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When simplified, $(-\frac{1}{216})^{-2/3}$ calculate the result. | 36 | 0.75 |
Given the number $873$, express it in the form $873 = b_1 + b_2 \times 2! + b_3 \times 3! + \ldots + b_n \times n!$, where $0 \le b_k \le k$, and solve for $b_4$. | 1 | 0.916667 |
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, let $S_n$ be the sum of the elements in the nth set. Find the value of $S_{15}$. | 1695 | 0.916667 |
If Arjun has a $5 \times 7$ index card, and if he shortens one side of this card by 2 inches, the card would have an area of 21 square inches, find the area of the card in square inches if instead he shortens the length of the other side by 2 inches. | 25 | 0.916667 |
A 4x4x4 cube is constructed from 64 standard dice, where opposite dice faces sum to 7. Determine the smallest possible sum of all visible values on the 6 faces of this larger cube. | 144 | 0.166667 |
Given a rectangle where the perimeter is $p$ and the diagonal is $d$, calculate the area of the rectangle. | \frac{p^2 - 4d^2}{8} | 0.666667 |
Using the calculator with only two keys [+1] and [x2], starting from "1", determine the fewest number of keystrokes needed to reach a final display of "250". | 12 | 0.416667 |
Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch reads 12:58 and 12 seconds. Assuming that her watch loses time at a constant rate, determine the actual time when her watch first reads 8:00 PM. | 8:14:51 \text{ PM} | 0.083333 |
If the area of $\triangle ABC$ is $100$ square units and the geometric mean between sides $AB$ and $AC$ is $15$ inches, find $\sin A$. | \frac{8}{9} | 0.916667 |
Determine the relationship between the arithmetic mean of $x$, $y$, and $z$ and their geometric mean. | \frac{x+y+z}{3} \geq \sqrt[3]{xyz} | 0.75 |
Given the numbers $1357$, $3571$, $5713$, and $7135$, calculate their sum. | 17776 | 0.583333 |
Given the expression $\left[(a + 2b)^3 (a - 2b)^3\right]^2$, determine the number of distinct terms when the expression is simplified. | 7 | 0.916667 |
Given that Paul owes Paula $50$ cents and has access to $5$-cent, $10$-cent, $20$-cent, and $25$-cent coins, determine the difference between the largest and the smallest number of coins he can use to pay her. | 8 | 0.666667 |
Given points B and C lie on AD, and the length of AB is 3 times the length of BD, and the length of AC is 7 times the length of CD, determine the fraction of the length of AD that is represented by the length of BC. | \frac{1}{8} | 0.416667 |
Determine the number of different total scores an athlete could achieve with 8 attempts, where each attempt results in either a 2-pointer or a 3-pointer. | 9 | 0.833333 |
A positive integer $N$ with three digits in its base ten representation is chosen at random, with each three-digit number having an equal chance of being chosen. Calculate the probability that $\log_3 N$ is an integer. | \frac{1}{450} | 0.916667 |
In the number $86549.2047$, calculate the ratio of the value of the place occupied by the digit 6 to the value of the place occupied by the digit 2. | 10,000 | 0.083333 |
Given that Jessie moves from 0 to 24 in six steps, and travels four steps to reach point x, then one more step to reach point z, and finally one last step to point y, calculate the value of y. | 24 | 0.5 |
The population of Hypothetical Town at one time was a perfect square. Later, with an increase of $150$, the population was one more than a perfect square. Now, with an additional increase of $150$, the population is again a perfect square. | 5476 | 0.916667 |
Given that Abby and Bridget are seated at a circular table with 6 seats, with Abby and Bridget occupying one of the 6 seats, calculate the probability that they are seated next to each other. | \frac{2}{5} | 0.916667 |
How many distinct triangles can be drawn using three of the dots below as vertices, where the dots are arranged in a grid of 2 rows and 4 columns? | 48 | 0.333333 |
Moe has a new, larger lawn which is a rectangular area of $120$ feet by $200$ feet. His mower has a swath width of $30$ inches and he overlaps each swath by $6$ inches. Moe walks at a pace of $4000$ feet per hour while mowing. Calculate the time it will take him to mow the entire lawn. | 3 | 0.916667 |
