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Given that a light bulb is located $15$ centimeters below the ceiling in Bob's living room, the ceiling is $2.8$ meters above the floor, Bob is $1.65$ meters tall and can reach $55$ centimeters above his head, and Bob standing on a chair can just reach the light bulb, calculate the height of the chair, in centimeters. | 45 | 0.666667 |
Given the ceiling is 300 centimeters above the floor, Alice is 160 centimeters tall and can reach 50 centimeters above her head, and standing on a box she can just reach a light bulb located 15 centimeters below the ceiling and 10 centimeters above a decorative shelf, calculate the height of the box in centimeters. | 75 | 0.833333 |
Let ($a_1$, $a_2$, ... $a_{20}$) be a list of the first 20 positive integers such that for each $2\le$ $i$ $\le20$, either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. Determine the number of such lists. | 524,288 | 0.083333 |
How many ordered pairs of integers \((x, y)\) satisfy the equation \(x^4 + y^2 = 2y + 3\)? | 2 | 0.416667 |
An uncrossed belt is fitted without slack around two circular pulleys with radii of $12$ inches and $3$ inches. If the distance between the points of contact of the belt with the pulleys is $20$ inches, calculate the distance between the centers of the pulleys in inches. | \sqrt{481} | 0.5 |
Let H represent the number of holes dug by the hamster and G represent the number of holes dug by the guinea pig. The hamster stores 5 seeds per hole, and the guinea pig stores 6 seeds per hole. If both animals stored the same total number of seeds, and the guinea pig needed 3 fewer holes, calculate the total number of seeds stored by the hamster. | 90 | 0.833333 |
Determine the value of $1 + 3 + 5 + \cdots + 2023 - (2 + 4 + 6 + \cdots + 2020)$. | 3034 | 0.75 |
Let $\overline{AB}$ be a diameter in a circle with a radius of $10$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at point $E$ such that $BE=3$ and $\angle AEC = 45^{\circ}$. Find $CE^2+DE^2$. | 200 | 0.166667 |
Given that Alice, Bob, Cindy, Dave, and Emma have internet accounts, some but not all of whom are internet friends with each other, and none have friends outside this group, and each has the same number of internet friends, determine the number of ways the configuration of friendships can occur. | 12 | 0.333333 |
A right triangle with legs of length a and b is drawn. A new right triangle is formed by joining the midpoints of the hypotenuse of the first one. Then a third right triangle is formed by joining the midpoints of the hypotenuse of the second; and so on forever. Find the limit of the sum of the perimeters of all the triangles thus drawn. | 2(a+b+\sqrt{a^2 + b^2}) | 0.833333 |
The sum of the first 15 terms and the sum of the first 90 terms of a given arithmetic progression are 150 and 15, respectively. Find the sum of the first 105 terms. | -189 | 0.416667 |
Given that the price of each piece of yellow candy is 24 cents, and that Lena has exactly enough money to buy 18 pieces of red candy, 16 pieces of green candy, 20 pieces of blue candy, or n pieces of yellow candy, calculate the smallest possible value of n. | 30 | 0.666667 |
Suppose that $12 \cdot 3^x = 7^{y+5}$. Determine $x$ when $y = -4$. | \log_3 \left(\frac{7}{12}\right) | 0.416667 |
Three A's, three B's, and three C's are placed in the nine spaces so that each row and column contains one of each letter. If B is placed in the upper left corner, calculate the number of arrangements possible. | 4 | 0.666667 |
In a right triangle XYZ, where XZ=15, YZ=8, and angle Z is a right angle, a circle is inscribed completely inside the triangle. Determine the radius of this circle. | 3 | 0.916667 |
Julia has a large white cube with an edge of 15 feet. She also possesses enough blue paint to cover 500 square feet. Julia decides to paint a blue square centered on each face of the cube, leaving a white border around it. Find the area of one of the white squares in square feet. | \frac{425}{3} | 0.583333 |
