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Jane, Bob, and Alice are playing a game in which they take turns throwing a ball at a bottle, with Jane throwing first, then Bob, then Alice, and then back to Jane, in a cyclic order. The probability that Jane or Alice knocks the bottle off on their turn is $\frac{1}{3}$, while the probability that Bob knocks it off is $\frac{2}{3}$. Calculate the probability that Jane wins the game. | \frac{9}{23} | 0.416667 |
Given Emily converts her $e$ U.S. dollars into Euros at an exchange bureau that offers $5$ Euros for every $4$ U.S. dollars, and after spending $75$ Euros she has exactly half of her original amount of U.S. dollars converted into Euros, calculate the value of $e$. | 120 | 0.75 |
What is the value of $\dfrac{13! - 12!}{11!}$? | 144 | 0.916667 |
A set $T$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, the line $y=x$, and the line $y=-x$. If $(3,4)$ is in $T$, determine the smallest number of points in $T$. | 8 | 0.75 |
For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. Find the number of positive integers $n$ for which $P(n) = \sqrt{n}$ and $P(n+60) = \sqrt{n+60}$. | 0 | 0.75 |
If ten friends split a bill equally, but nine friends paid an additional $3.00 each to cover the share of one friend, calculate the total bill. | 270 | 0.25 |
For how many values of $d$ is $2.00d5 > 2.007$? | 3 | 0.583333 |
Bella begins to walk from her house toward her friend Ella's house which is 3 miles away. At the same time, Ella begins to ride her bicycle toward Bella's house. Ella rides at a speed that is 4 times as fast as Bella walks. If Bella covers 3 feet with each step, how many steps will Bella take by the time she meets Ella? | 1056 | 0.833333 |
What is the tens digit of $9^{2023}$? | 2 | 0.75 |
Given the repeating decimal $3.71717171\ldots$, express this number as a fraction in lowest terms and find the sum of the numerator and denominator. | 467 | 0.833333 |
A cube has faces whose surface areas are $36$, $36$, $36$, $36$, $36$, and $36$ square units. Each side length is reduced uniformly by $1$ unit. Find the sum of the new dimensions of the cube. | 15 | 0.333333 |
How many primes less than $50$ have $3$ as the ones digit? | 4 | 0.666667 |
Given $M = 35^5 + 5\cdot35^4 + 10\cdot35^3 + 10\cdot35^2 + 5\cdot35 + 1$, calculate how many positive integers are factors of $M$. | 121 | 0.916667 |
Given the expression $3000(3000^{2500}) \cdot 2$, evaluate the expression. | 2 \cdot 3000^{2501} | 0.583333 |
Suppose that 1 euro is now worth 1.5 dollars. Diana has 600 dollars and Etienne has 350 euros. Additionally, there is a transaction fee of 2% when converting euros to dollars. Calculate the percent by which the value of Etienne's money is greater than or less than the value of Diana's money after accounting for the conversion fee. | 14.25\% | 0.583333 |
In a new arrangement of letters and numerals, one wants to spell "AMC10". Starting from an 'A' in the center, move only to adjacent letters (above, below, left, or right, but not diagonal) to spell out the sequence. Given that there are 4 'M’s around the central 'A', 4 'C’s next to each 'M', and now 5 '10’s reachable from each 'C', find the total number of different paths one can spell "AMC10". | 80 | 0.833333 |
A rectangular floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 57, how many tiles cover the floor. | 841 | 0.333333 |
Let \( x = -2023 \). Evaluate the expression \( \Bigg\vert 2 \times \Big\vert |x|-x\Big\vert - |x| \Bigg\vert - x \). | 8092 | 0.833333 |
Given the product of two positive integers $a$ and $b$ is $143$, where Alice mistakenly reversed the digits of the two-digit number $a$ to obtain this value, calculate the correct value of the product of $a$ and $b$. | 341 | 0.083333 |
Given the equations $x^2 + y^2 = m^2$ and $x + y = m$, determine the value of $m$ such that the graph of $x^2 + y^2 = m^2$ is tangent to the graph of $x + y = m$. | 0 | 0.416667 |
Given that $20\%$ of the students scored $60$ points, $25\%$ scored $75$ points, $25\%$ scored $85$ points, and the remaining students scored $95$ points, calculate the difference between the mean and median score of the students' scores on this test. | 4.5 | 0.166667 |