Tom buys either a $60$-cent apple or a $90$-cent orange each day over a seven-day week, and his total expenditure for the week was a whole number of dollars. Find the number of oranges he bought. | 6 | 0.916667 |
Given that the mean of three numbers is $20$ more than the least of the numbers and $25$ less than the greatest, and the median of the three numbers is $10$, calculate their sum. | The three numbers are a = -5, b = 10, and c = 40. Their sum is: -5 + 10 + 40 = 45 | 0.916667 |
Evaluate the expression $1 - \frac{1}{1 + \sqrt{5}} + \frac{1}{1 - \sqrt{5}}$. | 1 - \frac{\sqrt{5}}{2} | 0.166667 |
Carlos took $60\%$ of a whole pie. Maria then took one quarter of the remainder. Find the remaining portion of the whole pie. | 30\% | 0.25 |
Assume that x is a positive real number. Simplify the expression \( \sqrt[5]{x\sqrt[4]{x}} \). | x^{1/4} | 0.833333 |
The circle having $(0,0)$ and $(10,8)$ as the endpoints of a diameter intersects the $x$-axis at a second point. Find the $x$-coordinate of this point. | 10 | 0.916667 |
Given an average population of the towns in the Region of Maplefield is halfway between $4,800$ and $5,300$, calculate the total population of all $25$ towns. | 126,250 | 0.666667 |
A farmer plans to fence a rectangular garden using 60 meters of fencing material. One side of the garden borders a river, and thus does not require fencing. If the length of the garden alongside the river is twice as long as its width, calculate the area of the garden. | 450 | 0.666667 |
While Sophia is driving her car on a highway, she notices Sam riding his motorbike in the same direction 1 mile ahead of her. After overtaking him, she can continue to see him in her rearview mirror until he is 1 mile behind her. Sophia drives at a constant speed of 20 miles per hour, and Sam rides at a constant speed of 14 miles per hour. Calculate the time in minutes that Sophia can see Sam. | 20 | 0.833333 |
Let C represent the number of cows, D represent the number of ducks, and H represent the number of chickens. If the number of legs was 20 more than twice the number of heads, then $2C+2D+2H=2(H+C+D)+20$. Also, if the ducks and chickens together were twice the number of cows, then $D+H=2C$. | 10 | 0.583333 |
Given the condition that the product of the digits of a 3-digit positive integer equals 36, find the number of such integers. | 21 | 0.333333 |
Using only pennies, nickels, dimes, quarters, and half-dollars, determine the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar. | 9 | 0.416667 |
A bag contains $\frac{3}{4}$ red kernels and $\frac{1}{4}$ green kernels. Only $\frac{3}{5}$ of the red kernels will pop, whereas $\frac{3}{4}$ of the green ones will pop. Determine the probability that the kernel selected was red. | \frac{12}{17} | 0.666667 |
Bella has two tanks. Initially, the first tank is $\frac{3}{4}$ full of oil and the second tank is completely empty. Bella pours all the oil from the first tank into the second tank, afterward, the second tank is $\frac{2}{5}$ full of oil. Determine the ratio of the volume of the first tank to the volume of the second tank. | \frac{8}{15} | 0.916667 |
Given the expression $\left(\frac{x^4+1}{x^2}\right)\left(\frac{y^4+1}{y^2}\right)+\left(\frac{x^4-1}{y^2}\right)\left(\frac{y^4-1}{x^2}\right)$, where $xy \not= 0$, simplify the expression. | 2x^2y^2 + \frac{2}{x^2y^2} | 0.083333 |
A rectangular box has a total surface area of 166 square inches and the sum of the lengths of all its edges is 64 inches. Find the sum of the lengths in inches of all of its interior diagonals. | 12\sqrt{10} | 0.916667 |
Given that Teresa's age is a prime number, and at least half of the students guessed too low, three students guessed exactly 43, and the incorrect guesses are off by at least two from Teresa's age, find Teresa's age. | 43 | 0.25 |
Suppose the estimated $40$ billion dollar cost to send a scientific team to the moon is shared equally by $200$ million people in a country. Calculate each person's share. | 200 | 0.916667 |
