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|---|---|---|
Given that Ava's odometer displayed $14941$ miles, and four hours later another palindrome appeared on the odometer, calculate Ava's average speed, in miles per hour, during these 4 hours.
|
27.5
| 0.666667 |
If two congruent circles centered at points $P$ and $Q$ each pass through the other circle's center, a line containing both $P$ and $Q$ is extended to intersect the circles at points $F$ and $G$, and the circles intersect at point $H$, where $\triangle PFQ$ is such that $PF = FQ$ and $PQ = QH = PH$, find the degree measure of $\angle FHG$.
|
120^\circ
| 0.75 |
Given Tara bought some notebooks for a total of $\textdollar 5.20$ at the school supplies store and Lea bought some of the same notebooks for $\textdollar 7.80$, assuming the notebooks cost more than a dollar each, determine the difference in the number of notebooks bought by Lea and Tara.
|
2
| 0.166667 |
Given the quadratic equation \( Ax^2 + Bx + C = 0 \) with roots \( r \) and \( s \), formulate a new quadratic equation \( x^2 + px + q = 0 \) such that its roots are \( r^3 \) and \( s^3 \). Determine the value for \( p \).
|
\frac{B^3 - 3ABC}{A^3}
| 0.666667 |
If $1989 + 1991 + 1993 + 1995 + 1997 + 1999 + 2001 = 14000 - M$, calculate the value of M.
|
35
| 0.916667 |
How many primes less than $100$ have $3$ as the ones digit?
|
7
| 0.666667 |
Two concentric circles have radii of 15 meters and 30 meters. An aardvark starts at point $A$ on the smaller circle and runs along the path that includes half the circumference of each circle and each of the two straight segments that connect the circumferences directly (radial segments). Calculate the total distance the aardvark runs.
|
45\pi + 30
| 0.75 |
Two circles are placed outside a square $ABCD$. The first circle is tangent to side $\overline{AB}$, and the second is tangent to side $\overline{CD}$. Both circles are tangent to the lines extended from $BC$ and $AD$. Calculate the ratio of the area of the first circle to that of the second circle.
|
1
| 0.916667 |
Mr. A owns a home worth $15,000. He sells it to Mr. B at a $20\%$ profit. Mr. B then sells the house back to Mr. A at a $15\%$ loss. Calculate the amount Mr. A gains or loses in the transaction.
|
2700
| 0.416667 |
Calculate the value of $\dfrac{13! - 12! + 11!}{10!}$.
|
1595
| 0.916667 |
At a math contest, $75$ students are wearing blue shirts, and another $105$ students are wearing yellow shirts. The $180$ students are assigned into $90$ pairs. In exactly $30$ of these pairs, both students are wearing blue shirts. Calculate the number of pairs in which both students are wearing yellow shirts.
|
45
| 0.416667 |
Given that Nayla has an index card measuring $5 \times 7$ inches, and she shortens the length of one side by $2$ inches, resulting in a card with an area of $21$ square inches, determine the area of the card if instead, she shortens the length of the other side by the same amount.
|
25
| 0.25 |
Consider all triangles ABC such that AB = AC, and point D is on AC such that BD is perpendicular to AC. Given that AC and CD are integers and BD^2 = 85, find the smallest possible value of AC.
|
11
| 0.25 |
Given the function $g(x) = 3x^2 + x - 4$, calculate the expression $[g(x+h) - g(x)] - [g(x) - g(x-h)]$.
|
6h^2
| 0.916667 |
How many pairs of positive integers (a, b) satisfy the equation \(\frac{a+b^{-1}}{a^{-1}+b} = 9,\) given that \(a+b \leq 200\)?
|
20
| 0.5 |
The number of points common to the graphs of $(x+2y-3)(2x-y+1)=0$ and $(x-2y+4)(3x+4y-12)=0$ is what?
|
4
| 0.916667 |
Three cyclists start from the same point on a 600-meter circular track and travel clockwise with constant speeds of 3.6, 3.9, and 4.2 meters per second. Determine the time in seconds until they meet again at the same starting point.
|
2000
| 0.75 |
An urn initially contains two red balls and one blue ball. A supply of extra red and blue balls is available nearby. George performs the following operation five times: he draws a ball from the urn at random and then adds two balls of the same color from the supply before returning all balls to the urn. After the five operations, the urn contains thirteen balls. What is the probability that the urn contains nine red balls and four blue balls?
