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Given the integer $2083$, express it as the sum of two positive integers whose difference is as small as possible and calculate this difference. | 1 | 0.916667 |
What is the maximum possible product of three different numbers from the set $\{-9, -5, -3, 1, 4, 6, 8\}$? | 360 | 0.75 |
Given the sets of consecutive integers $\{1\}$, $\{2, 3\}$, $\{4,5,6\}$, $\{7,8,9,10\}$, $\ldots$, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set, let $S_n$ be the sum of the elements in the nth set. Find $S_{30}$. | 13515 | 0.833333 |
Given $2025$ is written as a product of two positive integers whose difference is as small as possible, calculate the difference. | 0 | 0.833333 |
An item, when sold for $x$ dollars, results in a loss of 20%. However, when sold for $y$ dollars, it leads to a gain of 25%. Find the ratio of $y:x$. | \frac{25}{16} | 0.916667 |
Determine the expression for $g(-x)$ in terms of $g(x)$, given that $g(x) = \frac{x^2 - x - 1}{x^2 + x - 1}$. | \frac{1}{g(x)} | 0.833333 |
George observed a train crossing a bridge and began timing it as it passed. He counted 8 cars in the first 12 seconds of the train's passage. The entire train took 3 minutes and 30 seconds to completely pass the bridge at a constant speed. Estimate the number of cars in the train. | 140 | 0.833333 |
LeRoy paid $L$ dollars, Bernardo paid $B$ dollars, and Cecilia paid $C$ dollars, where $L < B < C$. Calculate the amount in dollars that LeRoy must give to Cecilia so that they share the costs equally. | \frac{B + C - 2L}{3} | 0.333333 |
Big Bea, the bear, ate 196 bananas from April 1 through April 7. Each day she ate eight more bananas than on the previous day. Calculate the number of bananas Big Bea ate on April 7. | 52 | 0.916667 |
Given a rhombus $EFGH$ with a perimeter of $64$ meters and the length of diagonal $\overline{EH}$ of $30$ meters, calculate the area of the rhombus $EFGH$. | 30\sqrt{31} | 0.75 |
Determine the area of a quadrilateral with vertices at the coordinates $(2,1)$, $(1,6)$, $(4,5)$, and $(9,9)$. | 13 | 0.583333 |
Given that Sarah had the capacity to paint 45 rooms and 4 cans of paint were dropped, determine the number of cans of paint used to paint 35 rooms. | 14 | 0.166667 |
How many positive integers less than $500$ are $8$ times the sum of their digits? | 1 | 0.833333 |
Given the data set $[8, 15, 21, 29, 29, 35, 39, 42, 50, 68]$, where the median $Q_2 = 32$, the first quartile $Q_1 = 25$, and the third quartile $Q_3 = 45$, find the number of outliers present if an outlier is defined as a value that is more than $2.0$ times the interquartile range below the first quartile or above the third quartile. | 0 | 0.833333 |
The length of track required to rise 800 feet at a 4% grade is approximately 20,000 feet. Calculate the additional length of track required to reduce this grade to 3%. | 6667 | 0.666667 |
Given two distinct dice: Die A with numbers from $1$ to $6$, and Die B with numbers from $1$ to $7$, if Die B shows a $7$, its face value counts as zero in the sum, calculate the probability that the sum of the numbers on the two top faces is exactly $7$. | \frac{1}{7} | 0.5 |
Given a $5\times 5$ block of calendar dates from 1 to 25 arranged consecutively, reverse the order of the numbers in the second, third, and fifth rows, and then calculate the positive difference between the sums of the numbers on the two diagonals. | 4 | 0.5 |
Jo and Blair take turns counting, starting with Jo who says "3". Each subsequent number said by either Jo or Blair is two more than the last number said by the other person. What is the $100^{\text{th}}$ number said? | 201 | 0.75 |
Given the data set $[2, 11, 23, 23, 25, 35, 41, 41, 55, 67, 85]$, median $Q_2 = 35$, first quartile $Q_1 = 23$, and third quartile $Q_3 = 55$, determine how many outliers this data set has. | 0 | 0.583333 |
Determine the number of positive integers $x$ that satisfy the inequality $(10x)^4 > x^8 > 2^{16}$. | 5 | 0.833333 |
A circle is inscribed within a right angle triangle with sides $3$, $4$, and $5$. Simultaneously, a square is concentric with the circle and has side length $s$. Determine the value of $s$ and compute the area inside the circle but outside the square. | \pi - 2 | 0.583333 |