Consider a parabola with its vertex $V$ and focus $F$. Let there be a point $B$ on this parabola such that the distance $BF$ is 25 and $BV$ is 26. Determine the sum of all possible values for the length $FV$. | \frac{50}{3} | 0.5 |
The largest number by which the expression $n^4 - n^2$ is divisible for all possible integral values of $n$. | 12 | 0.416667 |
Given that the city's water tower stands 60 meters high and holds 150,000 liters of water, and Logan's miniature water tower holds 0.15 liters, determine the height, in meters, that Logan should make his tower. | 0.6 | 0.083333 |
Given that the floor of the hall is tiled in a repeated pattern of an 8x8 layout, with the configuration in a 4x4 section including 10 dark tiles, calculate the fraction of the hall's floor that is composed of darker tiles. | \frac{5}{8} | 0.833333 |
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $15!$? | 10 | 0.75 |
A collector offers to buy state quarters for 1500% of their face value. If Bryden has 10 state quarters, calculate the total amount he will receive. | 37.5\text{ dollars} | 0.916667 |
Chubby designs another nonstandard checkerboard, but this time it has 33 squares on each side. The board still follows the pattern of having a black square in every corner and alternates red and black squares along every row and column. Determine how many black squares are there on this checkerboard. | 545 | 0.416667 |
Given that $528$ be written as the sum of an increasing sequence of two or more consecutive positive integers, starting with an even number, determine the total number of such ways. | 0 | 0.583333 |
If one side of a triangle is $16$ inches and the opposite angle is $30^{\circ}$, calculate the diameter of the circumscribed circle. | 32\text{ inches} | 0.916667 |
Given that the Highest Common Divisor (HCD) of $12348$ and $2448$ is multiplied by $3$ and then diminished by $14$, determine the result. | 94 | 0.666667 |
Alice, Bob, Carol, and Dave repeatedly take turns tossing a die, with Alice beginning, followed by Bob, then Carol, and finally Dave before returning to Alice. Find the probability that Dave will be the first one to toss a six. | \frac{125}{671} | 0.583333 |
A shopkeeper purchases 2000 pens at a cost of $0.15 each. If the shopkeeper wants to sell them for $0.30 each, calculate the number of pens that need to be sold to make a profit of exactly $120.00. | 1400 | 0.833333 |
Alice and Bob play a game around a circle divided into 15 equally spaced points, numbered 1 through 15. Alice moves 7 points clockwise per turn, and Bob moves 4 points counterclockwise per turn. Determine how many turns will be required for Alice and Bob to land on the same point for the first time. | 15 | 0.916667 |
Jo and Blair take turns counting. Jo starts by saying 1, and each time Blair follows by saying the last number Jo said plus 2, while Jo says the last number Blair said plus 3. Determine the 20th number said. | 48 | 0.166667 |
Given the dimensions of a rectangular lawn $120$ feet by $180$ feet, and a mower swath of $30$ inches with a $6$-inch overlap, calculate the total time it will take Moe to mow the entire lawn while walking at a rate of $4000$ feet per hour. | 2.7 | 0.75 |
How many primes less than $50$ have $3$ as the ones digit? | 4 | 0.583333 |
Calculate the value of $(3(3(3(3(3(3+2)+2)+2)+2)+2)+2)$. | 1457 | 0.5 |
In our number system, the base is ten. If the base were changed to seven, count the twenty-fifth number in the new base. | 34_7 | 0.666667 |
Determine the number of distinct points common to the curves $x^2 + y^2 = 4$ and $x^2 + 2y^2 = 2$. | 0 | 0.916667 |
Let \( x = -2023 \). Determine the value of \(\Bigg\vert\Big\vert |x|-x\Big\vert-|x|+5\Bigg\vert-x+3\). | 4054 | 0.833333 |
Simplify the expression $(-\frac{1}{343})^{-2/3}$. | 49 | 0.916667 |