Lara and Darla are racing on a 500-meter circular track. Starting simultaneously, Lara runs at a speed 20% faster than Darla. Calculate the number of laps Lara has completed when she first overtakes Darla. | 6 | 0.916667 |
A half-sector of a circle of radius $6$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. Calculate the volume of the cone in cubic inches. | 9\pi \sqrt{3} | 0.833333 |
A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-third of the total number of faces of the unit cubes are red. What is $n$? | 3 | 0.833333 |
A small bottle of lotion can hold 60 milliliters, while a large bottle can hold 750 milliliters. Additionally, pad material in the large bottle absorbs 5% of the lotion from the small bottles during transfer. Determine the minimum number of small bottles Kyla must buy to fill the large bottle completely. | 14 | 0.916667 |
How many primes less than $100$ have $3$ as the ones digit? | 7 | 0.666667 |
In a city of 400 adults, 370 adults own scooters and 80 adults own bikes, calculate the number of the scooter owners who do not own a bike. | 320 | 0.916667 |
Evaluate the value of $N$ if $988 + 990 + 992 + 994 + 996 = 5000 - N$. | 40 | 0.833333 |
How many whole numbers are between $\sqrt{50}$ and $\sqrt{200}$? | 7 | 0.916667 |
In a class, $20\%$ of the students are juniors and $80\%$ are seniors. The overall average score of the class on a recent test was $85$. Juniors all received the same score, and the average score of the seniors was $84$. What score did each of the juniors receive on this test? | 89 | 0.916667 |
Given an 81-number grid ranging from 1 to 81 is written on a 9x9 checkerboard, calculate the sum of the numbers in the four corners of the checkerboard. | 164 | 0.916667 |
Mr. $X$ owns a property worth $15,000. He sells it to Mr. $Y$ with a 15% profit, then Mr. $Y$ sells it back to Mr. $X$ at a 5% loss. Determine the net outcome of these transactions for Mr. $X$. | 862.50 | 0.5 |
How many whole numbers between $200$ and $500$ contain the digit $1$? | 57 | 0.416667 |
Given positive integers $a$ and $b$ are each less than $8$, calculate the smallest possible value for $2 \cdot a - a \cdot b$. | -35 | 0.666667 |
Given a pentagon inscribed in a circle, determine the sum of the angles inscribed in the five arcs cut off by the sides of the pentagon. | 180^\circ | 0.916667 |
Given the number 2550, calculate the sum of its prime factors. | 27 | 0.666667 |
What is the least possible value of (x+1)(x+3)(x+5)(x+7) + 2050, where x is a real number? | 2034 | 0.833333 |
Given that the monogram consists of three initials in alphabetical order with a last initial of 'X', and the first and middle initials must be distinct, determine the total number of possible monograms. | 253 | 0.083333 |
In an isosceles triangle $\triangle ABC$, base angles $A$ and $B$ are such that $A = B$. Let's say $A = 30^\circ$. The altitude from vertex $C$ to the base $AB$ divides the angle $C$ into two parts $C_1$ and $C_2$, with $C_2$ adjacent to side $a$. Calculate $C_1 - C_2$. | 0^\circ | 0.916667 |
Given that for every $4^\circ$ rise in temperature, the volume of a certain gas expands by $3$ cubic centimeters, and when the temperature is $30^\circ$, the volume of the gas is $40$ cubic centimeters, determine the volume of the gas when the temperature drops first to $22^\circ$ and then further to $14^\circ$. | 28 | 0.833333 |
For how many positive integer values of $n$ are both $\frac{n}{4}$ and $4n$ three-digit whole numbers? | 0 | 0.916667 |
Four fair dice with 8 faces each (numbered 1 through 8) are tossed. What is the probability that the numbers on four of the dice can be arranged to form an arithmetic progression with a common difference of two? | \frac{3}{256} | 0.833333 |
The base three representation of $x$ is $1122001_3$. Determine the first digit (on the left) of the base nine representation of $x$. | 1 | 0.916667 |
Given that $x^2 + 2bx + c = 0$ and $x^2 + 2cx + b = 0$ do not have two distinct real solutions, calculate the number of ordered pairs of positive integers $(b, c)$. | 1 | 0.833333 |
If $4(5y + 3\pi) = Q$, then express the value of $8(10y + 6\pi + 2\sqrt{3})$ in terms of $Q$. | 4Q + 16\sqrt{3} | 0.833333 |