Determine the area of a quadrilateral with vertices at $(0, 0)$, $(4, 0)$, $(4, 3)$, and $(2, 5)$. | 13 | 0.916667 |
Every high school in the city of Newton sent a team of $4$ students to a math contest. Andrea scored the highest among all contestants, and her teammates Beth, Carla, and David placed $50$th, $75$th, and $100$th, respectively. Determine how many schools are in the city of Newton. | 25 | 0.666667 |
If Alice sells an item at $15 less than the list price and receives 15% of her selling price as her commission, and Charles sells the item at $18 less than the list price and receives 18% of his selling price as his commission, and if Alice and Charles both receive the same commission amount, determine the list price of the item. | 33 | 0.75 |
In the complex plane, let $A$ be the set of solutions to $z^3 - 27 = 0$ and let $B$ be the set of solutions to $z^3 - 9z^2 - 27z + 243 = 0,$ find the distance between the point in $A$ closest to the origin and the point in $B$ closest to the origin. | 3(\sqrt{3} - 1) | 0.75 |
Two angles of an isosceles triangle measure $60^\circ$ and $x^\circ$. What is the sum of the three possible values of $x$? | 180^\circ | 0.583333 |
Three numbers have a mean that is $20$ more than the smallest number and $30$ less than the largest number. The median of the three numbers is $10$. Find their sum. | 60 | 0.75 |
What is the value of $1235 + 2351 + 3512 + 5123$? | 12221 | 0.416667 |
Given $x \spadesuit y = (x+y+1)(x-y)$, calculate $2 \spadesuit (3 \spadesuit 6)$. | -864 | 0.916667 |
Given $x$ is real and positive and grows beyond all bounds, and $y$ is a positive constant, determine the limit of $\log_4{(8x + 3y - 7)} - \log_4{(4x + y + 2)}$. | \frac{1}{2} | 0.833333 |
Given the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers, how many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $20\%$? | 0 | 0.166667 |
Shop owner purchases 2000 pencils at $0.15 each and sells them at $0.30 each, determine how many pencils he must sell to make a profit of exactly $150. | 1500 | 0.833333 |
Consider a set of 16 integers from -8 to 7, inclusive, arranged to form a 4-by-4 square. Find the value of the common sum of the numbers in each row, column, and main diagonal. | -2 | 0.75 |
When George places his marbles into bags with 8 marbles per bag, he has 5 marbles left over. When John does the same with his marbles, he has 6 marbles left over. George then acquires 5 more marbles. Announce the number of marbles leftover when their total number of marbles is placed into as many bags as possible, carrying 8 marbles per bag. | 0 | 0.916667 |
Given the polynomial expansion of $(1+3x-x^2)^5$, find the coefficient of $x^9$. | 15 | 0.416667 |
Al and Carol start their new jobs on the same day. Al's schedule is 4 work-days followed by 2 rest-days, and Carol's schedule is 5 work-days followed by 1 rest-day. Determine the number of their first 1000 days in which both have rest-days on the same day. | 166 | 0.916667 |
A 4x4x4 cube is made of $64$ normal dice. Each die's opposite sides sum to $7$. Calculate the smallest possible sum of all the values visible on the $6$ faces of the large cube. | 144 | 0.166667 |
A parabolic arch has a height of $20$ inches and a span of $30$ inches. Determine the height of the arch at $3$ inches from the center. | 19.2 | 0.833333 |
Given that Store P offers a 20% discount on the sticker price plus a $120 rebate, and Store Q offers a 30% discount on the same sticker price with no rebate, and Clara saves $30 more by purchasing at Store P, determine the sticker price of the laptop. | 900 | 0.916667 |
What is $\sum_{n = 1}^{1023}{f(n)}$, where $f(n) = \left\{\begin{matrix}\log_{4}{n}, &\text{if }\log_{4}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{matrix}\right.$ | \frac{45}{2} | 0.083333 |
Consider a sequence of consecutive integer sets where each set starts one more than the last element of the preceding set and each set has one more element than the one before it. For a specific n where n > 0, denote T_n as the sum of the elements in the nth set. Find T_{30}. | 13515 | 0.583333 |