A cell phone plan charges $20$ dollars each month, plus $0.03$ dollars per text message sent, plus $0.10$ dollars for each minute used over $30$ hours, plus $0.15$ dollars for each megabyte of data used over $500$ MB. In January, Michelle sent $200$ text messages, talked for $31$ hours, and used $550$ MB of data. Calculate the total amount she has to pay. | 39.50 | 0.416667 |
Given that the stationary shop owner purchased 2000 pencils at $0.15 each and sells them for $0.30 each, determine how many pencils she must sell to achieve a profit of exactly $180.00. | 1600 | 0.583333 |
Sharon usually drives from her house to her office and it takes her 180 minutes at a constant speed. On a particular day, after she has covered half the distance, she runs into a heavy snowstorm and has to reduce her speed by 30 miles per hour for the remainder of the trip. Due to the weather conditions, her journey takes a total of 300 minutes today. Find the total distance of Sharon's journey from her house to her office. | 157.5 | 0.75 |
Given a cylindrical cheese log with a height of 8 cm and a radius of 5 cm, determine the volume of a wedge representing one-third of the cylinder's total volume. | \frac{200\pi}{3} | 0.833333 |
A large equilateral triangle with a side length of 20 cm is to be completely covered by non-overlapping equilateral triangles of side length 2 cm. Determine the number of smaller triangles needed if each row of small triangles is rotated by 180 degrees relative to the row immediately below it. | 100 | 0.916667 |
The positive integers $x$ and $y$ are the two smallest positive integers such that the product of $540$ and $x$ is a square and the product of $540$ and $y$ is a cube, determine the sum of $x$ and $y$. | 65 | 0.833333 |
A fair coin is tossed 4 times. What is the probability of at least three consecutive heads? | \frac{3}{16} | 0.5 |
Let $Q$ equal the product of 1,000,000,001 and 10,000,000,007. Calculate the number of digits in $Q$. | 20 | 0.75 |
Let $A$ be the set of the $1500$ smallest positive multiples of $7$, and let $B$ be the set of the $1500$ smallest positive multiples of $9$. What is the number of elements common to $A$ and $B$? | 166 | 0.833333 |
Given two right isosceles triangles are placed inside a square of side length $2$ such that their hypotenuses are the opposite sides of the square. Their intersection forms a rhombus. Calculate the area of this rhombus. | 2 | 0.833333 |
Given that Carlos took 60% of a whole pie and Maria took half of the remainder, find the portion of the whole pie that was left. | 20\% | 0.083333 |
Points P and Q are on a circle of radius 7 and the chord PQ=8. Point R is the midpoint of the minor arc PQ. Calculate the length of the line segment PR. | \sqrt{98 - 14\sqrt{33}} | 0.083333 |
Given that Paula initially had enough paint for 40 identically sized rooms and had reduced capacity to just enough paint for 30 rooms after losing 5 cans of paint, calculate the original number of cans of paint she had. | 20 | 0.916667 |
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 $y+7$ cans of milk. | \frac{y(y+3)(y+7)}{(y+2)(y+4)} | 0.25 |
Determine the value of $2 - (-3) - 4 + (-5) + 6 - (-7) - 8$. | 1 | 0.916667 |
Calculate the number of circular pipes with an inside diameter of 2 inches required to carry twice the amount of water as one pipe with an inside diameter of 8 inches. | 32 | 0.916667 |
Given the function \(\frac{(4^t - 2t)t}{16^t}\), find the maximum value for real values of \(t\). | \frac{1}{8} | 0.416667 |
Given two poles 30'' and 60'' high are 120'' apart, calculate the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole. | 20 | 0.583333 |
Given real numbers $m$ and $n$, define $m \star n = (3m - 2n)^2$. Evaluate $(3x-2y)^2 \star (2y-3x)^2$. | (3x-2y)^4 | 0.416667 |
Given $g(n) = \frac{1}{4} n(n+1)(n+2)(n+3)$, find $g(r) - g(r-1)$. | r(r+1)(r+2) | 0.833333 |
Calculate the sum of the sequences $(2+14+26+38+50) + (12+24+36+48+60) + (5+15+25+35+45)$. | 435 | 0.916667 |
Let $a_1,a_2,\dots,a_{2021}$ be a strictly increasing sequence of positive integers such that \[a_1+a_2+\cdots+a_{2021}=2021^{2021}.\] Find the remainder when $a_1^3+a_2^3+\cdots+a_{2021}^3$ is divided by $6$. | 5 | 0.25 |
