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How many primes less than $100$ have $3$ as the ones digit? | 7 | 0.666667 |
What is the result of $\sqrt{16-8\sqrt{3}} + \sqrt{16+8\sqrt{3}}$? | 4\sqrt{3} | 0.75 |
A parabolic arch has a height of $20$ inches and a span of $50$ inches. Find the height, in inches, of the arch at the point $10$ inches from the center $M$. | 16.8 | 0.916667 |
Lisa bakes a $24$-inch by $30$-inch pan of brownies and cuts the brownies into pieces that measure $3$ inches by $4$ inches. Calculate the total number of pieces of brownies that the pan contains. | 60 | 0.666667 |
Given that a car ran exclusively on its battery for the first $60$ miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of $0.03$ gallons per mile, and averaging $50$ miles per gallon for the whole trip, determine the total distance of the trip in miles. | 180 | 0.666667 |
What is the greatest number of consecutive integers whose sum is $36$? | 72 | 0.25 |
Determine the number of points C such that the perimeter of triangle ABC is 60 units and the area of triangle ABC is 144 square units, given that points A and B are 12 units apart. | 0 | 0.25 |
Given that Jeff spins three spinners labeled $X$, $Y$, and $Z$, where Spinner $X$ has numbers 1, 4, 5, Spinner $Y$ has numbers 1, 2, 3, and Spinner $Z$ has numbers 2, 4, 6, find the probability that the sum of the numbers Jeff gets from spinning $X$, $Y$, and $Z$ is even. | \frac{5}{9} | 0.166667 |
In a coordinate plane, points A and B are 8 units apart. Find the number of points C such that the perimeter of triangle ABC is 40 units and the area of triangle ABC is 80 square units. | 0 | 0.25 |
Let the sequence $x, 3x+3, 5x+5, \dots$ be in geometric progression. What is the fourth term of this sequence? | -\frac{125}{12} | 0.916667 |
Evaluate the value of the expression $3(3(3(3(3x+2)+2)+2)+2)+2$ where $x=5$. | 1457 | 0.833333 |
If $m$ men can complete a task in $d$ days, calculate the number of days it will take for $m + r^2$ men to complete the same task. | \frac{md}{m + r^2} | 0.916667 |
Calculate the sum of the sequence:
\[3\left(1 - \frac{1}{3}\right) + 4\left(1 - \frac{1}{4}\right) + 5\left(1 - \frac{1}{5}\right) + \cdots + 12\left(1 - \frac{1}{12}\right).\] | 65 | 0.5 |
The sum of the interior angles of a polygon is (k-2) × 180°. Determine the relationship between the sum of the interior angles and the number of sides k of the polygon. | (k-2) \times 180^\circ \text{ and Increases} | 0.833333 |
Given Tamara redesigns her garden with four rows of three $8$-feet by $3$-feet flower beds separated and surrounded by $2$-foot-wide walkways, calculate the total area of the walkways in square feet. | 416 | 0.083333 |
A worker's wage is reduced by 30%. To regain his original wage, what percentage raise is required? | 42.86\% | 0.916667 |
The sum of the first 1234 terms of the sequence $1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 3,\ldots$ can be found. | 3604 | 0.166667 |
Miki now has 15 oranges and 15 pears of the same size. In her new juicer, Miki can extract 10 ounces of pear juice from 4 pears and 7 ounces of orange juice from 1 orange. She decides to make a juice blend using 12 oranges and 8 pears. Calculate the percentage of the blend that is pear juice. | 19.23\% | 0.916667 |
A basketball league has introduced new rules. Each team now consists of 25 players. A player must earn a minimum of $15,000, and the team's total salary cap is $850,000. Calculate the maximum possible salary for a single player on a team under these new regulations. | 490,000 | 0.833333 |
The total number of black squares on the nonstandard checkerboard consists of 29 squares on each side. | 421 | 0.916667 |
Let $x=-1000$. Evaluate the expression $\bigg|$ $||x|-x|-|x|$ $\bigg|$ $+x$. | 0 | 0.916667 |
Given a rectangular enclosure that is $15$ feet long, $12$ feet wide, and $6$ feet high with the floor and walls each $1.5$ feet thick, calculate the number of one-foot cubical blocks needed to complete the enclosure. | 594 | 0.25 |
The ratio of $a$ to $b$ in the arithmetic sequence $a, y, b, 3y$ is what fraction. | 0 | 0.75 |
