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A rectangle with a width of 5 meters and a length of 15 meters is cut diagonally from one corner to the opposite corner. Determine the dimensions of one of the resulting right-angled triangles. | 5\ \text{m}, 15\ \text{m}, 5\sqrt{10}\ \text{m} | 0.833333 |
If the original price of a jacket is $\textdollar 250$, find the final price after consecutive discounts of 40% and 15% and a sales tax of 5%. | 133.88 | 0.166667 |
A product originally priced at \$120 receives a discount of 8%. Calculate the percentage increase needed to return the reduced price to its original amount. | 8.7\% | 0.833333 |
Given that $\log_{3x}729 = x$, where $x$ is a real number, find the value of $x$. | 3 | 0.75 |
Given that $x$ varies directly as the square of $y$, and $y$ varies directly as the cube root of $z$, determine the power $n$ such that $x$ varies as $x^n$. | \frac{2}{3} | 0.916667 |
Given eleven books consisting of three Arabic, two English, four Spanish, and two French, calculate the number of ways to arrange the books on the shelf keeping the Arabic books together, the Spanish books together, and the English books together. | 34560 | 0.166667 |
A school has $120$ students and $6$ teachers. The class sizes are $40, 35, 25, 10, 5, 5$. Calculate the difference $t - s$ where $t$ is the average number of students a randomly selected teacher sees, and $s$ is the average class size a randomly selected student experiences. | -10 | 0.166667 |
Given the binomial $\left(\frac{x^3}{3} - \frac{1}{x^2}\right)^9$, find the coefficient of $x^3$ in its expansion. | 0 | 0.75 |
A 4x4 grid is to be filled with four A's, four B's, four C's, and four D's such that each row and column contains one of each letter. If 'A' is placed in the upper left corner, calculate the number of possible arrangements. | 144 | 0.083333 |
If \(\frac{x}{x-1} = \frac{y^3 + 2y - 1}{y^3 + 2y - 3}\), calculate the value of \(x\). | \frac{y^3 + 2y - 1}{2} | 0.916667 |
Using the digits 1, 2, 3, 7, 8, and 9, find the smallest sum of two 3-digit numbers that can be obtained by placing each of these digits in the addition problem shown:
____ ____ + ____ = _______ | 417 | 0.583333 |
Given that an item is sold for $x$ dollars with a $20\%$ loss based on the cost and $y$ dollars with a $25\%$ profit based on the cost, find the ratio of $y:x$. | \frac{25}{16} | 0.916667 |
Given that a group of students sit around a large circular table and a bag containing $120$ pieces of candy is passed around, with each person taking two candies and the bag starting and ending with Chris, determine the possible number of students at the table. | 60 | 0.666667 |
Given that there are 30 students in the school club with 15 participating in chess, 18 in soccer, and 12 in music, and 14 participating in at least two activities, determine the number of students participating in all three activities. | 1 | 0.916667 |
Given y = x^2 - px + q, and the least possible value of y is 1, determine q. | 1 + \frac{p^2}{4} | 0.5 |
Given that the mean weight of the rocks in A is 45 pounds, the mean weight of the rocks in B is 55 pounds, the mean weight of the rocks in the combined piles A and B is 48 pounds, and the mean weight of the rocks in the combined piles A and C is 50 pounds, calculate the greatest possible integer value for the mean in pounds of the rocks in the combined piles B and C. | 66 | 0.083333 |
Given that Route X is 8 miles long with an average speed of 40 miles per hour, and Route Y is 7 miles long with 6.5 miles traveled at 50 miles per hour and a 0.5-mile stretch traveled at 10 miles per hour, determine the difference in time taken between the two routes. | 1.2 | 0.666667 |
