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Positive integers $a$ and $b$ are each less than $8$. Find the smallest possible value for $3 \cdot a - 2 \cdot a \cdot b$. | -77 | 0.333333 |
Determine the smallest positive integer $n$ such that $n$ is divisible by $36$, $n^2$ is a perfect cube, and $n^3$ is a perfect square. | 46656 | 0.416667 |
What is the least possible value of $(xy+1)^2+(x-y)^2$ for real numbers $x$ and $y$? | 1 | 0.916667 |
The product of two positive numbers is 16. The reciprocal of one of these numbers is 3 times the reciprocal of the other number. What is the sum of the two numbers? | \frac{16\sqrt{3}}{3} | 0.833333 |
A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, determine the fraction of the tiled floor that is made of darker tiles, considering that the repeating pattern is a $4 \times 4$ grid and the top $2 \times 2$ area contains 3 dark tiles. | \frac{3}{4} | 0.333333 |
Given that Shauna takes five tests, each worth a maximum of 120 points, and her scores on the first three tests are 86, 102, and 97, in order to average 90 for all five tests, calculate the lowest score she could earn on one of the other two tests. | 45 | 0.916667 |
What is the maximum value of n for which there is a set of distinct positive integers k_1, k_2, ..., k_n such that k_1^2 + k_2^2 + ... + k_n^2 = 2500? | 19 | 0.666667 |
Four fair dice are tossed at random. What is the probability that the four numbers turned up can be arranged to form an arithmetic progression with a common difference of one? | \frac{1}{18} | 0.916667 |
Given a list of $3000$ positive integers with a unique mode that occurs exactly $15$ times, determine the least number of distinct values that can occur in the list. | 215 | 0.416667 |
Given a basketball player made 7 baskets during a game, with each basket worth 1, 2, or 3 points, determine the number of different numbers that could represent the total points scored by the player. | 15 | 0.833333 |
A 4x4x4 cube is made of 64 normal dice. Each die's opposite sides still sum to 7. Calculate the smallest possible sum of all the values visible on the 6 faces of the large cube. | 144 | 0.083333 |
What percent of the birds that were not swans were sparrows? | 50\% | 0.083333 |
Simplify the expression: $\sqrt{1 + \left(\frac{x^6 - 1}{3x^3}\right)^2}$. | \frac{\sqrt{x^{12} + 7x^6 + 1}}{3x^3} | 0.666667 |
A 5x5 block of calendar dates is given. First, the order of the numbers in the second and the fifth rows are reversed. Then, the numbers on each diagonal are added. Calculate the positive difference between the two diagonal sums. | 4 | 0.583333 |
What is the hundreds digit of $(25! - 20!)?$ | 0 | 0.833333 |
Given a rectangle with sides of length 3 cm and 4 cm, construct a new rectangle where one dimension equals the diagonal of the original rectangle, and the other dimension equals the sum of the original rectangle's sides. Determine the area of the new rectangle. | 35 | 0.25 |
Given a polynomial function \( f \) of degree \( \ge 1 \) such that \( f(x^2) = [f(x)]^3 \) and \( f(f(x)) = [f(x)]^2 \), determine the number of possible polynomial functions \( f \). | 0 | 0.166667 |
Given $d$ is a digit, determine the number of values of $d$ for which $3.0d05 > 3.005$. | 9 | 0.916667 |
Calculate the area of a polygon with vertices at $(2,1)$, $(4,3)$, $(6,1)$, and $(4,6)$. | 6 | 0.75 |
Determine the median length of the names of 25 individuals recorded in a survey. | 5 | 0.166667 |
Given that \(x\) is a real number, find the least possible value of \((x+2)(x+3)(x+4)(x+5)+3033\). | 3032 | 0.833333 |
The doughnut machine is turned on at 9:00 AM and finishes one fourth of the day's job by 12:00 PM, determine the time it will complete the job. | 9:00 PM | 0.833333 |
A basketball player made 7 baskets during a game. Each basket was worth either 1, 2, or 3 points. Calculate how many different numbers could represent the total points scored by the player. | 15 | 0.75 |