Given that the probability of drawing a red card from a deck is initially $\frac{1}{4}$ and changes to $\frac{1}{6}$ after adding $6$ black cards, determine the original number of cards in the deck. | 12 | 0.916667 |
The polygon enclosed by solid lines consists of 5 congruent squares joined edge-to-edge in a cross shape. One more congruent square is attached to an edge at one of the twelve positions indicated. Determine the number of the resulting polygons that can be folded to form a cube. | 4 | 0.25 |
Given that $E(n)$ denotes the sum of the even digits of $n$, find the sum of $E(1) + E(2) + \cdots + E(999)$. | 6000 | 0.5 |
Given that old license plates consist of a letter followed by three digits and new license plates consist of four letters followed by four digits, determine the ratio of the number of possible new license plates to the number of possible old license plates. | 175760 | 0.25 |
Express the number $700$ in a factorial base of numeration, that is, $700=a_1+a_2\times2!+a_3\times3!+a_4\times4!+ \ldots a_n \times n!$ where $0 \le a_k \le k,$ for each $k$. Find the coefficient $a_4$. | 4 | 0.833333 |
The sum of the numbers on each pair of opposite faces is equal, and the middle number in this range is the largest number on one of the faces, where the numbers on the faces are consecutive whole numbers between 15 and 20. Calculate the sum of all the numbers on this cube. | 105 | 0.666667 |
Find the sum of all prime numbers between $1$ and $120$ that are simultaneously $1$ greater than a multiple of $3$ and $1$ less than a multiple of $5$. | 207 | 0.583333 |
Evaluate the expression $(3(3(3(3+2)+2)+2)+2)$. | 161 | 0.833333 |
Jeremy had exactly enough money to buy 24 posters when he visited a store, but he found a special deal: buy one poster at regular price and get the second poster for half off. Determine the total number of posters Jeremy could buy under this sale. | 32 | 0.916667 |
A rectangular park is to be fenced on three sides using a 150-meter concrete wall as the fourth side. Fence posts are to be placed every 15 meters along the fence, including at the points where the fence meets the concrete wall. Calculate the minimal number of posts required to fence an area of 45 m by 90 m. | 13 | 0.25 |
Two circles are associated with a regular pentagon $ABCDE$. The first circle is tangent to $\overline{AB}$ and the extended sides, while the second circle is tangent to $\overline{DE}$ and also the extended sides. What is the ratio of the area of the second circle to that of the first circle? | 1 | 0.833333 |
Find the number of distinct points in the $xy$-plane common to the graphs of $(x+2y-7)(2x-y+4)=0$ and $(x-2y+3)(4x+3y-18)=0$. | 4 | 0.916667 |
Lilian has two older twin sisters, and the product of their three ages is 162. Find the sum of their three ages. | 20 | 0.416667 |
Let $\frac {42x - 37}{x^2 - 4x + 3} = \frac {N_1}{x - 1} + \frac {N_2}{x - 3}$ be an identity in $x$. Find the numerical value of $N_1N_2$. | -\frac{445}{4} | 0.833333 |
In a regional athletics competition, $275$ sprinters participate in a $100-$meter dash. The track can accommodate $8$ runners at a time, and in each race, the top 2 runners qualify for the next round while the others are eliminated. Determine the number of races required to declare the overall winner. | 49 | 0.25 |
A pair of standard $6$-sided dice is rolled to determine the side length of a square. What is the probability that the numerical value of the area of the square is less than the numerical value of the perimeter? | \frac{1}{12} | 0.416667 |
Given a two-digit number whose tens' digit is $a$ and units' digit is $b$, find the expression for the new number formed when the digit $2$ is placed after it. | 100a+10b+2 | 0.666667 |
Given that the perimeter of rectangle PQRS is 40 cm, find the maximum value of the diagonal PQ in centimeters. | 20 | 0.25 |
Given the number $7.47474747\ldots$, express it as a fraction in lowest terms and find the sum of the numerator and denominator. | 839 | 0.916667 |
Find the largest number by which the expression $n^3 - n - 6$ is divisible for all possible integral values of $n$. | 6 | 0.916667 |