What is the smallest whole number larger than the perimeter of any triangle with sides of length $5$ and $12$? | 34 | 0.25 |
What is the value of $\dfrac{13! - 12!}{10!}$? | 1584 | 0.75 |
When simplified and expressed with negative exponents, the expression $(x^2 + y^2)^{-1}(x^{-2} + y^{-2})$ equals what value in terms of $x$ and $y$. | x^{-2}y^{-2} | 0.416667 |
Given rocks of three different types are available: 6-pound rocks worth $18 each, 3-pound rocks worth $9 each, and 2-pound rocks worth $3 each, and there are at least 15 rocks available for each type. Carl can carry at most 20 pounds. Determine the maximum value, in dollars, of the rocks Carl can carry out of the cave. | 57 | 0.25 |
If the first jar has a volume $V$ and an alcohol to water ratio of $2p:3$, and the second jar has a volume $2V$ and an alcohol to water ratio of $q:2$, determine the ratio of the volume of alcohol to the volume of water in the mixture. | \frac{\frac{2p}{2p+3} + \frac{2q}{q+2}}{\frac{3}{2p+3} + \frac{4}{q+2}} | 0.666667 |
Jasmine is a contractor who had paint for 50 identically sized rooms. On her route to the project site, she loses 4 cans which reduces her capacity to now cover only 36 rooms. Each can of another type she has adds a capacity of 2 rooms. If she finally manages to cover all 50 rooms using additional cans of another type, determine the number of the second type of cans she used. | 7 | 0.833333 |
In a classroom of 32 students, a majority bought the same number of pens, with each student purchasing more than 1 pen, and the cost of each pen in cents exceeded the number of pens bought by each student. If the total sum spent on the pens was $21.16, determine the cost of one pen in cents. | 23 | 0.083333 |
If a purchase is made at the gadget shop, calculate the final price of an electronic item whose original price is $\textdollar 240$, given a $30\%$ discount before noon, an additional $15\%$ discount on the already reduced price, and a $8\%$ tax applied to the final price. | 154.22 | 0.083333 |
For how many integers $n$ is $\dfrac{n}{24-n}$ the square of an integer? | 2 | 0.833333 |
A triangle has sides measuring $40$, $90$, and $100$ units. If an altitude is dropped to the side measuring $100$ units, find the length of the longer segment created on this side. | 82.5 | 0.083333 |
Let $N = 17^3 + 3 \cdot 17^2 + 3 \cdot 17 + 1$. Determine the number of positive integers that are factors of $N$. | 28 | 0.916667 |
Suppose that $f(x+5) = 4x^2 + 9x + 2$ and $f(x) = ax^2 + bx + c$. Calculate the value of $a+b+c$. | 30 | 0.583333 |
Find the Greatest Common Divisor (GCD) of 7254 and 156, then reduce the result by 10. | 68 | 0.333333 |
Let a sequence $\{v_n\}$ be defined by $v_1 = 7$ and the relationship $v_{n+1} - v_n = 2 + 5(n-1), n = 1, 2, 3, \dots$. If $v_n$ is expressed as a polynomial in $n$, find the algebraic sum of its coefficients. | 7 | 0.833333 |
Given the expression $n^3 - n + 2$, determine the largest number by which this expression is divisible for all possible integral values of $n$. | 2 | 0.916667 |
One dimension of a cube is tripled, another is decreased by `a/2`, and the third dimension remains unchanged. The volume gap between the new solid and the original cube is equal to `2a^2`. Calculate the volume of the original cube. | 64 | 0.333333 |
If the sum of all the angles except two of a convex polygon is $2340^{\circ}$, calculate the number of sides of the polygon. | 16 | 0.583333 |