Circle $C$ has a radius of $144$. Circle $D$ has an integer radius $s < 144$ and remains internally tangent to circle $C$ as it rolls once around the circumference of circle $C$. The circles have the same points of tangency at the beginning and end of circle $D$'s trip. Find the number of possible values of $s$. | 14 | 0.666667 |
How many ordered pairs (m,n) of positive integers, with m ≥ n, have the property that their squares differ by 72? | 3 | 0.75 |
Given that Bag A contains the chips labeled 0, 1, 3, and 5, and Bag B contains the chips labeled 0, 2, 4, and 6, determine the number of different values that are possible for the sum of the two numbers on the chips drawn from each bag. | 10 | 0.916667 |
Given that $OPQR$ is a rectangle where $O$ is the origin and point $Q$ has coordinates $(4,2)$, calculate the coordinates for a point $T$ on the x-axis so that the area of triangle $PQT$ is half the area of rectangle $OPQR$. | (8,0) | 0.333333 |
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 taping along the two radii. Find the volume of the cone in cubic inches. | 9\pi \sqrt{3} | 0.833333 |
Emily a cell phone plan costs $30 each month, including unlimited calls up to the first 50 hours, and each additional hour is charged at $15. Additionally, each text message costs $0.10, but after the first 150 messages, the cost per message doubles. In February, Emily sent 200 text messages and talked for 52 hours. Calculate the total amount Emily had to pay. | 85 | 0.25 |
Mrs. Anna Quick needs to pick up her friend from the airport every Friday. The airport is a certain distance from her home. If she drives at an average speed of 50 miles per hour, she arrives 5 minutes late, and if she drives at an average speed of 75 miles per hour, she arrives 5 minutes early. Determine the speed at which Mrs. Anna Quick should drive to arrive exactly on time. | 60 | 0.916667 |
A rectangular picture is framed with a border three inches wide on all sides. The picture itself measures $12$ inches in height and $15$ inches in width. Calculate the total area of the border. | 198 | 0.916667 |
Given the initial percentage of girls in the group is $50\%$, let $g$ represent the number of girls initially in the group. Shortly thereafter, three girls leave and three boys join the group, and then $40\%$ of the group are girls. Calculate the initial number of girls in the group. | 15 | 0.916667 |
Given that a positive integer consists of three digits whose product equals 30, calculate the number of 3-digit positive integers. | 12 | 0.5 |
Given that a cell phone plan costs $25$ dollars each month, plus $0.10$ dollars per text message sent, plus $0.15$ dollars for each minute used over $25$ hours, and an extra $2$ dollars per GB for data exceeding $3$ GB, calculate the total cost for Jackie in February given that she sent $200$ text messages and talked for $26.5$ hours, using $4.5$ GB of data. | 61.50 | 0.833333 |
Given that a water tower stands 60 meters high and contains 150,000 liters of water, and a model of the tower holds 0.15 liters, determine the height of Liam's model tower. | 0.6 | 0.333333 |
If one side of a triangle is $15$ inches and the opposite angle is $45^{\circ}$, calculate the diameter of the circumscribed circle. | 15\sqrt{2} \text{ inches} | 0.916667 |
Two pitchers, one with a capacity of 800 mL and the other with 500 mL, contain apple juice. The first pitcher is 1/4 full and the second pitcher is 3/8 full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is apple juice? | \frac{31}{104} | 0.916667 |
Find the number of pairs (m, n) of integers which satisfy the equation $m^3 + 10m^2 + 11m + 2 = 81n^3 + 27n^2 + 3n - 8$. | 0 | 0.5 |
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, with the parallel sides of the trapezoid having lengths $20$ meters and $30$ meters. Determine the fraction of the yard occupied by the flower beds. | \frac{1}{6} | 0.25 |
A pitcher is filled to five-sixths of its capacity with orange juice. The juice is then poured equally into 3 cups. What is the percent of the total capacity of the pitcher that each cup received? | 27.78 \% | 0.75 |