Given the expression $\left(\frac{x^2+2}{x}\right)\left(\frac{y^2+2}{y}\right)+\left(\frac{x^2-2}{y}\right)\left(\frac{y^2-2}{x}\right)$, where $xy \neq 0$, simplify the expression. | 2xy + \frac{8}{xy} | 0.916667 |
If $1764$ is written as a product of two positive integers whose difference is as small as possible, calculate the difference of these two positive integers. | 0 | 0.916667 |
Given $A$ and $B$ together can do a job in $3$ days; $B$ and $C$ can do it in $6$ days; and $A$ and $C$ in $3.6$ days, calculate the number of days $C$ needs to do the job alone. | 18 | 0.916667 |
Given \( n = x - y^{x - (y+1)} \), find the value of \( n \) when \( x = 3 \) and \( y = -3 \). | 246 | 0.916667 |
Queen High School has $1500$ students, and each student takes $6$ classes per day. Each teacher teaches $5$ classes, with each class having $25$ students and $1$ teacher. How many teachers are there at Queen High School? | 72 | 0.666667 |
Given that the digits of the integers must be from the set $\{1,3,4,5,6,8\}$ and the integers must be between $300$ and $800$, determine how many even integers have all distinct digits. | 40 | 0.166667 |
Given a pentagon is inscribed in a circle, calculate the sum of the angles that are inscribed into each of the five segments outside the pentagon, each segment being the region outside the pentagon but inside the circle. | 720^{\circ} | 0.083333 |
Let \( g(x) \) be a polynomial with leading coefficient 1, whose four roots are the reciprocals of the four roots of \( f(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( 1 < a < b < c < d \). Find \( g(1) \) in terms of \( a, b, c, \) and \( d \). | \frac{1 + a + b + c + d}{d} | 0.166667 |
Calculate the value of
\[
\frac{(.5)^4}{(.05)^3}
\] | 500 | 0.916667 |
How many 4-digit numbers greater than 1000 can be formed using each of the digits 3, 0, 3, 1 exactly once? | 9 | 0.75 |
A positive integer $n$ has $72$ divisors and $5n$ has $90$ divisors. What is the greatest integer $j$ such that $5^j$ divides $n$? | 3 | 0.583333 |
If $\log_{10}2 = a$ and $\log_{10}3 = b$, calculate $\log_{4}18$. | \frac{a + 2b}{2a} | 0.833333 |
Each of 3000 boxes in a line contains a single red marble, and for $1 \le k \le 3000$, the box in the $k\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. Determine the smallest value of $n$ for which $P(n) < \frac{1}{3000}$. | 55 | 0.916667 |
A box contains 30 red balls, 24 green balls, 22 yellow balls, 15 blue balls, 12 white balls, 10 black balls, and 5 purple balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 16 balls of a single color will be drawn? | 88 | 0.083333 |
Given the listed price of an item is greater than $\textdollar 150$, and the shopper can use one of three coupons: Coupon A offers 15% off the listed price, Coupon B provides a $\textdollar 30$ flat discount, and Coupon C offers 20% off the amount by which the listed price exceeds $\textdollar 150$, determine the difference between the smallest and largest prices for which Coupon A saves more dollars than both Coupon B and Coupon C. | 400 | 0.916667 |
A frog starts at point $(2, 3)$ on a grid and makes jumps of length $1$ parallel to the coordinate axes. Each jump direction (up, down, right, or left) is chosen randomly. The jumping sequence stops when the frog reaches any point on the boundary of the rectangle defined by vertices $(0,0), (0,5), (5,5),$ and $(5,0)$. Calculate the probability that the sequence of jumps ends at either the top or bottom horizontal side of the rectangle. | \frac{1}{2} | 0.25 |
Two-thirds of a pitcher is filled with orange juice. The pitcher is then emptied by pouring an equal amount of juice into each of 6 cups. Calculate the percent of the total capacity of the pitcher that each cup received. | 11.11\% | 0.666667 |
Two boxes contain colored balls. Box 1 has $40$ red balls, $30$ green balls, $25$ yellow balls, and $15$ blue balls. Box 2 has $35$ red balls, $25$ green balls, and $20$ yellow balls. Find the minimum number of balls that must be drawn from these boxes (without replacement, drawing from any box at will) to guarantee that at least $20$ balls of a single color are drawn. | 73 | 0.333333 |
