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Given six test scores have a mean of $85$, a median of $88$, and a mode of $90$. Calculate the sum of the two lowest test scores. | 154 | 0.75 |
A cube has faces numbered with consecutive integers, including negative numbers. The sum of the numbers on each pair of opposite faces is the same. If the smallest number is -3, find the sum of all six numbers on this cube. | -3 | 0.916667 |
Given that point B is the intersection of three lines, forming angles $\angle \text{ABC}$, $\angle \text{ABD}$, and $\angle \text{CBD}$, and that $\angle \text{CBD}$ is a right angle and $\angle \text{ABD} = 30^\circ$, find the measure of $\angle \text{ABC}$ given that the sum of the angles around point B is $180^\circ$. | 60 | 0.833333 |
How many $3$-digit positive integers have digits whose product equals $36$? | 21 | 0.083333 |
A cell phone plan costs $20$ dollars each month, plus $0.07$ dollars per text message sent, plus $0.15$ dollars for each minute used over 30 hours. In February, Alex sent 150 text messages and talked for 32 hours. Calculate the total amount he paid. | \$48.50 | 0.833333 |
Given that 6-pound rocks are valued at $18 each, 3-pound rocks are valued at $9 each, and 2-pound rocks are valued at $6 each, and that Alice can carry at most 21 pounds and has at least 15 of each size available, determine the maximum value of the rocks Alice can carry out of the cave. | 63 | 0.666667 |
Given \(2^{2000} - 2^{1999} - 3 \times 2^{1998} + 2^{1997} = l \cdot 2^{1997}\), determine the value of \(l\). | -1 | 0.333333 |
A container holds $26$ orange balls, $21$ purple balls, $20$ brown balls, $15$ gray balls, $12$ silver balls, and $10$ golden balls. Calculate the minimum number of balls that must be drawn from the container without replacement to ensure that at least $17$ balls of one color are retrieved. | 86 | 0.333333 |
Given that $\log_{2}{m}= 2b-\log_{2}{(n+1)}$, solve for $m$. | \frac{2^{2b}}{n+1} | 0.166667 |
Given the data set $[4, 21, 34, 34, 40, 42, 42, 44, 52, 59]$, calculate the number of outliers. | 1 | 0.583333 |
Jayden purchased four items for $\textdollar 2.93$, $\textdollar 7.58$, $\textdollar 12.49$, and $\textdollar 15.65$. Find the total amount spent, rounded to the nearest ten cents. | 38.70 | 0.833333 |
Given a list of seven numbers, the average of the first four numbers is $6$, and the average of the last four numbers is $9$. If the average of all seven numbers is $7\frac{6}{7}$, find the value of the number common to both sets of four numbers. | 5 | 0.833333 |
During her six-day workweek, Mary buys a beverage either coffee priced at $60$ cents or tea priced at $80$ cents. Twice a week, she also buys a $40$-cent cookie along with her beverage. Her total cost for the week amounts to a whole number of dollars. How many times did she buy tea? | 3 | 0.833333 |
Consider a sequence of hexagons where each new hexagon adds two layers of dots instead of one. The first hexagon has 1 dot. The second hexagon has 1 central dot, 6 dots in the first layer, and 12 dots in the second layer, making a total of 19 dots. Determine the total number of dots in the third hexagon. | 61 | 0.083333 |
Determine the largest number by which the expression $n^4 - n^2$ is divisible for all possible integral values of $n$. | 12 | 0.416667 |
Compute the value of $\frac{8}{4 \times 25}$. | 0.08 | 0.083333 |
The length of rectangle $XYZW$ is 8 inches and its width is 6 inches. Diagonal $XW$ is divided in the ratio 2:1 by point $P$. Calculate the area of triangle $YPW$. | 8 | 0.916667 |
Elena drives 45 miles in the first hour, but realizes that she will be 45 minutes late if she continues at the same speed. She increases her speed by 20 miles per hour for the rest of the journey and arrives 15 minutes early. Determine the total distance from Elena's home to the convention center. | 191.25 | 0.25 |
Given the expression $n$ represents a positive integer, determine the number of integer solutions for which both $\frac{n}{4}$ and $4n$ are four-digit whole numbers. | 0 | 0.916667 |
Four runners start at the same point on a 400-meter circular track and run clockwise with constant speeds of 3 m/s, 3.5 m/s, 4 m/s, and 4.5 m/s, respectively. Determine the time in seconds they will run before they meet again anywhere on the course. | 800 | 0.333333 |
Let $a + 2 = b + 4 = c + 6 = d + 8 = a + b + c + d + 10$. Calculate the value of $a + b + c + d$. | -\frac{20}{3} | 0.75 |