**A) $\frac{1920}{10395}$**
**B) $\frac{1024}{10395}**
**C) $\frac{192}{10395}**
**D) $\frac{960}{10395}**
|
\frac{1920}{10395}
| 0.666667 |
Given the graphs of $y = -2|x-a| + b$ and $y = 2|x-c| + d$, and the points of intersection at $(1, 6)$ and $(5, 2)$, find the value of $a+c$.
|
6
| 0.583333 |
Casper can buy $16$ pieces of red candy, $18$ pieces of green candy, $20$ pieces of blue candy, or $n$ pieces of yellow candy. A piece of yellow candy costs $\30$ cents. Determine the smallest possible value of $n$.
|
24
| 0.166667 |
Given Alex lists the whole numbers $1$ through $50$ once, and Tony copies Alex's numbers replacing each occurrence of the digit $3$ by the digit $2$, calculate how much larger Alex's sum is than Tony's.
|
105
| 0.25 |
If the side of one square is the diagonal of a second square, and the side of the second square is the diagonal of a third square, calculate the ratio of the area of the first square to the area of the third square.
|
4
| 0.166667 |
Given the equations $x^3 + bx + c = 0$ and $x^3 + cx + b = 0$, determine the number of ordered pairs $(b,c)$ of positive integers for which neither equation has any real solutions.
|
0
| 0.833333 |
Given the word $COMMITTEE$, calculate the number of distinguishable rearrangements of its letters with both the vowels first.
|
360
| 0.916667 |
A computer generates a 5-digit confirmation code where each digit ranges from $0$ to $9$ with repeated digits allowed. However, no code may end with the sequence $0,0,5$. Calculate the total number of valid confirmation codes possible.
|
99900
| 0.5 |
Determine the area of a rectangular garden plot which measures 1.5 meters on one side and 0.75 meters on the other side.
|
1.125 \, \text{m}^2
| 0.083333 |
Two integers have a sum of 28. When two more integers are added to these, the sum is 45. Finally, when two more integers are added to the sum of the previous 4 integers, the sum is 63. Determine the minimum number of odd integers among the 6 integers.
|
1
| 0.833333 |
Triangle ABC has a right angle at C and ∠A = 30°. If BD (D in AC) is the bisector of ∠ABC, determine ∠BDC.
|
60^\circ
| 0.916667 |
Find the smallest nonprime integer greater than $25$ with no prime factor less than $15$ and the sum of its digits greater than $10$.
|
289
| 0.666667 |
Given a classroom of $k > 10$ students with an average quiz score of 8. If the average score of 10 students is 15, calculate the mean score of the remaining $k - 10$ students.
|
\frac{8k-150}{k-10}
| 0.5 |
A skateboard rolls down a ramp, traveling $8$ inches in the first second and accelerating such that during each successive $1$-second time interval, it travels $9$ inches more than during the previous $1$-second interval. The skateboard takes $20$ seconds to reach the bottom of the ramp. Calculate the total distance it travels, in inches.
|
1870
| 0.416667 |
Two pitchers, each with a capacity of 800 mL, contain juices. The first pitcher is 1/4 filled with orange juice and 1/8 with apple juice. The second pitcher is 1/5 filled with orange juice and 1/10 with apple juice. The rest of each pitcher is filled with water. Both pitchers are then poured into one large container. What fraction of the mixture in the large container is orange juice?
|
\frac{9}{40}
| 0.916667 |
How many positive even multiples of $5$ less than $1000$ are perfect squares?
|
3
| 0.416667 |
Given a thin sheet of uniform density aluminum in the shape of a square measures 4 inches per side and weighs 8 ounces, find the weight of a similar aluminum sheet in the shape of an equilateral triangle with a side length of 6 inches.
|
\frac{9\sqrt{3}}{2}
| 0.583333 |
Given a deck consisting of four red cards labeled $A$, $B$, $C$, $D$, four green cards labeled $A$, $B$, $C$, $D$, and four blue cards labeled $A$, $B$, $C$, $D$, calculate the probability of drawing a winning pair, where a winning pair is defined as either two cards of the same color or two cards of the same letter.
|
\frac{5}{11}
| 0.333333 |
Zoey took four tests with varying numbers of problems. She scored 85% on a 30-problem test, 75% on a 50-problem test, 65% on a 20-problem test, and 95% on a 40-problem test. What is the percentage of all problems that Zoey answered correctly?
|
81.43\%
| 0.583333 |
How many sequences of $0$s and $1$s of length $21$ begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s?
|
114
| 0.083333 |
Several students are competing in a series of four races. A student earns $6$ points for winning a race, $4$ points for finishing second, and $2$ points for finishing third. There are no ties. What is the smallest number of points that a student must earn in the four races to be guaranteed of earning more points than any other student?
|
22
| 0.083333 |
A box contains 35 red balls, 30 green balls, 25 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?