In a tournament, there are eight teams that play each other twice. A team earns 3 points for a win, 1 point for a draw, and 0 points for a loss. Furthermore, the top four teams earned the same number of total points. Calculate the greatest possible number of total points for each of the top four teams. | 33 | 0.083333 |
Calculate Mr. $X$'s net gain or loss from the transactions, given that he sells his home valued at $12,000$ to Mr. $Y$ for a $20\%$ profit and then buys it back from Mr. $Y$ at a $15\%$ loss. | 2160 | 0.416667 |
Given that Josanna has six test scores: $95, 85, 75, 65, 90, 70$, and she wants to achieve an exact average of $5$ points higher than her current average with her next test, determine the score she needs on her seventh test. | 115 | 0.166667 |
Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. The integers $x$ and $y$ satisfy the equation $x^2 - y^2 = 9m^2$ for some positive integer $m$. Determine $x + y + 2m$. | 143 | 0.583333 |
Given that the speed of sound is 1100 feet per second and a wind is slowing down the sound by 60 feet per second, and considering the time elapsed between Charlie Brown seeing the lightning and hearing the thunder is 15 seconds, determine the distance at which Charlie Brown was from the lightning, rounded off to the nearest half-mile. | 3 | 0.833333 |
Calculate $\log 216$ and determine its value in terms of simpler logarithmic expressions. | 3(\log 2 + \log 3) | 0.416667 |
A cylindrical water tank, placed horizontally, has an interior length of 15 feet and an interior diameter of 8 feet. If the surface area of the water exposed is 60 square feet, find the depth of the water in the tank. | 4 - 2\sqrt{3} | 0.166667 |
Given that a quadratic equation of the form $x^2 + bx + c = 0$ has real roots, where $b$ is chosen from the set $\{-6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6\}$ and $c$ from the set $\{1, 2, 3, 4, 5, 6\}$, determine the number of such equations. | 38 | 0.166667 |
Let Martha the moose take 50 equal steps between consecutive lamp posts on a city street. Let Percy the pronghorn cover the same distance in 15 equal bounds. The lamp posts are evenly spaced, and the 51st post is exactly 2 miles or 10560 feet from the first post. Calculate the difference in length between Percy's bound and Martha's step. | 9.856 | 0.75 |
How many whole numbers between $200$ and $500$ contain the digit $2$? | 138 | 0.083333 |
Given the sequence defined by $g(n) = \dfrac{2 + \sqrt{2}}{4}\left(1 + \sqrt{2}\right)^n + \dfrac{2 - \sqrt{2}}{4}\left(1 - \sqrt{2}\right)^n$, determine $g(n+1) - g(n-1)$ in terms of $g(n)$. | 2g(n) | 0.666667 |
In the republic of Midland, statisticians estimate there is a baby born every $6$ hours and a death every $2$ days. Additionally, an average of $1$ person migrates into Midland every day. Calculate the net change in population of Midland per year, rounded to the nearest ten. | 1640 | 0.833333 |
Given the function \( f \) such that \( f\left(\frac{x}{4}\right) = x^2 - x + 2 \), find the sum of all values of \( z \) for which \( f(4z) = 8 \). | \frac{1}{16} | 0.916667 |
A ferry boat transports visitors to a resort starting at 9 AM until the final ride at 5 PM, on an hourly schedule. On the initial 9 AM trip, there are 120 visitors and with each subsequent trip, the number of visitors decreases by 2. Determine the total number of visitors carried to the resort that particular day. | 1008 | 0.75 |
Given that eight students from Dover High worked for 4 days, five students from Edison High worked for 7 days, and six students from Franklin High worked for 6 days, and a total of 884 dollars was paid to the students, determine the total amount earned by the students from Edison High. | 300.39\ \text{dollars} | 0.083333 |