Given that x is a perfect square, find the second next larger perfect square. | x + 4\sqrt{x} + 4 | 0.75 |
There are two integers greater than $1$ which, when divided by any integer $k$ such that $2 \le k \le 13$, has a remainder of $1$. Find the difference between the two smallest such integers. | 360360 | 0.75 |
What is the degree measure of the angle formed by the hands of a clock at 1:30? | 135^{\circ} | 0.916667 |
Let S be the set of the 1500 smallest positive multiples of 5, and let T be the set of the 1500 smallest positive multiples of 7. Find the number of elements common to S and T. | 214 | 0.916667 |
Cheenu used to cycle 24 miles in 2 hours as a young man and now he can jog 18 miles in 3 hours as an older man. Calculate the difference in minutes per mile between his cycling speed and his jogging speed. | 5 | 0.916667 |
A man buys a house for $15,000 and wants to achieve a $6\%$ return on his investment while incurring a yearly tax of $450$, along with an additional $200$ yearly for owner's insurance. The percentage he sets aside from monthly rent for maintenance remains $12\frac{1}{2}\%$. Calculate the monthly rent. | 147.62 | 0.5 |
Given a polynomial \(g(x)\) with leading coefficient 1, whose four roots are the reciprocals of the four roots of \(f(x) = x^4 + ax^3 + bx^2 + cx + d\), find \(g(1)\) in terms of \(a, b, c,\) and \(d\). | \frac{1 + a + b + c + d}{d} | 0.166667 |
Given a device purchased at $40 less a 15% discount, and a goal of achieving a 25% selling profit based on the cost after allowing a 10% discount on the final sales price, determine the marked price of the device in dollars. | 47.22 | 0.25 |
Two friends, Alice and Bob, start cycling towards a park 80 miles away. Alice cycles 3 miles per hour slower than Bob. Upon reaching the park, Bob immediately turns around and starts cycling back, meeting Alice 15 miles away from the park. Find the speed of Alice. | 6.5 | 0.666667 |
Given that $x$ is a positive real number, simplify the expression $\sqrt[4]{x\sqrt[3]{x}}$. | x^{1/3} | 0.916667 |
Two subsets of the set $T = \{w, x, y, z, v\}$ need to be chosen so that their union is $T$ and their intersection contains exactly three elements. How many ways can this be accomplished, assuming the subsets are chosen without considering the order? | 20 | 0.416667 |
Given a rectangle $R_1$ with one side of length $3$ inches and an area of $18$ square inches, and a similar rectangle $R_2$ with a diagonal of length $20$ inches, calculate the area of $R_2$. | 160 | 0.916667 |
Given that $P(n)$ denotes the greatest prime factor of $n$ and $P(n+36) = \sqrt{n+36}$, determine the number of positive integers $n$ for which $P(n) = \sqrt{n}$. | 0 | 0.666667 |
Consider the expression $\sqrt{\frac{4}{5}} - \sqrt[3]{\frac{5}{4}}$. Evaluate its value. | \frac{2\sqrt{5}}{5} - \frac{\sqrt[3]{5}}{\sqrt[3]{4}} | 0.583333 |
$5y$ varies inversely as the cube of $x$. When $y=4, x=2$. Find the value of $y$ when $x=4$. | \frac{1}{2} | 0.916667 |
Given the number 3080, determine the sum of its prime factors. | 25 | 0.083333 |
Given that there are 2023 boxes in a line, with the box in the kth position containing k white marbles, determine the smallest value of n for which the probability that Isabella stops after drawing exactly n marbles is less than 1/2023. | 45 | 0.083333 |
Find the number of distinct ordered pairs $(x,y)$ where $x$ and $y$ have positive integral values satisfying the equation $x^4 y^4 - 16x^2 y^2 + 15 = 0$. | 1 | 0.75 |
Each of two boxes contains four chips numbered $1$, $2$, $3$, and $4$. Calculate the probability that the product of the numbers on the two chips is a multiple of $4$. | \frac{1}{2} | 0.083333 |
Alice, Bob, and Chris went on a fishing trip and decided to split all costs equally. Alice paid $135, Bob paid $165, and Chris paid $225. To equalize their shares, Alice gave Chris $x$ dollars, and Bob gave Chris $y$ dollars. Calculate $x - y$. | 30 | 0.916667 |
A large cube $n$ units on each side is fully painted blue on all six faces and then cut into $n^3$ smaller cubes. Exactly one-third of the total number of faces of these smaller cubes are blue. What is $n$? | 3 | 0.833333 |
Given that $i^2 = -1$, for how many integers $n$ is $(n+i)^5$ an integer? | 0 | 0.916667 |
Calculate the product of the expressions $x^4$, $x - \frac{1}{x}$, and $1 + \frac{1}{x} + \frac{1}{x^2}$ and find its degree. | 5 | 0.333333 |