Given the equations $x^2 + kx + 12 = 0$ and $x^2 - kx + 12 = 0$. If, when the roots of the equations are suitably listed, each root of the second equation is $7$ more than the corresponding root of the first equation, determine the value of $k$. | 7 | 0.916667 |
Let $n$ be the smallest positive integer such that $n$ is divisible by $30$, $n^2$ is a perfect cube, and $n^3$ is a perfect square. What is the number of digits of $n$. | 9 | 0.916667 |
$\triangle ABC$ has a right angle at $C$ and $\angle A = 30^\circ$. If $BD$ ($D$ in $\overline{AC}$) is the bisector of $\angle ABC$, then find the measure of $\angle BDC$. | 60^\circ | 0.833333 |
Determine the sum of the digits of the greatest prime number that is a divisor of $2^{13} - 1$. | 19 | 0.25 |
What is the greatest number of consecutive integers whose sum is $48?$ | 96 | 0.25 |
Elena builds a rectangular enclosure using one-foot cubical blocks that measures 15 feet in length, 13 feet wide, and 6 feet high. Both the floor and the ceiling, as well as the four walls, are all one foot thick. Calculate the number of blocks used in the construction of this enclosure. | 598 | 0.583333 |
Elena intends to buy 7 binders priced at $\textdollar 3$ each. Coincidentally, a store offers a 25% discount the next day and an additional $\textdollar 5$ rebate for purchases over $\textdollar 20$. Calculate the amount Elena could save by making her purchase on the day of the discount. | 10.25 | 0.166667 |
Evaluate the expression \(\sqrt{\frac{4}{3} \left( \frac{1}{15} + \frac{1}{25} \right)}\). | \frac{4\sqrt{2}}{15} | 0.916667 |
Given $(729^{\log_3 2187})^{\frac{1}{3}}$, evaluate the expression. | 3^{14} | 0.083333 |
How many 3-digit positive integers have digits whose product equals 36? | 21 | 0.333333 |
What is the smallest whole number larger than the perimeter of any triangle with a side of length $7$ and a side of length $24$? | 62 | 0.416667 |
Given that all of Maria's telephone numbers have the form $555-abc-defg$, where $a$, $b$, $c$, $d$, $e$, $f$, and $g$ are distinct digits, in increasing order, and none is either $0$, $1$, or $9$, calculate the total number of different telephone numbers Maria can have. | 1 | 0.833333 |
What is the minimum possible product of three different numbers from the set $\{-9, -5, -3, 0, 4, 6, 8\}$? | -432 | 0.5 |
A rectangular garden that is $14$ feet wide and $19$ feet long is paved with $2$-foot square pavers. Given that a bug walks from one corner to the opposite corner in a straight line, determine the total number of pavers the bug visits, including the first and the last paver. | 16 | 0.166667 |
Given \(xy = 2b\) and \(\frac{1}{x^2} + \frac{1}{y^2} = a\), find the value of \((x+y)^2\). | 4ab^2 + 4b | 0.083333 |
The ratio of the length to the width of a rectangle is $5$ : $2$. If the rectangle has a diagonal of length $13$ units, find the constant $k$ such that the area of the rectangle can be expressed as $kd^2$. | \frac{10}{29} | 0.916667 |
Let $g(x) = \frac{x^6 + x^3}{x + 2}$, find $g(i)$ where $i = \sqrt{-1}$. | -\frac{3}{5} - \frac{1}{5}i | 0.583333 |
For a set of three distinct lines in a plane, there are exactly $M$ distinct points that lie on two or more of these lines. What is the sum of all possible values of $M$? | 6 | 0.583333 |
Find the minimum value of the function $f(x) = 2x^2 + 4x + 6 + 2\sqrt{x}$, assuming $x \geq 0$. | 6 | 0.916667 |
Given that the base-ten representation for $15!$ is $1,307,674,H00,M76,T80$, determine the sum of the digits $H$, $M$, and $T$. | 17 | 0.75 |
If $x$ cows give $x+2$ cans of milk in $x+1$ days, determine how many cows are needed to give $x+4$ cans of milk in $x+3$ days. | \frac{x(x+1)(x+4)}{(x+2)(x+3)} | 0.5 |
Points $B$ and $C$ lie on $\overline{AD}$. The length of $\overline{AB}$ is $3$ times the length of $\overline{BD}$, and the point $C$ is the midpoint of $\overline{AD}$. Determine the fraction of the length of $\overline{AD}$ that is occupied by $\overline{BC}$. | \frac{1}{4} | 0.916667 |
Given that Mrs. Germain has $15$ students, Mr. Newton has $12$ students, and Mrs. Young has $9$ students taking the AMC $8$, and $3$ students from Mrs. Young's class are also in Mr. Newton's class, determine the total number of students at Euclid Middle School who are taking the contest. | 33 | 0.916667 |