Given that a rectangle $R$ has dimensions $a$ and $b$ with $a < b$, determine the number of possible values of $x$ and $y$ such that the perimeter is half of $R$'s, and the area is half of $R$'s. | 2 | 0.833333 |
Given a 16-slice pizza, Anna eats 2 slices and shares another slice equally with Ben and John, her two cousins. Determine the fraction of the pizza that Anna eats. | \frac{7}{48} | 0.083333 |
Given circle $O$, point $C$ is on one side of diameter $\overline{AB}$, and point $D$ is on the opposite side. If $\angle AOC = 40^{\circ}$, and $\angle DOB = 30^{\circ}$, calculate the ratio of the area of the smaller sector $COD$ to the area of the circle. | \frac{17}{36} | 0.083333 |
Given the expression $\left(\frac{x^2+2}{x^2}\right)\left(\frac{y^2+2}{y^2}\right)+\left(\frac{x^2-2}{y^2}\right)\left(\frac{y^2-2}{x^2}\right)$, with $xy \not= 0$, simplify the expression. | 2 + \frac{8}{x^2y^2} | 0.916667 |
For how many positive integers $m$ is $\frac{180}{m^2 - 3}$ a positive integer? | 2 | 0.666667 |
Given that a group initially consists of $30\%$ girls, and after three girls leave and three boys join, the group consists of $25\%$ girls, determine the initial number of girls in the group. | 18 | 0.916667 |
Given that the sum of the interior angles of a convex polygon is 2797 degrees, find the combined degree measure of the two forgotten angles that Ben initially missed in his calculations. | 83 | 0.5 |
In a 60-question multiple choice math contest, students receive 5 points for a correct answer, 0 points for an answer left blank, and -2 points for an incorrect answer. Jamie's total score on the contest was 150. Determine the maximum number of questions that Jamie could have answered correctly. | 38 | 0.75 |
How many $3$-digit positive integers have digits whose product equals $30$? | 12 | 0.25 |
Given Jeremy's father drives him to school under normal conditions in 30 minutes, and on a particularly clear day, he can drive 15 miles per hour faster in 18 minutes, calculate the distance to school in miles. | 11.25 | 0.916667 |
Given that a ticket to a school concert costs a whole number of dollars, and a group of 11th graders buys tickets costing a total of $72, and a group of 12th graders buys tickets costing a total of $90, find the number of possible values for the cost of a ticket. | 6 | 0.916667 |
A positive integer divisor of $10!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 10 | 0.916667 |
Lucas's apartment consists of 2 rooms, each 15 feet long, 12 feet wide, and 10 feet high. Doorways and windows, which will not be painted, occupy 75 square feet in total per room. Calculate the total number of gallons of paint Lucas needs to buy if he wants to apply 2 coats of paint, given that each gallon of paint covers 350 square feet for one coat. | 6 | 0.666667 |
Given that $36$ students are participating in the summer training program, with equal numbers of undergraduate and graduate students on the coding team, and $20\%$ of the undergraduate students and $25\%$ of the graduate students are on the coding team, determine the number of undergraduate students in the program. | 20 | 0.833333 |
Karl's car uses a gallon of gas every 30 miles, and his gas tank holds 16 gallons when it is full. One day, Karl started with a full tank of gas, drove 420 miles, bought 10 gallons of gas, and continued driving until his gas tank was three-quarters full. Determine the total distance Karl drove that day. | 420 | 0.583333 |
Given that $b = 2$, evaluate the expression $\frac{3b^{-1} + \frac{b^{-1}}{3} + 3}{b + 1}$. | \frac{14}{9} | 0.916667 |
Calculate the number of terms in the arithmetic sequence $17, 21, 25, \dots, 101, 105$ such that an additional term $49$ also belongs to the sequence. | 23 | 0.833333 |
Three years ago, Tom was four times as old as his brother Jerry, and five years before that, Tom was five times as old as Jerry. Determine how many years from now the ratio of their ages will be $3 : 1$. | 7 | 0.083333 |