Given Orvin had enough money to buy 24 balloons at the regular price, and he can buy another balloon at half off for each balloon purchased at the regular price, determine the maximum number of balloons Orvin can now purchase. | 32 | 0.916667 |
In an $h$-meter race, Sunny is exactly $2d$ meters ahead of Windy when Sunny finishes the race. The next time they race, Sunny sportingly starts $2d$ meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. Calculate the distance that Sunny is ahead when Sunny finishes the second race. | \frac{4d^2}{h} | 0.166667 |
Given that the speed of sound is $1100$ feet per second, estimate, to the nearest quarter-mile, how far Charlie was from the flash of lightning, given that fifteen seconds passed between the lightning flash and the sound of thunder. | 3.25 | 0.833333 |
A baseball league consists of two four-team divisions. Each team plays every other team in its division N games. Each team plays every team in the other division M games with $N>2M$ and $M>6$. Each team plays a $92$ game schedule. Determine the number of games that a team plays within its own division. | 60 | 0.666667 |
Given that Lucy's odometer initially showed the palindrome $123321$ and after driving for $4$ hours it displayed another palindrome, calculate her average speed during these $4$ hours. | 275 \text{ mph} | 0.916667 |
A circle of radius 3 is centered at point $A$. An equilateral triangle with side length 6 has a vertex at $A$. Find 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 - \sqrt{3}) | 0.25 |
A triangle-shaped grid of points is labeled with vertices A, B, C, and an additional point D, the centroid, inside the triangle. Determine the number of non-congruent triangles that can be formed by selecting vertices from these four points. | 2 | 0.166667 |
John took $80\%$ of a whole pizza. Emma took one fourth of the remainder. Calculate the portion of the whole pizza that was left. | 15\% | 0.416667 |
A farmer needs to fence off a completely rectangular grazing area measuring 48 meters by 72 meters. Fence posts are to be installed every 8 meters around the perimeter. What is the minimum number of fence posts required? | 30 | 0.416667 |
If $x+2$ cows give $x+4$ cans of milk in $x+3$ days, determine the number of days it will take $x+5$ cows to give $x+9$ cans of milk. | \frac{(x+9)(x+2)(x+3)}{(x+5)(x+4)} | 0.75 |
A square with integer side length is cut into 12 squares, 9 of which have an area of 1 and 3 of which have an area of 4 each. Calculate the smallest possible side length of the original square. | 5 | 0.833333 |
Jamie's smartphone battery lasts for 20 hours if the phone is not used but left on, and 4 hours if used constantly. After being on for 10 hours, with 90 minutes of use, how many more hours will the battery last if it is not used but left on? | 4 | 0.166667 |
Determine the value of the product $\left(1-\frac{1}{3^{2}}\right)\left(1-\frac{1}{4^{2}}\right)...\left(1-\frac{1}{12^{2}}\right)$. | \frac{13}{18} | 0.916667 |
The centers of two circles are $50$ inches apart. One circle has a radius of $7$ inches while the other has a radius of $10$ inches. Determine the length of the common internal tangent between the two circles. | \sqrt{2211} | 0.416667 |
Given that the distance EG along Elm Street is 8 miles, the height EF from point F to Elm Street is 4 miles, another triangle EFG' is within triangle EFG where G' is on the line FG, EG' is 4 miles, and the height EF' from F to Elm Street is 2 miles. Determine the area of the triangular plot EFG, excluding the area of triangle EFG'. | 12 | 0.916667 |
Numbers from 1 through 81 are written sequentially on a 9 by 9 grid checkerboard, where each successive row continues the sequence from the previous row. Calculate the sum of the numbers in the four corners of this 9x9 checkerboard. | 164 | 0.666667 |
Evaluate $[\log_{2}(3\log_{2}8)]^2$. | (\log_{2}9)^2 | 0.166667 |
Triangle XYZ has a right angle at Y. Point W is the foot of the altitude from Y, XW=4, and WZ=9. Find the area of triangle XYZ. | 39 | 0.583333 |
The number obtained from the last two nonzero digits of $100!$ is equal to $n$. What is $n$? | 76 | 0.333333 |