Given integers between $2000$ and $9999$ have four distinct digits, calculate the total count. | 4032 | 0.916667 |
The sum of 30 consecutive odd integers is 7,500. Find the value of the largest of these 30 consecutive integers. | 279 | 0.916667 |
A frog makes $4$ jumps, each exactly $0.5$ meters long. Determine the probability that the frog's final position is no more than $0.5$ meters from its starting position. | \frac{1}{8} | 0.25 |
Determine the number of points $C$ that exist such that the perimeter of $\triangle ABC$ is $60$ units and the area of $\triangle ABC$ is $144$ square units, given that points $A$ and $B$ are $12$ units apart. | 0 | 0.083333 |
Given a triangle with a side of length 7 and a side of length 21, what is the smallest whole number larger than its perimeter? | 56 | 0.166667 |
A rectangular barn with a roof is 12 yd. wide, 15 yd. long and 7 yd. high. Calculate the total number of sq. yd. to be painted inside and outside on the walls and the ceiling, and only on the outside of the roof but not on the floor. | 1116 | 0.25 |
Given that Spinner 1 has sectors numbered 1, 2, 3 and Spinner 2 has sectors numbered 4, 5, 6, determine the probability that the sum of the numbers on which the spinners land is prime. | \frac{4}{9} | 0.833333 |
The mean, median, and unique mode of the positive integers 2, 3, 4, 8, 8, 9, 10, and $y$ are all equal. What is the value of $y$? | 20 | 0.166667 |
Given that a room initially requires 2 gallons of paint, on the first day one-fourth of the paint is used, and on the second day one-half of the remaining paint is used, and an additional half gallon is needed to finish painting, calculate the total amount of additional paint needed. | \frac{1}{2} | 0.666667 |
Calculate the fourth term in the expansion of \((\frac{a}{x} + \frac{x}{a^3})^7\). | \frac{35}{a^5x} | 0.916667 |
A sphere with center $O$ has radius $8$. A triangle with sides of length $13$, $13$, and $10$ is situated in space so that each of its sides is tangent to the sphere. Calculate the distance between $O$ and the plane determined by the triangle. | \frac{2\sqrt{119}}{3} | 0.25 |
Amanda divided her hiking trip into four segments. She hiked the first one-fourth of the total distance to a stream and then went another $25$ kilometers through a forest. The next segment involved crossing a mountain range, which took up another one-sixth of the total trip, before finishing the last portion on a plain. If the plain's segment was twice the length of the forest path, derive the length of Amanda's entire hiking trip. | \frac{900}{7} | 0.5 |
Packaged snacks are sold in containers of 5, 10, and 20 units. Find the minimum number of containers needed to acquire exactly 85 units of snacks. | 5 | 0.833333 |
Given in isosceles trapezoid $ABCD$, sides $AB$ and $CD$ are bases with $AB < CD$, the height of the trapezoid from $AB$ to $CD$ is 6 units, $AB$ is 10 units and $CD$ is 18 units, find the perimeter of $ABCD$. | 28 + 4\sqrt{13} | 0.833333 |
If a wooden cube of side length \( n \) units is entirely painted blue on all six faces and then divided into \( n^3 \) smaller cubes of unit volume each, and exactly one-third of the total number of faces of these unit cubes are blue, what is \( n \)? | 3 | 0.916667 |
Given a $24$-inch by $30$-inch pan of brownies, cut into pieces that measure $3$ inches by $4$ inches. Calculate the number of pieces of brownie the pan contains. | 60 | 0.416667 |
Given square $ABCD$ has side lengths of 4, segments $CM$ and $CN$ divide the square's area into three equal parts. Each triangle formed by these segments and square's sides shares one vertex $C$. Calculate the length of segment $CM$. | \frac{4\sqrt{13}}{3} | 0.416667 |
Jasmine now has access to three sizes of shampoo bottles: a small bottle that holds 25 milliliters, a medium bottle that holds 75 milliliters, and a large bottle that holds 600 milliliters. Determine the minimum total number of bottles Jasmine must buy to almost completely fill a large bottle, using small bottles for the remaining volume. | 8 | 0.333333 |