Given a basketball player made $7$ baskets during a game, with each basket worth either $2$ or $3$ points, determine the number of different numbers that could represent the total points scored by the player. | 8 | 0.916667 |
Points $A$ and $B$ are on a circle of radius $7$ and $\overline{AB}=10$. Point $C$ is the midpoint of the minor arc $AB$. What is the length of the line segment $AC$. | \sqrt{98 - 28\sqrt{6}} | 0.166667 |
Given the stronger Goldbach conjecture, for the even number 130, find the largest possible difference between two distinct primes that add up to 130. | 124 | 0.916667 |
Given every 8-digit whole number is a possible telephone number except those that begin with 0 or 1, or end in 9, determine the fraction of these telephone numbers that begin with 9 and end with 0. | \frac{1}{72} | 0.75 |
Define a sequence $\{b_n\}$ such that $b_1=3$ and $b_{n+1}=b_n + 2n + 1$ where $n \geq 1$. Find the value of $b_{100}$. | 10002 | 0.75 |
If Lin drove 100 miles on the highway and 20 miles on a forest trail, where she drove four times as fast on the highway as on the forest trail, and the time taken to drive on the forest trail was 40 minutes, calculate the total time taken for the entire trip. | 90 | 0.916667 |
Given that each student scored at least $70$ points on a $120$-point test, seven students scored a perfect $120$, and the mean score of the class was $85$, calculate the smallest possible number of students in the class. | 24 | 0.916667 |
How many different four-digit numbers can be formed by rearranging the four digits in $3008$? | 6 | 0.916667 |
Given that $a * b$ means $4a - 2b$, find the value of $x$ if $3 * (6 * x) = -2$. | \frac{17}{2} | 0.583333 |
Given rectangle ABCD, 30% of its area overlaps with square EFGH. Square EFGH shares 40% of its area with rectangle ABCD. If AD equals one-tenth of the side length of square EFGH, determine the ratio of AB to AD. | \frac{400}{3} | 0.916667 |
Given that the perimeter of a rectangle is 60 units, and the side lengths are integers, find the difference between the largest and smallest possible areas of the rectangle. | 196 | 0.833333 |
Cagney can frost a cupcake every 18 seconds and Lacey can frost a cupcake every 40 seconds. Lacey starts working 1 minute after Cagney starts. Calculate the number of cupcakes that they can frost together in 6 minutes. | 27 | 0.75 |
Given $A, B$, and $C$ are pairwise disjoint sets of animals, the average weights of animals in sets $A, B, C, A \cup B, A \cup C$, and $B \cup C$ are $45, 35, 55, 40, 50$, and $45$ respectively. Find the average weight of the animals in set $A \cup B \cup C$. | 45 | 0.833333 |
Triangle $ABC$ has a right angle at $B$. Point $D$ is the foot of the altitude from $B$, $AD=5$, and $DC=3$. Calculate the area of $\triangle ABC$. | 4\sqrt{15} | 0.083333 |
Given that the car ran exclusively on its battery for the first 50 miles, used gasoline at a rate of 0.03 gallons per mile for the rest of the trip, and averaged 50 miles per gallon for the entire trip, calculate the total length of the trip in miles. | 150 | 0.833333 |
Expand and simplify the expression $(1+x^3)(1-x^4)$. | 1 + x^3 - x^4 - x^7 | 0.833333 |
Given that Brianna used one fourth of her money to buy one fourth of the CDs, determine the fraction of her money that she will have left after she buys all the CDs. | 0 | 0.833333 |
If \(y\) cows produce \(y+2\) cans of milk in \(y+3\) days, determine the number of days it will take \(y+4\) cows to produce \(y+6\) cans of milk. | \frac{y(y+3)(y+6)}{(y+2)(y+4)} | 0.166667 |
Consider two concentric circles where the width of the track formed by these circles is 15 feet. If the diameter of the smaller circle is 25 feet, calculate how long the outer circle's circumference is compared to the inner circle. | 30\pi | 0.583333 |
A magician has 3 red chips and 3 green chips in a hat. Chips are drawn randomly one at a time without replacement, until all 3 of the red chips are drawn or until all 3 green chips are drawn. Determine the probability that all the 3 red chips are drawn first. | \frac{1}{2} | 0.75 |
