reflecting back to point $A$. If the angle $\measuredangle CDA=10^\circ$, determine the maximum number of times the light can be reflected before striking $B$. | 8 | 0.083333 |
The line segment whose endpoints are $(5, 23)$ and $(60, 353)$ contains how many lattice points? | 56 | 0.916667 |
John drove 180 miles in 180 minutes. His average speed during the first 45 minutes was 40 mph, during the second 45 minutes was 50 mph, and during the third 45 minutes was 60 mph. Calculate John's average speed, in mph, during the last 45 minutes. | 90 \text{ mph} | 0.916667 |
Given the number $825$, convert it into a factorial base of numeration, that is, $825 = a_1 + a_2 \times 2! + a_3 \times 3! + \dots + a_n \times n!$ where $0 \le a_k \le k$. Find the value of $a_5$. | 0 | 0.75 |
Given the equations $2002C - 3003A = 6006$ and $2002B + 4004A = 8008$, as well as $B - C = A + 1$, calculate the average of A, B, and C. | \frac{7}{3} | 0.916667 |
Given Maya takes 45 minutes to ride her bike to a beach 15 miles away, and Naomi takes 15 minutes to cover half the distance on a bus, stops for 15 minutes, and then takes another 15 minutes to cover the remaining distance. Find the difference, in miles per hour, between Naomi's and Maya's average speeds. | 0 | 0.833333 |
A rectangle has length $AC = 40$ and width $AE = 25$. Point $B$ is the midpoint of $\overline{AC}$, making $AB = BC = 20$, but point $F$ is now three-fifths of the way along $\overline{AE}$ from $A$ to $E$, making $AF = 15$ and $FE = 10$. Find the area of quadrilateral $ABDF$. | 550 | 0.666667 |
A rectangular floor that is 12 feet wide and 18 feet long is tiled with 216 one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, calculate the number of tiles the bug visits. | 24 | 0.833333 |
The taxi fare in Metropolis City is $3.00 for the first $\frac{1}{2}$ mile and additional mileage charged at the rate $0.30 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate how many miles can you ride for $15. | 3.5 | 0.583333 |
Consider a sequence \(v_n = n^4 + 2n^2 + 2\). Define the sequence of differences \(\Delta^k(v_n) = \Delta^1(\Delta^{k-1}(v_n))\). Identify for which \(k\) the sequence \(\Delta^k(v_n)\) becomes zero for all \(n\). | 5 | 0.916667 |
Given Mia works 3 hours a day and is paid $0.40 per hour for each full year of her age, and during a nine-month period, Mia worked 80 days and earned $960, determine Mia's age at the end of the nine-month period. | 11 | 0.083333 |
Given a cyclic quadrilateral $ABCD$ with point $E$ such that $AB$ is extended to $E$. Assume $\measuredangle BAD = 85^\circ$ and $\measuredangle ADC = 70^\circ$, calculate the angle $\measuredangle EBC$. | 70^\circ | 0.416667 |
Consider those functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions. | 36 | 0.416667 |
In a circle with a radius of 8 units, points A and B are such that the length of chord AB is 10 units. Point C is located on the major arc AB such that the arc length AC is a third of the circumference of the circle. Calculate the length of the line segment AC. | 8\sqrt{3} | 0.75 |
Mr. C owns a home worth $15,000$. He sells it to Mr. D at a 15% profit based on the worth of the house. Mr. D then sells the house back to Mr. C at a 5% loss. Calculate Mr. C's net gain or loss from these transactions. | 862.50 \text{ dollars} | 0.083333 |
Let x be the total number of seventh graders and y be the total number of tenth graders. If $\frac{1}{4}$ of all tenth graders are paired with $\frac{3}{7}$ of all seventh graders, what fraction of the total number of seventh and tenth graders have a partner? | \frac{6}{19} | 0.416667 |
Let A, B, and C be three piles of rocks. The mean weight of the rocks in A is 30 pounds, and in B it is 55 pounds. The mean weight of the rocks in the combined piles A and B is 35 pounds, and in the combined piles A and C it is 32 pounds. Find the greatest possible integer value for the mean in pounds of the rocks in the combined piles B and C. | 62 | 0.166667 |
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