Given that in triangle ABC, segment AD divides the side BC into segments BD and DC such that $BD + DC = p$ and $BD \cdot DC = q^2$, find the polynomial whose roots are $BD$ and $DC$ under these conditions. | x^2 - px + q^2 | 0.333333 |
Given that Mr. and Mrs. Delta are deciding on a name for their baby such that its monogram (first, middle, and last initials) will be in alphabetical order with no letter repeated, and the middle initial must be 'M', calculate the number of such monograms possible. | 156 | 0.083333 |
At a certain park, 20% of the birds are pigeons, 30% are parrots, 15% are peacocks, and 35% are sparrows. Calculate what percent of the birds that are not sparrows are pigeons. | 30.77\% | 0.583333 |
Walter was one-third as old as his grandmother at the end of $1998$. Find Walter's age at the end of $2003$, given that the sum of the years in which they were born was $3860$. | 39 | 0.583333 |
Linda plans to participate in a mini-triathlon. She averages 1 mile per hour swimming for a 1/8-mile stretch and 8 miles per hour running for a 2-mile stretch. She aims to finish the mini-triathlon in 1.5 hours. Determine her required average speed in miles per hour for the 8-mile bike ride. | \frac{64}{9} | 0.916667 |
Pablo drove at an average speed of 60 km/h and then stopped for 30 minutes. After the stop, he drove at an average speed of 90 km/h. Altogether, he drove 270 km in a total trip time of 4 hours including the stop. Which equation could be used to solve for the time $t$ in hours that he drove before his stop? | 60t + 90\left(\frac{7}{2} - t\right) = 270 | 0.416667 |
A ferry boat starts its service to an island at 10 AM and ends at 4 PM, shuttling tourists every hour. On the first trip at 10 AM, there were 120 tourists, and on each subsequent trip, the number of tourists decreased by 2 from the previous trip. Calculate the total number of tourists transported by the ferry that day. | 798 | 0.5 |
Four fair dice are tossed at random. What is the probability that the numbers turned up can be arranged to form an arithmetic progression with common difference one or two? | \frac{1}{18} | 0.333333 |
A circle of radius 6 is inscribed in a rectangle. The ratio of the length of the rectangle to its width is 3:1. Find the area of the rectangle. | 432 | 0.916667 |
Determine the values of $a, b$, and $c$ that satisfy the equality $\sqrt{a^2 - \frac{b}{c}} = a - \sqrt{\frac{b}{c}}$, where $a, b$, and $c$ are positive integers. | b = a^2c | 0.083333 |
Given the in-store price for a laptop is $\textdollar 599.99$, the store offers a 5% discount. Additionally, an online advertisement offers the same laptop for five equal payments of $\textdollar 109.99$ plus a one-time shipping charge of $\textdollar 19.99$. Calculate the amount saved by purchasing the laptop online. | 0.05 | 0.416667 |
Points $A$ and $B$ are on a circle with radius $7$ and $AB=8$. Points $C$ and $E$ are the midpoints of the minor and major arcs $AB$, respectively. Find the length of the line segment $AC$. | \sqrt{98 - 14\sqrt{33}} | 0.166667 |
Given a right circular cone and a sphere with the same radius $r$, the volume of the cone is one-third that of the sphere. Find the ratio of the altitude of the cone to the radius of its base. | \frac{4}{3} | 0.916667 |
Given a circle is inscribed in a triangle with side lengths $9, 12$, and $15$, let the segments of the side of length $9$, made by a point of tangency, be $p$ and $q$, with $p < q$. Calculate the ratio $p:q$. | 1:2 | 0.25 |
Given the line whose equation is $y = 2x + 3$, and the points $(p, q)$ and $(r, s)$ satisfy $s = 2r + 6$, find the distance between $(p, q)$ and $(r, s)$ in terms of $p$ and $r$. | \sqrt{5(r - p)^2 + 12(r - p) + 9} | 0.666667 |
Lucia needs to save 35 files onto disks, each with a capacity of 1.6 MB. 5 of the files are 0.9 MB each, 10 of the files are 0.8 MB each, and the rest are 0.5 MB each. Calculate the smallest number of disks needed to store all 35 files. | 15 | 0.75 |
Calculate the value of $3^{\left(1^{\left(0^8\right)}\right)}+\left(\left(3^1\right)^0\right)^8$. | 4 | 0.916667 |