The pan containing 24-inch by 15-inch brownies is cut into pieces that measure 3 inches by 2 inches. Calculate the total number of pieces of brownies the pan contains. | 60 | 0.5 |
Given the equation $\frac{a+b^{-1}}{a^{-1}+b} = 9$, find the number of pairs of positive integers $(a, b)$ satisfying $a+b \leq 50$. | 5 | 0.5 |
Seven positive consecutive integers start with $a$. Find the average of $7$ consecutive integers that start with the average of the original seven integers. | a + 6 | 0.583333 |
Given that \(a\) is a positive real number and \(b\) is an integer between \(3\) and \(300\), inclusive, find the number of ordered pairs \((a, b)\) that satisfy the equation \((\log_b a)^{101} = \log_b(a^{101})\). | 894 | 0.5 |
How many whole numbers are between $\sqrt{50}$ and $\sqrt{200}$ | 7 | 0.916667 |
Find the number of dimes you must have given that you have eleven coins in total, a collection of pennies, nickels, dimes, and quarters totaling $1.18, and at least one coin of each type is present. | 3 | 0.333333 |
Given two positive numbers $x$ and $y$ in the ratio $a: b^2$, where $0 < a < b$. If $x+y = 2c$, determine the smaller of $x$ and $y$. | \frac{2ac}{a + b^2} | 0.583333 |
Given a set of $n$ numbers where $n > 1$, of which two are $1 + \frac{1}{n}$, one is $1 - \frac{1}{n}$, and all the others are $1$, compute the arithmetic mean of the $n$ numbers. | 1 + \frac{1}{n^2} | 0.916667 |
A half-sector of a circle with a radius of $6$ inches is rolled up to form the lateral surface area of a right circular cone by joining along the two radii. Determine the volume of the cone in cubic inches. | 9\pi \sqrt{3} | 0.916667 |
Given the equation $x^2 + px + 7 = 0$, determine the relationship between the absolute value of the sum of its distinct real roots $r_1$ and $r_2$. | |r_1 + r_2| > 2\sqrt{7} | 0.166667 |
Calculate the area of the polygon with vertices at $(2,1)$, $(4,3)$, $(6,1)$, $(4,-2)$, and $(3,4)$. | \frac{11}{2} | 0.166667 |
John drove 150 miles in 120 minutes. His average speed during the first 40 minutes was 70 mph, and for the next 40 minutes, he traveled at 75 mph. Determine the average speed during the last 40 minutes, in mph. | 80 | 0.916667 |
Given $A$ can finish a task in $12$ days, $B$ is $20\%$ more efficient than $A$, and $C$ works twice as efficiently as $A$, calculate how many days it will take for $B$ and $C$ together to complete the task. | 3.75 | 0.833333 |
Given that $x, y$ and $3x - \frac{y}{3}$ are not zero, evaluate the expression
\[
\left( 3x - \frac{y}{3} \right)^{-1} \left[ (3x)^{-1} + \left( \frac{y}{3} \right)^{-1} \right]^2
\] | \frac{(y + 9x)^2}{3x^2y^2(9x - y)} | 0.083333 |
Given that point $P$ moves perpendicularly to side $AB$ in $\triangle PAB$, $M$ and $N$ are the midpoints of sides $PA$ and $PB$ respectively, determine how many of the four quantities listed below change as $P$ is moved: the length of segment $MN$, the perimeter of the triangle $PAB$, the area of $\triangle PAB$, the area of trapezoid $ABNM$. | 3 | 0.666667 |
In Mr. Calculation's class, a majority of the 36 students purchased notebooks from the school store. Each of these students bought the same number of notebooks, and each notebook cost more in cents than the number of notebooks bought by each student. The total expenditure for all notebooks was 2275 cents. Determine the cost per notebook in cents. | 13 | 0.666667 |
A circle is inscribed in a triangle with side lengths $10, 15, 19$. Let the segments of the side of length $10$, made by a point of tangency, be $r'$ and $s'$, with $r'<s'$. Find the ratio $r':s'$. | 3:7 | 0.083333 |
Given Alex had 40 hits with 2 home runs, 3 triples, and 6 doubles, determine the percentage of his hits that were singles. | 72.5\% | 0.916667 |