John has two identical cups. Initially, he puts 6 ounces of tea into the first cup and 6 ounces of milk into the second cup. He then pours one-third of the tea from the first cup into the second cup and mixes thoroughly. After stirring, John then pours half of the mixture from the second cup back into the first cup. Finally, he pours one-quarter of the mixture from the first cup back into the second cup. What fraction of the liquid in the first cup is now milk? | \frac{3}{8} | 0.5 |
For how many integers x does a triangle with side lengths 15, 20 and x have all its angles acute? | 11 | 0.75 |
Jo and Blair take turns counting numbers starting from 1, with each person adding 2 to the last number said by the other person. Given that Jo starts by saying "1", find the $30^{\text{th}}$ number said. | 59 | 0.916667 |
Sharon drives from her house to her mother's house in a known time and distance relationship with a variable speed. Given the original travel time is 180 minutes, and if she decreases her speed by 30 miles per hour after driving $\frac{1}{2}$ of the distance, the total travel time extends to 300 minutes. Find the distance from Sharon's house to her mother's house. | 157.5 | 0.833333 |
Given two altered dice where one die has faces labeled with the numbers $1, 1, 3, 3, 5, 5$ and the other has faces labeled $2, 2, 4, 4, 6, 6$, find the probability that the sum of the numbers on the top faces will be $7$ after the two dice are rolled. | \frac{1}{3} | 0.166667 |
Given that the cost of a school play ticket is $x$ dollars, where $x$ is a whole number, and a group of 9th graders spends $36$, a group of 10th graders spends $54$, and a group of 11th graders spends $72$ dollars, determine the number of possible values for $x$. | 6 | 0.583333 |
For all real numbers $x$, evaluate the expression $x[x\{x(3-x)-5\}+12]+2$. | -x^4 + 3x^3 - 5x^2 + 12x + 2 | 0.75 |
Jeremy's father drives him to school at a normal speed of 15 minutes, but on a certain day, he drove at a speed 25 miles per hour faster and arrived at school in 9 minutes. Find the distance to school. | 9.375 | 0.916667 |
A total of 10 men can paint a house in 6 days. If 4 additional skilled men join them, and the skilled men work 25% more efficiently than the original group, calculate the number of days it will take for all the men to paint the same house. | 4 | 0.916667 |
Evaluate $\sqrt{\frac{x}{1 - \frac{3x - 2}{2x}}}.$ | \sqrt{\frac{2x^2}{2 - x}} | 0.75 |
Using only pennies, nickels, dimes, quarters, and half-dollars, find the smallest number of coins Freddie would need to pay any amount of money less than a dollar. | 9 | 0.583333 |
Alex sent 150 text messages and talked for 28 hours, given a cell phone plan that costs $25 each month, $0.10 per text message, $0.15 per minute used over 25 hours, and $0.05 per minute within the first 25 hours. Calculate the total amount Alex had to pay in February. | 142.00 | 0.583333 |
Given two circular pulleys with radii of $10$ inches and $6$ inches, and the distance between the points of contact of the belt with the pulleys is $30$ inches, calculate the distance between the centers of the pulleys in inches. | 2\sqrt{229} | 0.25 |
Given a basketball player made 7 baskets during a game, with each basket being worth either 2 or 3 points, determine the total number of different numbers that could represent the total points scored by the player. | 8 | 0.916667 |
Given \( y = \frac{1+i\sqrt{3}}{2} \), calculate \( \dfrac{1}{y^3-y} \). | -\frac{1}{2} + \frac{i\sqrt{3}}{6} | 0.083333 |
Each of two boxes contains three chips numbered $1$, $4$, $3$. A chip is drawn randomly from each box. Find the probability that the product of the numbers on the two chips is even. | \frac{5}{9} | 0.833333 |
Calculate the sum of $(3+13+23+33+43)+(7+17+27+37+47)$. | 250 | 0.916667 |
When $n$ standard 8-sided dice are rolled, the probability of obtaining a sum of 3000 is greater than zero and is the same as the probability of obtaining a sum of S. Find the smallest possible value of S. | 375 | 0.666667 |
What is the radius of a circle inscribed in a rhombus with diagonals of length 16 and 30? | \frac{120}{17} | 0.833333 |
Consider a 4x4x4 cube made out of 64 normal dice, where each die's opposite sides still sum to 7. Find the smallest possible sum of all of the values visible on the 6 faces of the larger cube. | 144 | 0.166667 |