Charlie and Dana start their new jobs on the same day. Charlie's schedule is 4 work-days followed by 2 rest-days. Dana's schedule is 9 work-days followed by 1 rest-day. Determine how many times both have rest-days on the same day in their first 1200 days of work. | 40 | 0.916667 |
Given the quadratic equation $ax^2 + bx + c = 0$, if one root is triple the other, determine the relationship among the coefficients $a,\,b,\,c$. | 3b^2 = 16ac | 0.916667 |
Given that Frank the flamingo takes $60$ equal steps to walk between consecutive beacons, and Peter the penguin covers the same distance in $15$ equal slides, and the $31$st beacon is exactly half a mile ($2640$ feet) from the first beacon, find the difference in length, in feet, between one of Peter's slides and one of Frank's steps. | 4.4 | 0.583333 |
Determine the number of integers $x$ that satisfy $x^6 - 52x^3 + 51 < 0$. | 2 | 0.5 |
Alice has taken three exams, each worth a maximum of 100 points and her scores are 85, 76, and 83. What is the lowest possible score she could achieve on one of her remaining two exams to still reach an average of 80 for all five exams? | 56 | 0.916667 |
Evaluate $(64)^{-2^{-3}}$. | \frac{1}{\sqrt[4]{8}} | 0.833333 |
In the diagram, $\overline{AB}$ and $\overline{CD}$ are diameters of a circle with center $O$, with $\overline{AB} \perp \overline{CD}$. Chord $\overline{DF}$ intersects $\overline{AB}$ at $E$. If $DE = 8$ and $EF = 4$, calculate the area of the circle. | 48\pi | 0.083333 |
Given the function $x \spadesuit y = (x+y)^2(x-y)$, calculate the value of $2 \spadesuit (3 \spadesuit 6)$. | 14229845 | 0.75 |
In a sign pyramid where a cell receives a "+" if the two cells below it have the same sign and a "-" otherwise, determine the number of ways to fill the five cells in the bottom row to produce a "+" at the top of a new pyramid with an increased number of levels. | 16 | 0.75 |
If $y$ cows give $y+2$ cans of milk in $y+1$ days, calculate how many days will it take $y+4$ cows to give $y+10$ cans of milk. | \frac{y(y+1)(y+10)}{(y+2)(y+4)} | 0.083333 |
Find the value of A when the minimum of $\sin\frac{A}{2} - \cos\frac{A}{2}$ is attained. | \frac{7\pi}{2} | 0.5 |
Consider the repeating decimal number $3.71717171\ldots$. When written in the fraction form and reduced to the lowest terms, find the sum of the numerator and denominator. | 467 | 0.916667 |
Given the sales of $110, 90, 50, 130, 100, and 60$ dollars for January, February, March, April, May, and June, respectively, calculate the average monthly sales in dollars, accounting for a 20% discount on all candy sales in June. | 88 | 0.833333 |
Given that there are 12 students in Mr. Gauss' class, 10 students in Ms. Euler's class, and 7 students in Mr. Fibonacci's class, with one student counted twice, calculate the total number of distinct mathematics students at Euclid Middle School taking the AMC 8 this year. | 28 | 0.916667 |
The centers of two circles are $50$ inches apart. The smaller circle has a radius of $7$ inches and the larger one has a radius of $10$ inches. Find the length of the common internal tangent. | \sqrt{2211} | 0.5 |
A merchant buys goods at 30% off the list price. He desires to mark the goods so that he can give a discount of 20% on the marked price and still clear a profit of 30% on the selling price. What percent of the list price must he mark the goods? | 125\% | 0.75 |
In a modified sequence of numbers, the first number is $1$, and for all $n \geq 2$, the product of the first $n$ numbers in the sequence is $(n+1)^3$. Find the sum of the third and the fifth numbers in this sequence. | \frac{13832}{3375} | 0.75 |
Given the probability that the red ball is tossed into bin k is $3^{-k}$, and the probability that the green ball is tossed into bin k is $2^{-k}$ for k = 1,2,3,..., find the probability that the red ball is tossed into a higher-numbered bin than the green ball. | \frac{1}{5} | 0.083333 |
Find the coefficient of $x^9$ in the expansion of $\left(\frac{x^3}{3} - \frac{3}{x^2}\right)^9$. | 0 | 0.666667 |
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