|
95
| 0.5 |
Calculate the sum $E(1)+E(2)+E(3)+\cdots+E(200)$ where $E(n)$ denotes the sum of the even digits of $n$, and $5$ is added to the sum if $n$ is a multiple of $10$.
|
902
| 0.166667 |
How many whole numbers are between $\sqrt{50}$ and $\sqrt{200}$?
|
7
| 0.916667 |
A circle with area \(A_1\) is contained in the interior of a larger circle with area \(A_1 + A_2\). If the radius of the larger circle is \(4\), and if \(A_1, A_1 + A_2, A_2\) is an arithmetic progression, then find the radius of the smaller circle.
|
\frac{4\sqrt{3}}{3}
| 0.416667 |
Calculate the sum of all numbers of the form $2k + 3$, where $k$ takes integer values from $0$ to $n-1$.
|
n^2 + 2n
| 0.833333 |
Alice and Bob each draw a number from a hat containing the numbers $1$ to $50$, both without replacement. If Alice cannot determine who has the larger number, and Bob can still determine who has the larger number after hearing Alice's statement, and Bob's number is prime, andAlice multiplies Bob's number by $150$ and adds her own number to get a perfect square, find the sum of the two numbers drawn from the hat.
|
26
| 0.083333 |
The hands of a clock will form an angle of $120^{\circ}$ at what times between $7$ and $8$ o'clock, correct to the nearest minute?
|
7:16 \text{ and 8:00}
| 0.75 |
If \( y \) varies inversely as \( x \), and if \( y = 6 \) when \( x = 3 \), find the value of \( y \) when \( x = 12 \).
|
\frac{3}{2}
| 0.75 |
Queen Secondary School has 1500 students. Each student takes 4 classes a day. Each class has 25 students and is taught by 1 teacher. Each teacher teaches 5 classes a day. Determine the number of teachers at Queen Secondary School.
|
48
| 0.583333 |
What is the largest quotient that can be formed using two numbers chosen from the set $\{-12, -4, -3, 1, 3, 9\}$?
|
9
| 0.916667 |
Aman, Bella, and Charlie went on a holiday trip and agreed to share the costs equally. During the holiday, they took turns paying for shared expenses like meals and accommodations. At the end of the trip, Aman had paid $X$ dollars, Bella had paid $Y$ dollars, and Charlie had paid $Z$ dollars, where $X < Y < Z$. How many dollars must Aman give to Charlie so that they all share the costs equally?
|
\frac{Y + Z - 2X}{3}
| 0.5 |
In the xy-plane, find the number of lines with a positive prime x-intercept and a composite y-intercept that pass through the point (5,4).
|
1
| 0.75 |
A square pattern is composed of 5 black and 20 white square tiles. Around the pattern, attach a border consisting of white tiles on all sides. What is the ratio of black tiles to white tiles in the extended pattern?
|
\frac{5}{44}
| 0.416667 |
Let $S$ be the set of the $1700$ smallest positive multiples of $5$, and let $T$ be the set of the $1700$ smallest positive multiples of $9$. Determine the number of elements that are common to both $S$ and $T$.
|
188
| 0.75 |
Given that $5y + 3$ varies inversely as the cube of $x$, and when $y = 8$, $x = 2$, determine the value of $y$ when $x = 4$.
|
\frac{19}{40}
| 0.75 |
A number $m$ is randomly selected from the set $\{21, 23, 25, 27, 29\}$, and a number $n$ is randomly selected from $\{2010, 2011, 2012, \ldots, 2030\}$. Calculate the probability that $m^n$ ends in the digit $1$.
|
\frac{2}{5}
| 0.416667 |
Determine the number of solutions to the equation $\tan(3x) = \cos(x - \frac{\pi}{4})$ on the interval $[0, \frac{3\pi}{2}]$.
|
5
| 0.25 |
Two poles, one 30 inches high and another 50 inches high, are 120 inches apart. Determine the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole.
|
18.75
| 0.416667 |
Twenty-five percent less than 80 is one-quarter more than what number?
|
48
| 0.916667 |
Given Alicia earns $25 per hour, of which $2\%$ is deducted to pay local taxes and an additional $0.5\%$ is deducted for state taxes, calculate the total amount of cents per hour of Alicia's wages that are used to pay these taxes combined.
|
62.5
| 0.5 |
Calculate the value of the expression $\frac{8 \times 10^{10}}{2 \times 10^5 \times 4}$.
|
100000
| 0.75 |
A store owner bought 2000 pencils at $0.15 each. If he sells them for $0.30 each, calculate the number of pencils he must sell to make a profit of exactly $180.00$.