Given that quadrilateral EFGH is a rhombus with a perimeter of 80 meters and the length of diagonal EG is 30 meters, calculate the area of rhombus EFGH in square meters. | 150\sqrt{7} | 0.916667 |
Given the digits 2, 0, 2, and 1, and the restriction that no zero can be in the first or last position, determine the number of different four-digit numbers which can be formed by rearranging these digits. | 6 | 0.666667 |
Suppose that $\tfrac{3}{4}$ of $16$ apples are worth as much as $10$ grapes. Calculate the number of grapes worth as much as $\tfrac{1}{3}$ of $9$ apples. | \frac{5}{2} | 0.5 |
Jane buys either a $60$-cent muffin or an $80$-cent bagel every day for a seven-day week. Given that her total weekly expenditure is a whole number of dollars, determine the number of bagels she bought this week. | 4 | 0.833333 |
Given $(xy+1)^2 + (x-y)^2$, find the smallest possible value for this expression for real numbers $x$ and $y$. | 1 | 0.75 |
A laser is placed at the point $(2,3)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, finally hitting the point $(6,3)$. Calculate the total distance the beam will travel along this path. | 10 | 0.75 |
Consider the arithmetic sequence defined by the set $\{2, 5, 8, 11, 14, 17, 20\}$. Determine the total number of different integers that can be expressed as the sum of three distinct members of this set. | 13 | 0.666667 |
If $y$ cows produce $y+2$ cans of milk in $y+3$ days, determine the number of days it will take for $y+4$ cows to give $y+7$ cans of milk. | \frac{y(y+3)(y+7)}{(y+2)(y+4)} | 0.166667 |
Given that 8 balls are randomly and independently painted either red or blue with equal probability, find the probability that exactly 4 balls are red and exactly 4 balls are blue, and all red balls come before any blue balls in the order they were painted. | \frac{1}{256} | 0.666667 |
Given that a collector offers to buy state quarters for 2500% of their face value, and for every set of five quarters, the collector adds a $2 bonus, calculate the amount Bryden will get for his seven state quarters. | 45.75 | 0.916667 |
Calculate the area of $\triangle AMC$ in rectangle $ABCD$ where $AB=10$, $AD=12$, and $AM=9$ meters. | 45 | 0.416667 |
How many 3-digit positive integers have digits whose product equals 36? | 21 | 0.333333 |
Given an arithmetic sequence with a first term of $7$, a last term of $88$, and the sum of all terms equal to $570$, determine the common difference. | \frac{81}{11} | 0.916667 |
Suppose the estimated cost to establish a colony on Mars is $50$ billion dollars. This amount is to be equally shared by $300$ million tax-paying citizens of a country. Calculate each person’s share. | 166.67 | 0.916667 |
A square-shaped floor is covered with congruent square tiles. Along with the two diagonals of the square and a horizontal line through its center that includes additional tiles, the total number of tiles along the two diagonals and this central line is 55. Determine the number of tiles that cover the entire floor. | 361 | 0.833333 |
Given that Three-digit powers of 3 and 7 are used in this "cross-number" puzzle, find the only possible digit for the outlined square. | 4 | 0.166667 |
Determine the number of distinct positive integral pairs (x, y) satisfying x^6y^6 - 13x^3y^3 + 36 = 0. | 0 | 0.333333 |
Given every 7-digit whole number is a possible telephone number except those that begin with 0, 1, or 2, determine the fraction of telephone numbers that begin with a digit from 3 to 9 and end with an even digit. | \frac{1}{2} | 0.916667 |
Given that each of two boxes contains three chips numbered $1$, $4$, and $5$, calculate the probability that the product of the numbers on the two chips is a multiple of $3$. | 0 | 0.583333 |
What is the greatest possible product of the digits in the base-seven representation of a positive integer less than $2300$? | 1080 | 0.083333 |
Given Ricardo has $3000$ coins comprised of pennies ($1$-cent coins), nickels ($5$-cent coins), and dimes ($10$-cent coins), with at least one of each type of coin, calculate the difference in cents between the highest possible and lowest total value that Ricardo can have. | 26973 | 0.083333 |
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