Given that Crystal runs due north for 1 mile, then due east for 2 miles, and then due south for 1 mile, determine the distance of the last portion of her run that takes her on a straight line back to where she started in miles. | 2 | 0.916667 |
Let X be the set of the 3000 smallest positive multiples of 5, and let Y be the set of the 3000 smallest positive multiples of 8. Find the number of elements common to sets X and Y. | 375 | 0.583333 |
Given soda is sold in packs of 6, 12, 24, and 30 cans, find the minimum number of packs needed to buy exactly 120 cans of soda. | 4 | 0.916667 |
The number of revolutions of a wheel, with a fixed center and an outside diameter of $8$ feet, required to cause a point on the rim to go two miles can be calculated. | \frac{1320}{\pi} | 0.5 |
Given the list of numbers $8-3\sqrt{10}$, $3\sqrt{10}-8$, $23-6\sqrt{15}$, $58-12\sqrt{30}$, $12\sqrt{30}-58$, identify the smallest positive number in the list. | 3\sqrt{10} - 8 | 0.916667 |
For how many integers $x$ is the number $x^4 - 53x^2 + 150$ negative? | 12 | 0.25 |
Given Clara's correct scores were doubled and summed, but one score of 15 was mistakenly tripled instead of doubled, calculate the difference between the correct sum and the incorrect sum of her scores. | 15 | 0.666667 |
A circle of radius 5 is centered at point $B$. An equilateral triangle with a vertex at $B$ has a side length of 10. 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. | 25(\pi - \sqrt{3}) | 0.25 |
If the ratio of $3x-2y$ to $2x+y$ is $\frac{4}{5}$, calculate the ratio of $x$ to $y$. | 2 | 0.75 |
Find the least common positive period p for any function f that satisfies f(x+5) + f(x-5) = f(x) for all real x. | 30 | 0.166667 |
Four runners start simultaneously from the same point on a 600-meter circular track and run clockwise. Their constant speeds are 4.2, 4.5, 4.8, and 5.1 meters per second respectively. Find the time in seconds when all runners are together again at some point on the track. | 2000 | 0.166667 |
Given a set of $2n$ numbers where $n > 1$, of which one is $1 + \frac{1}{n}$ and all the others are $1$, calculate the arithmetic mean of these $2n$ numbers. | 1 + \frac{1}{2n^2} | 0.916667 |
Determine the difference between the sum of the first one hundred positive even integers and the sum of the first one hundred positive multiples of 3. | -5050 | 0.5 |
Given a square piece of paper is folded so that point (0,4) is matched with (4,0) using a fold that also involves a $45^\circ$ rotation, find the aligned coordinates of point $(8,6)$ and determine the sum of its coordinates. | 14 | 0.833333 |
Calculate the value of $(3(3(3(3(3(3+2)+2)+2)+2)+2)+2)$. | 1457 | 0.416667 |
A four-digit palindrome is chosen at random. What is the probability that it is divisible by 11?
A) $\dfrac{1}{10}$
B) $\dfrac{1}{9}$
C) $\dfrac{1}{2}$
D) $\dfrac{1}{5}$
E) $1$ | 1 | 0.666667 |
A particle begins at a point P on the parabola y = x^2 - 2x - 8 where the y-coordinate is 8. It rolls along the parabola to the nearest point Q where the y-coordinate is -8. Calculate the horizontal distance traveled by the particle, defined as the absolute difference between the x-coordinates of P and Q. | \sqrt{17} - 1 | 0.666667 |
Calculate the product $\left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right)\ldots\left(1-\frac{1}{12^{2}}\right)$. | \frac{13}{24} | 0.5 |
Given that a rhombus $PQRST$ has side length $5$ and $\angle Q = 90^{\circ}$, find the area of region $W$ that consists of all points inside the rhombus that are closer to vertex $Q$ than any of the other three vertices. | 6.25 | 0.333333 |
Given a three-digit number with digits $x, y, z$, where $x$ is the first digit, $z$ is the third digit, and $y$ is the second digit, the number is not divisible by $5$, has digits that sum to less than $15$, and $x=z>y$, calculate the number of such numbers. | 14 | 0.166667 |
Lara in her car notices Leo on his motorcycle $1$ mile ahead moving in the same direction. After she overtakes him, she can still see him in her rearview mirror until he is $1$ mile behind her. Lara drives at a constant speed of $60$ miles per hour, and Leo cruises at a constant speed of $40$ miles per hour. Calculate the time in minutes that Lara can see Leo. | 6 | 0.916667 |