Given a student travels from her university to her hometown, a distance of 150 miles, in a sedan that averages 25 miles per gallon, and for the return trip she uses a hybrid car averaging 50 miles per gallon due to a different route of 180 miles, calculate the average gas mileage for the entire trip. | 34.375 | 0.083333 |
Given a box initially containing 3 red, 3 green, and 3 yellow marbles, determine the probability that Cheryl takes 3 marbles of the same color after Carol takes 3 marbles at random from the box and then Cheryl takes the last 3 marbles. | \frac{1}{28} | 0.333333 |
Given that $\frac{2}{3}$ of the marbles are blue and the rest are red, determine the fraction of marbles that will be red after the number of blue marbles is tripled and the number of red marbles stays the same. | \frac{1}{7} | 0.916667 |
A bug starts crawling on a number line at point $3$. It first moves left to $-4$, then turns around and crawls right to $8$. Calculate the total distance the bug crawls. | 19 | 0.916667 |
Given that $(n+1)! + (n+3)! = n! \cdot 1320$, calculate the sum of the positive integer $n$. | 9 | 0.25 |
A right triangle has one angle measuring $30^\circ$. This triangle shares its hypotenuse with a second triangle that is also right-angled. The two triangles together form a quadrilateral. If the other acute angle in the second triangle is $45^\circ$, find the area of the quadrilateral given that the hypotenuse common to both triangles is 10 units long. | \frac{25\sqrt{3} + 50}{2} | 0.416667 |
Given a box contains $4$ shiny pennies and $5$ dull pennies, determine the probability that the third shiny penny appears on the sixth draw. | \frac{5}{21} | 0.166667 |
Given the circle described by $x^2 + y^2 = m$ and the line $x - y = \sqrt{m}$, determine the value of $m$ such that the circle is tangent to the line. | 0 | 0.083333 |
Let $\clubsuit(x)$ denote the sum of the digits of a positive integer $x$. Find the number of two-digit values of $x$ for which $\clubsuit(\clubsuit(x))=4$. | 10 | 0.833333 |
The total price of the microwave from the in-store price is $\textdollar 149.95$, and the microwave from the television commercial is paid in five easy payments of $\textdollar 27.99$ and includes a shipping and handling charge of $\textdollar 14.95$ and an extended warranty fee of $\textdollar 5.95$. Calculate the total amount saved in cents. | 1090 | 0.5 |
A circle of radius 7 is inscribed in a rectangle. The ratio of the length to the width of the rectangle is 3:1. Calculate the area of the rectangle. | 588 | 0.833333 |
Given the expression $3000(3000^{1500} + 3000^{1500})$, evaluate this expression. | 2 \cdot 3000^{1501} | 0.75 |
Joe has a rectangular lawn measuring 120 feet by 180 feet. His lawn mower has a cutting swath of 30 inches, and he overlaps each cut by 6 inches to ensure no grass is missed. Joe mows at a rate of 4000 feet per hour. Calculate the time it will take Joe to mow his entire lawn. | 2.7 | 0.583333 |
In a right triangle, the sides other than the hypotenuse are $a$ and $b$, with $a:b = 2:5$. If the hypotenuse is $c$, and a perpendicular from the right angle vertex to $c$ is drawn, dividing it into segments $r$ and $s$, adjacent respectively to $a$ and $b$, find the ratio of $r$ to $s$. | \frac{4}{25} | 0.833333 |
A box contains 4 red marbles, 3 green marbles, and 3 yellow marbles. Carol takes 3 marbles from the box at random; then Claudia takes 3 marbles from the remaining 7 marbles at random; and then Cheryl takes the last 4 marbles. What is the probability that Cheryl gets at least 2 marbles of the same color?