A checkerboard of 15 rows and 15 columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered 1,2,...,15, the second row 16,17,...,30, and so on down the board. If the board is renumbered so that the left column, top to bottom, is 1,2,...,15, the second column 16,17,...,30 and so on across the board, find the sum of the numbers in the squares that have the same numbers in both numbering systems. | 1695 | 0.916667 |
What is the sum of the digits of the greatest prime number that is a divisor of 32,767? | 7 | 0.75 |
Determine how many lattice points are located on the line segment whose endpoints are $(5, 11)$ and $(35, 221)$. | 31 | 0.916667 |
Let $y = 3(x-a)^2 + (x-b)^2 + kx$, where $a, b$ are constants and $k$ is a new constant. Find the value of $x$ at which $y$ is a minimum. | \frac{6a + 2b - k}{8} | 0.083333 |
Evaluate the expression $2^{-(3k+1)}-3 \cdot 2^{-(3k-1)}+4 \cdot 2^{-3k}$. | -\frac{3}{2} \cdot 2^{-3k} | 0.166667 |
If a jar contains $5$ different colors of gumdrops, where $40\%$ are blue, $15\%$ are brown, $10\%$ are red, $20\%$ are yellow, and the rest are green, and there are $50$ green gumdrops in total, calculate the number of yellow gumdrops after a third of the red gumdrops are replaced with yellow gumdrops. | 78 | 0.916667 |
Given $(9x)^{18} = (18x)^9$, determine the non-zero real value for $x$. | \frac{2}{9} | 0.833333 |
Given that the Sunshine Café sold 310 cups of coffee to 120 customers in one morning, and each customer purchased at least one cup of coffee, determine the maximum possible median number of cups of coffee bought per customer that morning. | 4.0 | 0.333333 |
A circle with a radius of 7 is inscribed in a rectangle, and the ratio of the rectangle's length to its width is 3:1, whereas a square is inscribed inside the same circle. Calculate the total area of the rectangle and the square. | 686 | 0.916667 |
A positive integer $n$ has $72$ divisors and $5n$ has $90$ divisors. Find the greatest integer $k$ such that $5^k$ divides $n$. | 3 | 0.5 |
The minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30}\cdot 5^3}$ as a decimal. | 30 | 0.333333 |
A 20-quart radiator is filled completely with water. Five quarts are removed and replaced with pure antifreeze liquid. Then five quarts of the mixture are removed and replaced with pure antifreeze. This process is repeated three more times, for a total of five replacements. Calculate the fractional part of the final mixture that is water. | \frac{243}{1024} | 0.75 |
A conical container with a base radius of $14 \mathrm{cm}$ and height of $20 \mathrm{cm}$ is completely filled with water. This water is then carefully poured into a cylindrical container with a radius of $28 \mathrm{cm}$. If 10% of the water is spilled during transfer, calculate the height of the remaining water in the cylindrical container. | 1.5 | 0.833333 |
Consider a rhombus with diagonals such that one diagonal is three times the length of the other. Express the side length of the rhombus in terms of the area K. | \sqrt{\frac{5K}{3}} | 0.666667 |
Given that Crystal runs due north for 2 miles, then northwest for 1 mile, and southwest for 1 mile, find the distance of the last portion of her run that returns her directly to her starting point. | \sqrt{6} | 0.583333 |
Given that Big Al, the ape, ate 140 bananas from May 1 through May 6, with each day's total being five more than the previous day, but on May 4 he did not eat any, calculate the number of bananas Big Al ate on May 6. | 38 | 0.833333 |
Six people are sitting in a row of 8 seats with the first and last seats remaining unoccupied. Each person must reseat themselves in either their original seat or the seat immediately adjacent to their original seat. Determine the number of ways the people can be reseated. | 13 | 0.333333 |
Three fair dice are thrown, and each die has six faces, numbered 1 through 6. Calculate the probability that the numbers shown can be arranged to form an arithmetic progression with a common difference of two. | \frac{1}{18} | 0.916667 |
A convex polyhedron S has vertices U1, U2, …, Um, and 120 edges. This polyhedron is intersected by planes Q1, Q2, …, Qm, where each plane Qk intersects only those edges that are connected to vertex Uk. No two planes intersect within the volume or on the surface of S. As a result, m pyramids are formed along with a new polyhedron T. Determine the number of edges that polyhedron T now possesses. | 360 | 0.25 |