The number of eighth graders and fifth graders who bought a pencil can be represented as variables x and y, respectively. Given that some eighth graders each bought a pencil and paid a total of $2.34 dollars, and some of the $50$ fifth graders each bought a pencil and paid a total of $3.25 dollars, determine the difference in the number of fifth graders and eighth graders who bought a pencil. | 7 | 0.083333 |
Given a right triangle with legs $a$ and $b$, and the hypotenuse $c$, where $a:b = 2:5$, find the ratio of $r$ to $s$ if a perpendicular is dropped from the right-angle vertex to the hypotenuse, dividing it into segments $r$ and $s$. | \frac{4}{25} | 0.75 |
Given that Mary is $25\%$ older than Sally, and Sally is $30\%$ younger than Danielle, and the sum of their ages is $36$ years, determine Mary's age on her next birthday. | 13 | 0.333333 |
If the digits $12$ are placed after a three-digit number whose hundreds' digit is $h$, tens' digit is $t$, and units' digit is $u$, express the new formed number as an algebraic expression. | 10000h + 1000t + 100u + 12 | 0.75 |
What is the probability that a three-digit number is formed from 5 pieces of paper labeled with the digits 2, 3, 4, 5, and 6, such that the number is even? | \frac{3}{5} | 0.916667 |
Carl has a collection of $6$-pound rocks worth $$16$ each, $3$-pound rocks worth $$9$ each, and $2$-pound rocks worth $$3$ each, with at least $30$ of each size. Determine the maximum value, in dollars, of the rocks he can carry out of the cave, given that he can carry at most $24$ pounds and no more than $4$ rocks of any size. | \$68 | 0.333333 |
What is the hundreds digit of the expression $(25! - 20! + 10!)$? | 8 | 0.916667 |
Suppose the estimated €25 billion (Euros) cost to send a person to the planet Mars is shared equally by the 300 million people in a consortium of countries. Given the exchange rate of 1 Euro = 1.2 dollars, calculate each person's share in dollars. | 100 | 0.75 |
The original price of the article in terms of p and q can be expressed as $\frac{10000}{10000 + 100(p - q) - pq}$. | \frac{10000}{10000 + 100(p - q) - pq} | 0.75 |
Given that Alex received $150$ points in a $60$-question multiple choice math contest, where $5$ points are awarded for a correct answer, $0$ points for an answer left blank, and $-2$ points for an incorrect answer, find the maximum number of questions that Alex could have answered correctly. | 38 | 0.666667 |
How many ways are there to write $3060$ as the sum of twos and threes, ignoring order? | 511 | 0.833333 |
Given a convex polygon with the sum of its interior angles equal to $3140^\circ$, calculate the degree measure of the missing angle. | 100 | 0.166667 |
Given the sum of the first twenty terms of an arithmetic progression is five times the sum of the first ten terms, determine the ratio of the first term to the common difference. | -\frac{7}{6} | 0.916667 |
Find the product of the three smallest prime factors of 180. | 30 | 0.416667 |
Given real numbers $a$ and $b$, $\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a} - \frac{1}{b}} = 1001.$ Find the value of $\frac{a+b}{a-b}$. | -1001 | 0.666667 |
Let \( p, q, r, s, \) and \( t \) be distinct integers such that \((8-p)(8-q)(8-r)(8-s)(8-t) = 120\). Calculate the value of \( p+q+r+s+t\). | 25 | 0.5 |
If 104 is divided into three parts which are proportional to 1, 1/2, and 1/3, calculate the value of the middle part. | 28\frac{4}{11} | 0.083333 |
Triangle $ABC$ has a right angle at $B$. Point $D$ is the foot of the altitude from $B$, $AD=5$, and $DC=12$. Calculate the area of $\triangle ABC$. | 17\sqrt{15} | 0.083333 |
For every dollar Ben spent on coffee, David spent $0.5 less. Ben paid $15 more than David. Determine the total amount they spent in the coffee shop together. | 45 | 0.916667 |
Consider a scenario where a cubical die is labeled with the numbers $2$, $3$, $3$, $4$, $4$, and $5$. Another die is marked with $1$, $2$, $3$, $5$, $6$, and $7$. Calculate the probability that rolling these two dice simultaneously results in a sum on their top faces of either $6$, $8$, or $10$. | \frac{7}{18} | 0.166667 |