Country $X$ has $40\%$ of the world's population and $60\%$ of the world's wealth. Country $Y$ has $20\%$ of the world's population but $30\%$ of its wealth. Country $X$'s top $50\%$ of the population owns $80\%$ of the wealth, and the wealth in Country $Y$ is equally shared among its citizens. Determine the ratio of the wealth of an average citizen in the top $50\%$ of Country $X$ to the wealth of an average citizen in Country $Y$. | 1.6 | 0.416667 |
If $2b$ men can lay $3f$ bricks in $c$ days, calculate the number of days it will take $4c$ men to lay $6b$ bricks. | \frac{b^2}{f} | 0.916667 |
Ahn chooses a three-digit integer, subtracts it from 300, triples the result, and then adds 50. What is the largest number Ahn can obtain? | 650 | 0.75 |
What is the sum of the two smallest prime factors of the number $540$? | 5 | 0.916667 |
How many pairs $(m,n)$ of integers satisfy the equation $(m-2)(n-2)=4$? | 6 | 0.583333 |
Evaluate $\frac{7}{3} + \frac{11}{5} + \frac{19}{9} + \frac{37}{17} - 8$. | \frac{628}{765} | 0.25 |
Given that $\frac{2}{3}$ of the marbles are blue, calculate the fraction of the marbles that will be red if the number of red marbles is tripled and the number of blue marbles stays the same. | \frac{3}{5} | 0.666667 |
Given the store's sales tax rate is 8%, the original price of the jacket is $120, and the promotional discount is 25%, calculate the difference between the total prices computed by Pete and Polly. | 0 | 0.666667 |
Determine how many hours it will take Carl to mow the lawn, given that the lawn measures 120 feet by 100 feet, the mower's swath is 30 inches wide with an overlap of 6 inches, and Carl walks at a rate of 4000 feet per hour. | 1.5 | 0.666667 |
Seven test scores have a mean of $85$, a median of $88$, and a mode of $90$. Calculate the sum of the three lowest test scores. | 237 | 0.5 |
Given a square initially painted black, with $\frac{1}{2}$ of the square black and the remaining part white, determine the fractional part of the original area of the black square that remains black after six changes where the middle fourth of each black area turns white. | \frac{729}{8192} | 0.333333 |
A set of 36 square blocks is arranged into a 6 × 6 square. Calculate the number of different combinations of 4 blocks that can be selected from that set so that no two are in the same row or column. | 5400 | 0.916667 |
Suppose that \( f(x-2) = 4x^2 + 9x + 5 \) and \( f(x) = ax^2 + bx + c \). Find the value of \( a+b+c \). | 68 | 0.666667 |
Bella and Ella start moving towards each other from a distance of 3 miles apart. Bella is walking while Ella is riding a scooter at a speed 4 times as fast as Bella's walking speed. If Bella covers 3 feet with each step, determine the number of steps she has taken by the time they meet. | 1056 | 0.916667 |
Let $\triangle ABC$ be a triangle with angles $\alpha = 59^\circ$, $\beta = 60^\circ$, and $\gamma = 61^\circ$. For each positive integer $n$, define $\alpha_n = 59^\circ + n\cdot 0.02^\circ$, $\beta_n = 60^\circ$, and $\gamma_n = 61^\circ - n\cdot 0.02^\circ$. What is the least positive integer $n$ for which $\triangle ABC$ with angles $\alpha_n$, $\beta_n$, and $\gamma_n$ is obtuse? | 1551 | 0.916667 |
In a square grid composed of $121$ uniformly spaced points, calculate the probability that a line connecting point $P$, at the center of the square, to a randomly chosen point $Q$ from the other $120$ points is a line of symmetry for the square. | \frac{1}{3} | 0.666667 |
A triangle with vertices at $A=(2,4)$, $B=(7,2)$, and $C=(6,5)$ is plotted on an $8 \times 6$ grid. Calculate the fraction of the grid covered by the triangle. | \frac{13}{96} | 0.916667 |
Find the simplest form of $1 - \frac{2}{1 + \frac{b}{1 - 2b}}$ where $b \neq \frac{1}{2}$. | \frac{3b - 1}{1 - b} | 0.666667 |
Suppose that $\frac{3}{4}$ of $12$ bananas are worth as much as $6$ oranges. Determine the number of oranges that are worth as much as $\frac{2}{3}$ of $9$ bananas. | 4 | 0.75 |
For how many three-digit whole numbers does the sum of the digits equal $26$? | 3 | 0.5 |
Given a basketball player made 7 baskets during a game, with each basket worth either 1, 2, or 3 points, determine the number of different total scores the player could achieve. | 15 | 0.916667 |
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