A) $\frac{1}{10}$
B) $\frac{1}{6}$
C) $\frac{3}{4}$
D) $\frac{9}{10}$
E) $1$ | 1 | 0.583333 |
Each of two boxes contains three chips numbered 1, 2, 3. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. Calculate the probability that their product is a multiple of 3. | \frac{5}{9} | 0.916667 |
What is the sum of the digits of the square of 1111111? | 49 | 0.666667 |
Given that a line extends from $(c, 0)$ to $(2, 2)$ and divides a region into two regions of equal area in a coordinate plane, where four unit squares form a larger square with a side length of 2 units with the lower left corner at the origin, determine the value of $c$. | 0 | 0.25 |
Given that the ratio of $8^\text{th}$-graders to $7^\text{th}$-graders is $7:4$ and the ratio of $7^\text{th}$-graders to $6^\text{th}$-graders is $10:9$, find the smallest number of students that could be participating if all grades are involved. | 73 | 0.916667 |
Simplify $\left(\sqrt[4]{81} - \sqrt{8\frac{1}{2}}\right)^2$. | \frac{35}{2} - 3\sqrt{34} | 0.833333 |
Given that Nadia makes a fruit punch using 150 grams of orange juice with 45 calories in 100 grams, 200 grams of apple juice with 52 calories in 100 grams, 50 grams of sugar with 385 calories in 100 grams, and 600 grams of water, calculate the total calories in 300 grams of her fruit punch. | 109.2 | 0.916667 |
Two circles, each of radius 2, are constructed as follows: The center of circle A is chosen uniformly at random from the line segment joining (0,0) and (2,0). The center of circle B is chosen uniformly at random from the line segment joining (0,2) to (2,2), with the additional condition that the center of B must lie above or on the line y = 1. Calculate the probability that circles A and B intersect. | 1 | 0.166667 |
Given the expression $\left(x\sqrt[4]{3} + y\sqrt[3]{5}\right)^{400}$, calculate the number of terms with rational coefficients in its expansion. | 34 | 0.833333 |
Consider a regular tetrahedron with vertices \(A\), \(B\), \(C\), and \(D\), where each edge of the tetrahedron is of length 1. Let \(P\) be a point on edge \(AB\) such that \(AP = t \times AB\) and \(Q\) be a point on edge \(CD\) such that \(CQ = s \times CD\), where \(0 \leq t, s \leq 1\). Find the least possible distance between points \(P\) and \(Q\). | \frac{\sqrt{2}}{2} | 0.416667 |
In an isosceles trapezoid, if the smaller base equals the length of a diagonal and the larger base equals twice the altitude, find the ratio of the altitude to the larger base. | \frac{1}{2} | 0.75 |
The point is chosen at random within the rectangle in the coordinate plane whose vertices are $(0, 0), (3030, 0), (3030, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{3}$. Find $d$ to the nearest tenth. | 0.3 | 0.333333 |
How many 4-digit numbers greater than 3000 are there that use the four digits of 2033? | 6 | 0.25 |
Determine the 1234th digit to the right of the decimal point in the number \( x = .123456789101112...498499500 \). | 4 | 0.75 |
Given a square EFGH with side length 2, an inscribed circle Ω intersects GH at point N. Line EN intersects Ω at a point Q different from N. Determine the length of EQ. | \frac{\sqrt{5}}{5} | 0.166667 |
How many distinct triangles can be formed by selecting three vertices from a set of points arranged in a 2×4 grid (2 rows and 4 columns)? | 48 | 0.5 |
Let \( x = -2023 \). Find the value of \(\left\vert\left\vert |x|^2 - x\right\vert - |x|\right\vert - x\). | 4094552 | 0.083333 |
Given a square in the coordinate plane with vertices at \((0, 0)\), \((3030, 0)\), \((3030, 3030)\), and \((0, 3030)\), find the value of \(d\) to the nearest tenth, where the probability that a randomly chosen point within the square is within \(d\) units of a lattice point is \(\frac{3}{4}\). | 0.5 | 0.25 |
In a class of $n > 15$ students, the overall mean score on a test is $10$. A group of $15$ students from the class has a mean score of $16$. What is the mean of the test scores for the rest of the students in terms of $n$? | \frac{10n - 240}{n-15} | 0.916667 |
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