Given bed A has 600 plants, bed B has 500 plants, bed C has 400 plants, beds A and B share 60 plants, beds A and C share 80 plants, beds B and C share 40 plants, and beds A, B, and C share 20 plants collectively, calculate the total number of unique plants when considering just beds A, B, and C. | 1340 | 0.916667 |
Given that $A$ and $B$ together can complete a job in 3 days, $B$ and $C$ can complete it in 3 days, and $A$ and $C$ together can complete it in 4.8 days, find the number of days required for $A$ to do the job alone. | 9.6 | 0.833333 |
A merchant buys an item for a discount of 30% of the list price. He seeks to set a new marked price that allows him to provide a discount of 20% on that marked price and still obtain a profit of 30% on the selling price. Calculate the percent of the list price at which he should mark the item. | 125\% | 0.833333 |
Given the expression \[\frac{k(P+Q)}{k(P-Q)} - \frac{k(P-Q)}{k(P+Q)},\] where \( P = a + b \) and \( Q = a - b \), and \( k \) is a non-zero constant, simplify this expression. | \frac{a^2 - b^2}{ab} | 0.416667 |
Four runners start running simultaneously from the same point on a 600-meter circular track. They each run clockwise around the track at constant speeds of 5 m/s, 6.5 m/s, 7.5 m/s, and 8 m/s. Determine the time in seconds before the runners regroup somewhere on the track. | 1200 | 0.583333 |
The taxi fare in New Metro City is $3.00 for the first 0.75 mile and additional mileage charged at the rate of $0.25 for each additional 0.05 mile. You decide to give the driver a $3 tip. Determine the total distance in miles that you can ride for $15. | 2.55 | 0.75 |
An equilateral triangle is originally painted black. Each time the triangle is changed, the middle ninth of each black triangle turns white. After three changes, what fractional part of the original area of the black triangle remains black. | \frac{512}{729} | 0.833333 |
Given that a positive multiple of $4$ is less than $3000$, is a perfect square, and is a multiple of $5$, calculate how many such numbers exist. | 5 | 0.25 |
Randy drove the first quarter of his trip on a gravel road, the next 30 miles on pavement, and the remaining one-sixth on a dirt road. Find the total distance of Randy's entire trip. | \frac{360}{7} | 0.5 |
Let $x$ be the weight of the heavier package and $y$ be the weight of the lighter package. The total weight of two packages is 7 times the difference in their weights. Find the ratio of the weight of the heavier package to the lighter package. | \frac{4}{3} | 0.916667 |
If $x$ women working $x$ hours a day for $x$ days produce $x^2$ units of a product, determine the number of units produced by $z$ women working $z$ hours a day for $z$ days. | \frac{z^3}{x} | 0.25 |
Given that Larry and Julius are playing a game where Larry throws first, and the probability of Larry knocking the bottle off the ledge is $\frac{1}{3}$ and the probability that Julius knocks the bottle off the ledge is $\frac{1}{4}$, calculate the probability that Larry wins the game. | \frac{2}{3} | 0.833333 |
John drove 150 miles in 3 hours. His average speed during the first hour was 45 mph, and he stopped for 30 minutes. His average speed during the next 45 minutes was 50 mph, and the remaining duration was driven at an unknown average speed v mph. | 90 | 0.583333 |
Given $\sqrt{\left(5-4\sqrt{2}\right)^2}+\sqrt{\left(5+4\sqrt{2}\right)^2}$, evaluate its value. | 8\sqrt{2} | 0.833333 |
Let $g(x) = x^2 + 4x + 3$ and let $T$ be the set of integers $\{0, 1, 2, \dots, 20\}$. Determine the number of members $t$ of $T$ such that $g(t)$ has remainder zero when divided by $5$. | 8 | 0.916667 |
A box contains $36$ red balls, $24$ green balls, $18$ yellow balls, $15$ blue balls, $12$ white balls, and $10$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $20$ balls of a single color will be drawn. | 94 | 0.083333 |
Given a circle of radius $4$ and a circle of radius $6$, find the total number of possible values of $m$, where $m$ is the number of lines that are simultaneously tangent to both circles. | 5 | 0.416667 |
Determine the number of distinct terms in the expansion of the expression $[(a+2b)^3(a-2b)^3]^2$ when fully simplified. | 7 | 0.833333 |
Let a sequence $\{u_n\}$ be defined by $u_1=7$ and the relationship $u_{n+1} - u_n = 5 + 3(n-1), n=1,2,3,\dots$. Express $u_n$ as a polynomial in $n$ and find the algebraic sum of its coefficients. | 7 | 0.916667 |
A cylindrical tank with radius $6$ feet and height $10$ feet is lying on its side. The tank is filled with water to a depth of $3$ feet. Calculate the volume of water in the tank. | 120\pi - 90\sqrt{3} | 0.75 |