Given the expression $3 + \sqrt{3} + \frac{1}{3 + \sqrt{3}} + \frac{1}{\sqrt{3} - 3}$, evaluate the expression. | 3 + \frac{2\sqrt{3}}{3} | 0.916667 |
Find the unique positive integer $n$ such that $\log_3{(\log_{27}{n})} = \log_9{(\log_3{n}). | 19683 | 0.583333 |
Given a round-robin tournament with 8 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. | 5 | 0.083333 |
The perimeter of an equilateral triangle exceeds the perimeter of a square by $108 \ \text{cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d+2 \ \text{cm}$. The square has a perimeter greater than 0. Calculate the number of positive integers that are NOT possible values for $d$. | 34 | 0.916667 |
The sum of the squares of the roots of the equation \(x^2 - 4hx = 8\) is 20. Find the value of \(h\). | \pm \frac{1}{2} | 0.666667 |
Given that the sum of any two numbers equals the third number, find the number of ordered triples $(a, b, c)$ of non-zero real numbers. | 0 | 0.916667 |
Thirty percent of the objects in the urn are beads and rings, and beads make up half of that volume. Additionally, thirty-five percent of the coins are silver. Determine the percentage of objects in the urn that are gold coins. | 45.5\% | 0.583333 |
Given that $3/8$ of the people in a room are wearing gloves and $5/6$ of the people are wearing hats, find the minimum number of people in the room wearing both a hat and a glove. | 5 | 0.916667 |
Given that Al, Bert, and Carl are to divide a pile of stickers in a ratio of $4:3:2$, calculate the fraction of stickers that go unclaimed when each of them takes his share, where Al takes the correct total amount, Bert mistakenly takes his share based on a total that is 10% less, and Carl takes his share based on what remains. | \frac{1}{30} | 0.166667 |
What is the tens digit of $9^{2023}$? | 2 | 0.75 |
Two angles of an isosceles triangle measure $80^\circ$ and $y^\circ$. Find the sum of the three possible values of $y$. | 150 | 0.083333 |
Lily uses a mower to cut her rectangular 120-foot by 180-foot lawn with a 30-inch wide swath and 6-inch overlap. Calculate the time it will approximately take her to mow the lawn, given that she walks at a rate of 6000 feet per hour. | 1.8 | 0.916667 |
A circle with a radius of 7 is perfectly inscribed in a rectangle. The ratio of the length of the rectangle to its width is 3:1. Calculate the area of the rectangle. | 588 | 0.916667 |
When $\left ( 1 - \frac{1}{a} \right ) ^8$ is expanded, calculate the sum of the last three coefficients. | 21 | 0.833333 |
Determine the length of the median of the trapezoid formed by combining two equilateral triangles with side lengths of 4 units and 3 units, where the triangles form the two bases. | 3.5 | 0.083333 |
Given a rectangular array of numbers with 50 rows and 100 columns, find the value of A/B, where A is the average of the 100 sums calculated by Andy (each sum being the sum of a column) and B is the average of the 50 sums calculated by Bethany (each sum being the sum of a row). | \frac{1}{2} | 0.75 |
What is the greatest possible sum of the digits in the base-six representation of a positive integer less than $1728$? | 20 | 0.416667 |
The number $22!$ has many positive integer divisors. If one is chosen at random, calculate the probability that it is odd. | \frac{1}{20} | 0.416667 |
A rectangular grazing area is to be fenced off on three sides using part of a $120$ meter rock wall as the fourth side. Fence posts are to be placed every $15$ meters along the fence, including the two posts where the fence meets the rock wall. Determine the fewest number of posts required to fence an area $45$ m by $75$ m. | 12 | 0.416667 |
Given points G, H, I, and J lie on a line, with GH = 2, HI = 3, IJ = 4, and points K, L, and M lie on another line, parallel to the first, with KL = 2 and LM = 3, determine the number of possible different values for the area of a triangle with positive area formed by three of these points. | 6 | 0.083333 |
Seven students, Abby, Bridget, Charlie, and four others are to be seated in two rows of four. If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column, and Charlie is seated in a different row from Abby?