|
1600
| 0.833333 |
Connie multiplies a number by 4 and gets 200 as her result. She realizes she should have divided the number by 4 and then added 10 to get the correct answer. Find the correct value of this number.
|
22.5
| 0.916667 |
Four friends make cookies from the same amount of dough with the same thickness. Art's cookies are circles with a radius of 2 inches, and Trisha's cookies are squares with a side length of 4 inches. If Art can make 18 cookies in his batch, determine the number of cookies Trisha will make in one batch.
|
14
| 0.833333 |
Given the binomial expansion of $\left(\frac{x^3}{3} - \frac{3}{x^2}\right)^{10} \cdot x^2$, find the coefficient of $x^5$.
|
0
| 0.75 |
Given the dimensions of a rectangular tile are reported as 4 inches by 6 inches, considering the uncertainties of at least $x - 1$ inches and at most $x + 1$ inch, find the minimum possible area of this rectangle in square inches.
|
15
| 0.666667 |
Find the total number of diagonals that can be drawn in two polygons, the first with 100 sides and the second with 150 sides.
|
15875
| 0.5 |
A rabbit sits at a corner of a square park with side length $12$ meters. It hops $7.2$ meters along a diagonal toward the opposite corner. It then makes a $90^{\circ}$ right turn and hops $3$ more meters. Calculate the average of the shortest distances from the rabbit to each side of the square.
|
6
| 0.916667 |
Determine how many positive even multiples of 3 less than 1500 are perfect squares.
|
6
| 0.916667 |
If $g(x) = 1 - 2x^2$ and $f(g(x))=\frac{1-2x^2}{x^2}$ when $x \neq 0$, find $f(1/3)$.
|
1
| 0.833333 |
How many whole numbers are there between $\sqrt{50}$ and $\sqrt{200}+1$?
|
8
| 0.833333 |
Driving at a constant speed, Tim usually takes 120 minutes to drive from his home to his workplace. One day Tim begins the drive at his usual speed, but after driving half of the way, he encounters heavy traffic and reduces his speed by 30 miles per hour. Due to this, the trip takes him a total of 165 minutes. Calculate the total distance from Tim's home to his workplace in miles.
|
140
| 0.916667 |
Given the equation $y = \sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+\cdots}}}}$, determine the range within which $y$ falls.
|
\frac{1 + \sqrt{17}}{2}
| 0.75 |
Calculate the area of a polygon with vertices at (2,1), (4,3), (6,1), and (4,6).
|
6
| 0.5 |
Given Gilda has given away $30\%$ of her marbles and $15\%$ of what is left to Ebony, and $30\%$ of what is now left to Jimmy, calculate the percentage of her original bag of marbles that Gilda has left for herself.
|
41.65\%
| 0.916667 |
Samuel traveled the first quarter of his journey on a muddy road, the next 30 miles on a tiled road, and the remaining one-sixth on a sandy road. Calculate the total length of Samuel's journey in miles.
|
\frac{360}{7}
| 0.833333 |
An upright cone filled with water has a base radius of $10 \mathrm{cm}$ and a height of $15 \mathrm{cm}$. This water is then poured into a cylinder with a base radius of $15 \mathrm{cm}$. If the cylindrical container is only $10 \mathrm{cm}$ high and any excess water overflows into a spherical container, calculate the volume of water in the spherical container.
|
0
| 0.916667 |
Given a triangle with vertices at points with integer coordinates (2, 3), (5, 7), and (3, 4), determine the area of this triangle.
|
\frac{1}{2}
| 0.833333 |
In a round-robin tournament with 7 teams, each team plays one game against each other team, and each game results in one team winning and one team losing, determine the maximum number of teams that could be tied for the most wins at the end of the tournament.
|
7
| 0.083333 |
The Lions beat the Eagles 3 out of the 4 times they played, then played N more times, and the Eagles ended up winning at least 98% of all the games played; find the minimum possible value for N.
|
146
| 0.166667 |
Calculate the sum of the sequence $3\left(1-\dfrac{1}{3}\right) + 4\left(1-\dfrac{1}{4}\right) + 5\left(1-\dfrac{1}{5}\right) + \cdots + 12\left(1-\dfrac{1}{12}\right)$.
|
65
| 0.5 |
In an annual school competition, the ratio of 9th-graders to 10th-graders is $7:4$, and the ratio of 9th-graders to 11th-graders is $21:10$. Determine the smallest number of students that could be participating in the competition from these grades.
|
43
| 0.75 |
Given the product of Kiana and her two older twin brothers' ages is 256, find the sum of their three ages.