Given 500 members voting, a city council initially defeated a proposal. In a subsequent vote, with the same members participating, the proposal was passed by one and a half times the margin by which it was originally defeated. The number voting for the proposal in the re-vote was $\frac{13}{12}$ of the number voting against it originally. Determine how many more members voted for the proposal in the re-vote compared to the initial vote. | 125 | 0.5 |
Samantha lives 3 blocks west and 3 blocks south of the southwest corner of City Park. Her school is 3 blocks east and 3 blocks north of the northeast corner of City Park. On school days, she bikes on streets to the southwest corner of City Park, then can choose between 2 different diagonal paths through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, calculate the number of different routes she can take. | 800 | 0.833333 |
Given A is $x\%$ greater than B and also $y\%$ greater than C, and A, B, and C are positive numbers such that A > B > C > 0, determine the expressions for x and y. | 100 \left(\frac{A-C}{C}\right) | 0.083333 |
Given that a piece of purple candy costs $30$ cents, and Casper has enough money to buy $16$ pieces of red candy, $18$ pieces of yellow candy, or $20$ pieces of black candy, determine the smallest possible value of $m$ if he can also buy $m$ pieces of purple candy. | 24 | 0.833333 |
The number $15!$ has several positive integer divisors. What is the probability that a randomly chosen divisor is odd? | \frac{1}{12} | 0.166667 |
Given quadrilateral $ABCD$ is a rhombus with a perimeter of 80 meters and the length of diagonal $\overline{AC}$ is 30 meters, calculate the area of rhombus $ABCD$ in square meters. | 150\sqrt{7} | 0.833333 |
Given that in right trapezoid ABCD, AD is perpendicular to DC, AD = AB = x, DC = 2x, and E is the midpoint of DC such that DE = EC = x. Find the area of triangle BEC when x = 5. | 12.5 | 0.083333 |
Given that there are 3-pounds stones worth $9 dollars each, 6-pounds stones worth $15 dollars each, and 1-pounds stones worth $1 dollar each, and that Tanya can carry at most 24 pounds, determine the maximum value, in dollars, of the stones Tanya can carry. | \$72 | 0.833333 |
Given the series $1 - 2 + 3 - 4 + \cdots - 100 + 101$, calculate the sum of the series. | 51 | 0.916667 |
In a 150-shot archery competition, Nora leads by 80 points after 75 shots. Each shot can score 10, 8, 5, 3, or 0 points. Nora always scores at least 5 points per shot. To ensure her victory, if Nora's next $n$ shots are all scoring 10 points each, calculate the minimum number of such shots required for her guaranteed victory. | 60 | 0.083333 |
Suppose three whole numbers in ascending order have pairwise sums of 18, 23, and 27, respectively. Find the middle number. | 11 | 0.5 |
Laura has an 8-hour workday in which she attends three meetings. The first meeting lasts 40 minutes, the second meeting lasts twice as long as the first, and overlaps the last 10 minutes of the first meeting. The third meeting is 30 minutes long and does not overlap with the others. Calculate the percentage of Laura's workday that was spent in meetings. | 29.17\% | 0.833333 |
Given the number of minutes studied by Clara and Mira over a period of one week, calculate the average difference in minutes per day in the number of minutes studied by Mira and Clara. | 5 | 0.083333 |
Each of two wheels contains numbers from 1 to 8. When the wheels are spun, a number is selected from each wheel. Find the probability that the sum of the two selected numbers is divisible by 4. | \frac{1}{4} | 0.666667 |
If four $\Delta$'s and two $\diamondsuit$'s balance twelve $\bullet$'s, and one $\Delta$ balances a $\diamondsuit$ and two $\bullet$'s, determine the number of $\bullet$'s that balance three $\diamondsuit$'s in this balance. | 2 | 0.916667 |
Orvin went to the store with just enough money to buy 40 balloons. During his visit, he found out that the store had a special sale on balloons: buy 1 balloon at the regular price and get a second at 50% off the regular price. What is the greatest number of balloons Orvin could buy? | 53 | 0.333333 |
Determine the minimum number of digits to the right of the decimal point required to express the fraction $\frac{987654321}{2^{30} \cdot 5^5}$ as a decimal. | 30 | 0.333333 |
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