A) $\frac{1}{4}$
B) $\frac{5}{21}$
C) $\frac{1}{3}$
D) $\frac{2}{7}$
E) $\frac{1}{5}$ | \frac{5}{21} | 0.333333 |
A particle moves such that its speed for each subsequent mile varies inversely as the square of the integral number of miles already traveled. If the speed for the first mile was $1$ mile per hour and the second mile was traversed in $2$ hours, determine the time in hours needed to traverse the $n$-th mile. | 2(n-1)^2 | 0.166667 |
Determine the roots of the polynomial \( p(x) = x(x+3)^2(5-x) = 0 \). | 0, -3, 5 | 0.916667 |
Given $g(3x) = \frac{3}{3 + 2x}$ for all $x > 0$, calculate the value of $3g(x)$. | \frac{27}{9 + 2x} | 0.916667 |
Given all four-digit palindromes that can be represented as $\overline{abba}$, where $a, b$ are digits and $a \neq 0$, find the sum of the digits of the total of these palindromes. | 18 | 0.75 |
What is the area of the polygon with vertices at $(2, 1)$, $(4, 3)$, $(6, 1)$, $(5, -2)$, and $(3, -2)$? | 13 | 0.666667 |
Twenty-five percent of the audience listened to the entire 90-minute talk, and fifteen percent did not pay attention at all. Of the remainder, 40% caught half of the talk, and the rest heard only one fourth of it. Calculate the average time in minutes the talk was heard by the audience members. | 41.4 | 0.416667 |
What is the probability that a randomly selected positive factor of $90$ is less than $8$? | \frac{5}{12} | 0.916667 |
Given that half of Julia's marbles are blue, one-third of them are red, and twelve of them are green, find the smallest number of yellow marbles that Julia could have. | 0 | 0.75 |
Given $a$ and $b$ are positive numbers such that $a^b = b^a$ and $b = 4a$, solve for the value of $a$. | \sqrt[3]{4} | 0.833333 |
Let $x=2023$. Find the value of $\bigg|$ $||x|-x|-|x|$ $\bigg|$ $-x$. | 0 | 0.833333 |
A cylindrical tank with a radius of $5$ feet and a height of $10$ feet is lying on its side. The tank is half-filled with water, reaching the top of the cylinder's horizontal diameter. Calculate the volume of water in cubic feet. | 125\pi | 0.916667 |
The two spinners are spun once. Spinner A has sectors numbered 0, 2, 3 while spinner B has sectors numbered 1, 3, 5. Determine the probability that the sum of the numbers on which the spinners land is a prime number. | \frac{5}{9} | 0.583333 |
A particle moves in such a way that its speed for each mile after the first is inversely proportional to the square of the number of miles already traveled. If it takes 4 hours to travel the third mile, find the time, in hours, needed to traverse the $n$th mile. | (n-1)^2 | 0.166667 |
Evaluate the expression $(-2)^{3^2} + 2^{-1^3}$. | -511.5 | 0.083333 |
When three different numbers from the set $\{-4, -3, -1, 5, 6\}$ are multiplied, calculate the largest possible product. | 72 | 0.833333 |
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