|
20
| 0.333333 |
Given that the first, middle, and last initials of the baby's monogram must start with 'O' and the remaining two initials must be distinct lowercase letters from the first half of the alphabet, and the initials must be in alphabetical order, determine the total number of possible monogram combinations.
|
78
| 0.5 |
Given the fraction $\frac{987654321}{2^{30}\cdot 5^6}$, determine the minimum number of digits to the right of the decimal point required to express this fraction as a decimal.
|
30
| 0.166667 |
Alice and Bob play a game on a circle divided into 12 equally spaced points, numbered 1 to 12 clockwise. Both start at point 12. Alice moves 7 points clockwise, and Bob moves 4 points counterclockwise each turn. Determine the number of turns after which Alice and Bob will meet on the same point.
|
12
| 0.916667 |
Jo and Blair take turns counting, but this time, each one says a number that is two more than the last number said by the other person, starting with $1$ as the first number. What is the $30^{\text{th}}$ number said?
|
59
| 0.833333 |
Suppose $x$ cows produce $y$ gallons of milk in $z$ days. Calculate the amount of milk that $2x$ cows will produce in $3z$ days if the rate of milk production increases by 10%.
|
6.6y
| 0.833333 |
Given the sequence where the first number is $1$, and, for all $n\ge 2$, the product of the first $n$ numbers in the sequence is $2n^2$, calculate the sum of the fourth and the sixth numbers in this sequence.
|
\frac{724}{225}
| 0.916667 |
Given the pan of brownies measures 24 inches by 15 inches and each brownie is cut into pieces measuring 3 inches by 2 inches, calculate the total number of equal-sized pieces of brownie the pan contains.
|
60
| 0.5 |
Given that the product of $450$ and $x$ is a square and the product of $450$ and $y$ is a cube, calculate the sum of $x$ and $y$.
|
62
| 0.75 |
Given the numbers $7350$ and $165$, find the number obtained when the Greatest Common Divisor is first decreased by $15$, and then multiplied by $3$.
|
0
| 0.833333 |
Evaluate the expression $\frac{8^5}{4 \times 2^5 + 16}$.
|
\frac{2^{15}}{4 \times 2^5 + 16} = \frac{2^{15}}{128 + 16} = \frac{2^{15}}{144} = \frac{2^{15-4}}{9} = \frac{2^{11}}{9} = \frac{2048}{9}
| 0.833333 |
Given that $\log_{b} 1024$ is a positive integer, how many positive integers $b$ satisfy this condition.
|
4
| 0.916667 |
The average age of all members including the pet dog is $22$ years, and the father is $50$ years old. The average age of the mother, children, and the pet dog is $18$ years. If the pet dog is $10$ years old, calculate the number of children in the Kasun family.
|
5
| 0.166667 |
Given $\angle \text{CBD}$ as a right angle and the sum of angles around point B, including $\angle \text{ABC}$, $\angle \text{ABD}$, and $\angle \text{CBD}$, totals $200^\circ$. If the measure of $\angle \text{ABD}$ is $70^\circ$, find the measure of $\angle \text{ABC}$.
|
40^\circ
| 0.916667 |
A semipro baseball league mandates that each team consists of 25 players. The league rule specifies that every player must receive a salary of at least $18,000, while the total salaries for all players on a team cannot exceed $1,000,000. Determine the maximum possible salary, in dollars, that a single player can attain under these rules.
|
568,000
| 0.75 |
How many ordered pairs of integers (x, y) satisfy the equation x^4 + y^2 = 4y?
|
2
| 0.5 |
Soda is sold in packs of 8, 16, and 32 cans. Find the minimum number of packs needed to buy exactly 120 cans of soda.
|
5
| 0.666667 |
Given that Alice mixes $150$ grams of lemon juice, $120$ grams of honey, and $330$ grams of water to make a beverage, and that the calorie content for each ingredient is $30$ calories per $100$ grams of lemon juice, $415$ calories per $100$ grams of honey, and water contains no calories, determine the total calorie content in $300$ grams of Alice's beverage.
|
271.5
| 0.916667 |
Given that the sum of $x$ and $y$ is $12$ and the product of $x$ and $3y$ is $108$, find the quadratic polynomial whose roots are $x$ and $y$.
|
t^2 - 12t + 36
| 0.833333 |
LeRoy, Bernardo, and Camila went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses. At the end of the trip, it turned out that LeRoy had paid A dollars, Bernardo had paid B dollars, and Camila had paid C dollars. If the sums paid were A < B < C, calculate the total amount LeRoy must give to Bernardo and Camila so that they share the costs equally.
|
\frac{B + C - 2A}{3}
| 0.5 |
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