Given Sean's current test scores of 82, 76, 88, 94, 79, and 85, determine the minimum test score he would need to achieve to increase his overall average by exactly 5 points and not drop below his current lowest score. | 119 | 0.75 |
Given Timothy's house has $4$ bedrooms, each one is $15$ feet long, $12$ feet wide, and $10$ feet high, and the doorways and windows in each bedroom occupy $75$ square feet, calculate the total square feet of walls that must be painted. | 1860 | 0.916667 |
Evaluate the expression \( 3000(3000^{2999})^2 \). | 3000^{5999} | 0.5 |
Given that bricklayer Alice can build a wall alone in 8 hours, and bricklayer Bob can build it alone in 12 hours, and they complete the wall in 6 hours when working together with a 15-brick-per-hour decrease in productivity, determine the number of bricks in the wall. | 360 | 0.583333 |
Jack walks to a park 1.5 miles away at a speed of 3 miles per hour, while Jill roller skates at a speed of 8 miles per hour. Calculate the time difference in minutes between their arrivals at the park. | 18.75 | 0.75 |
Rectangle ABCD and right triangle AEF share side AD and have the same area. Side AD = 8, and side AB = 7. If EF, which is perpendicular to AD, is denoted as x, determine the length of hypotenuse AF. | 2\sqrt{65} | 0.833333 |
A sphere with center O has a radius of 8. A triangle with sides of lengths 13, 13, 10 is situated such that all of its sides are tangent to the sphere. Calculate the distance from O to the plane determined by the triangle. | \frac{2\sqrt{119}}{3} | 0.416667 |
Given the equations $4x + ay + d = 0$ and $dx - 3y + 15 = 0$, determine the number of pairs of values of $a$ and $d$ such that these two equations have the same graph. | 2 | 0.916667 |
Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\textdollar 2.16$. Sharona bought some of the same pencils and paid $\textdollar 2.72$. Determine the number of more pencils Sharona bought than Jamar. | 7 | 0.833333 |
What is the probability that a randomly drawn positive factor of $72$ is less than $8$? | \frac{5}{12} | 0.916667 |
In a science fair, 64 students are wearing green shirts, and another 68 students are wearing red shirts. The 132 students are grouped into 66 pairs. In exactly 28 of these pairs, both students are wearing green shirts. Determine the number of pairs in which both students are wearing red shirts. | 30 | 0.5 |
An equilateral triangle ABC has a side length of 4. A right isosceles triangle DBE, where $DB=EB=1$ and angle $D\hat{B}E = 90^\circ$, is cut from triangle ABC. Calculate the perimeter of the remaining quadrilateral. | 10 + \sqrt{2} | 0.166667 |
Given that Ahn chooses a two-digit integer, subtracts it from 300, multiplies the result by 2/3, and doubles the whole result, find the largest number Ahn can get. | \frac{1160}{3} | 0.5 |
If \(x\) is real and positive and grows beyond all bounds, calculate the limit as \(x\) approaches infinity of \(\log_4{(8x-3)}-\log_4{(5x+2)}\). | \log_4{\left(\frac{8}{5}\right)} | 0.666667 |
Calculate the value of $\text{rem}(x, y)$ where $x = \frac{5}{7}$ and $y = \frac{3}{4}$, and then multiply the result by 2. | \frac{10}{7} | 0.916667 |
Evaluate \((x^x)^{(x^x)}\) at \(x = 3\). | 27^{27} | 0.833333 |
Given that Mrs. Martinez teaches math to $20$ students, and that the average grade for the class was $75$ when she graded everyone's test except Leah's and Jake's, was $76$ after grading Leah's, and $77$ after grading Jake's, determine Jake's score on the test. | 96 | 0.75 |
In multiplying two positive integers $a$ and $b$, Ron mistakenly reversed the digits of the two-digit number $b$. His erroneous product was $143.$ From this, find the correct value of the product of $a$ and $b$. | 341 | 0.416667 |
Given a store offering a promotion where all items are priced at "one-third off" the original price, and a special coupon grants an extra 25% discount on the promotional prices, determine the percentage off the original price a customer ultimately pays. | 50\% | 0.916667 |
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