Given the equation $4^{x+1} - 8 \cdot 2^{x+1} - 2^{x} + 8 = 0$, determine the number of real numbers $x$ that satisfy this equation. | 2 | 0.75 |
Given that a positive integer is a perfect square, an even multiple of 5, and less than 2500, determine the number of such positive even multiples of 5. | 4 | 0.25 |
What is the smallest integer larger than $(\sqrt{5}+\sqrt{3})^4$? | 248 | 0.833333 |
Given the equation $\frac{X}{8}=\frac{8}{Y}$, determine the number of ordered pairs of positive integers $(X, Y).$ | 7 | 0.916667 |
Find the minimum value of $\sqrt{x^2+y^2}$ given that $8x + 15y = 120$ and $x \geq 0$. | \frac{120}{17} | 0.583333 |
What is the value of $1324 + 2431 + 3142 + 4213 + 1234$? | 12344 | 0.416667 |
Let $g(x) = |x-3| + |x-5| - |2x-8|$ for $3 \leq x \leq 10$. Find the sum of the largest and smallest values of $g(x)$ on this interval. | 2 | 0.916667 |
The number of positive integers less than $1200$ that are divisible by neither $6$ nor $8$ is what? | 900 | 0.916667 |
Given that bricklayer Brenda would take $8$ hours to build a chimney alone, and bricklayer Brandon would take $12$ hours to build it alone, when they work together they reduce their combined output by $12$ bricks per hour, and that they complete the chimney in $6$ hours, calculate the total number of bricks in the chimney. | 288 | 0.666667 |
How many 7-digit palindromes can be formed using the digits 1, 1, 2, 2, 2, 4, 4? | 6 | 0.083333 |
The average age of the family members is $22$, the father is $50$ years old, and the average age of the mother and children is $15$. Given that the average age of all the family members is $22$, determine the number of children in the Alvez family. | 3 | 0.666667 |
Given that Square $ABCD$ has side length $5$, point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$, calculate the degree measure of $\angle AMD$. | 45 | 0.916667 |
Given that there are $n > 15$ students in a class with an average score of 10 and a subgroup of 15 students with an average score of 16, determine the average score of the remaining students in terms of $n$. | \frac{10n - 240}{n-15} | 0.916667 |
A rectangular box has a total surface area of 142 square inches and the sum of the lengths of all its edges is 60 inches. Find the sum of the lengths in inches of all of its interior diagonals. | 4\sqrt{83} | 0.916667 |
A regular hexagonal estate with a scale of 300 miles to 1 inch has a side length of 6 inches on the map. What is the area of this estate in square miles? | 4860000\sqrt{3} | 0.916667 |
If Leyla has a $3 \times 5$ index card and shortens the length of one side of this card by $2$ inches, the area becomes $9$ square inches, find the area of the card if she instead shortens the length of the other side by $2$ inches, keeping the first side's reduction as well. | 3 | 0.666667 |
Starting with the display "1," calculate the fewest number of keystrokes needed to reach "300" using the keys [+1] and [x2]. | 11 | 0.666667 |
Given that $\log_8 x = 3.5$, solve for the value of $x$. | 1024\sqrt{2} | 0.416667 |
The sum of two numbers is 3 times their difference. If the larger number is denoted by x and the smaller one by y, express the ratio x/y as a simplified fraction. | 2 | 0.666667 |
If ten students from Delta school worked for 4 days, six students from Echo school worked for 6 days, eight students from Foxtrot school worked for 3 days, and three students from Golf school worked for 10 days, and the total payment made for the students' work was $1233, with each student receiving the same amount for a day's work, calculate the total amount earned by the students from Echo school altogether. | 341.45\text{ dollars} | 0.333333 |
Given two numbers, their difference, their sum, and their product are to one another as $1:9:40$. Calculate the product of the two numbers. | 80 | 0.916667 |
$\triangle ABC$ has a right angle at $C$ and $\angle A = 30^{\circ}$. Given that $BD$ ($D$ in $\overline{AC}$) is the bisector of $\angle ABC$, determine $\angle BDC$. | 60 | 0.833333 |
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