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In an International track meet, 256 sprinters participate in a 100-meter dash competition. If the track has 8 lanes, and only the winner of each race advances to the next round while the others are eliminated, how many total races are needed to determine the champion sprinter? | 37 | 0.25 |
For how many integers $n$ is $(n+i)^6$ an integer? | 1 | 0.75 |
A triangle has an area of $24$, one side of length $8$, and the median to that side of length $7$. Let $\theta$ be the acute angle formed by that side and the median. Find $\cos{\theta}$. | \frac{\sqrt{13}}{7} | 0.583333 |
Change the digit of $0.123456$ to an $8$ to result in the smallest number. | 0.123458 | 0.75 |
Consider two events A and B where the probability that event A occurs is $\frac{5}{6}$, and the probability that event B occurs is $\frac{3}{4}$. Determine the smallest interval that necessarily contains the probability that both A and B occur. | \left[\frac{7}{12}, \frac{3}{4}\right] | 0.666667 |
The miniature tower holds 0.2 liters of water and is a scaled version of the original tower that holds 200,000 liters of water. Calculate the height of the miniature tower given that the original tower is 60 meters high. | 0.6 | 0.833333 |
Determine the area of the polygon with vertices at $(0,1)$, $(3,4)$, $(7,1)$, and $(3,7)$. | 10.5 | 0.75 |
The highest power of 3 that is a factor of the integer $N=181920\cdots9293$, formed by writing the 2-digit integers from 18 to 93 consecutively, must be calculated. | 1 | 0.25 |
A cylinder has a radius of 5 cm and a height of 10 cm. Find the height of another cylinder that has twice the volume of this original cylinder, assuming the radius is doubled. | 5 | 0.916667 |
A right rectangular prism has edge lengths \(\log_{3}x, \log_{5}x,\) and \(\log_{6}x,\) and its surface area and volume are numerically equal. Find the value of \(x\). | 8100 | 0.5 |
In a coordinate plane, points $A$ and $B$ are $12$ units apart. Determine the number of points $C$ such that for $\triangle ABC$, the perimeter is $60$ units, and the area is $240$ square units. | 0 | 0.5 |
It takes Mina 90 seconds to walk down an escalator when it is not operating, and 30 seconds to walk down when it is operating. Additionally, it takes her 40 seconds to walk up another escalator when it is not operating, and only 15 seconds to walk up when it is operating. Calculate the time it takes Mina to ride down the first operating escalator and then ride up the second operating escalator when she just stands on them. | 69 | 0.75 |
Given the expression $[(a+2b)^3(a-2b)^3]^3$, determine the number of terms in the simplified expansion of this expression. | 10 | 0.583333 |
At Archimedes Academy, there are three teachers, Mrs. Algebra, Mr. Calculus, and Ms. Statistics, teaching students preparing for the AMC 8 contest. Mrs. Algebra has 13 students, Mr. Calculus has 10 students, and Ms. Statistics has 12 students. If 3 students are enrolled in both Mrs. Algebra's and Ms. Statistics' classes, and no other overlaps exist, how many distinct students are preparing for the AMC 8 at Archimedes Academy? | 32 | 0.916667 |
Given a cashier mistakenly counts $y$ half-dollar coins as $1$ dollar bills and $y$ $5$ dollar bills as $10$ dollar bills, calculate how much the cashier should adjust the total cash amount to account for these mistakes. | 5.50y | 0.5 |
A rectangle has length $AC = 48$ and width $AE = 30$. Points $B$ and $F$ divide lines $\overline{AC}$ and $\overline{AE}$ in the ratio $1:3$ and $2:3$, respectively. Find the area of quadrilateral $ABDF$. | 468 | 0.5 |
Given the operation \( x \clubsuit y = (x^2 + y^2)(x - y) \), calculate \( 2 \clubsuit (3 \clubsuit 4) \). | 16983 | 0.916667 |
If $a @ b = \frac{a \times b}{a + b + 2}$, calculate the result of $7 @ 21$ after adding 3 to the result. | \frac{79}{10} | 0.416667 |
In $\triangle ABC$, point $F$ divides side $AC$ in the ratio $1:3$. Let $E$ be the point of intersection of side $BC$ and $AG$ where $G$ is the midpoint of $BF$. Determine the ratio in which the point $E$ divides side $BC$. | \frac{1}{4} | 0.666667 |
Consider a modified finite sequence of four-digit integers where the tens, hundreds, and units digits of each term are, respectively, the thousands, hundreds, and tens digits of the next term, and the tens, hundreds, and units digits of the last term are, respectively, the thousands, hundreds, and tens digits of the first term. Let S be the sum of all terms in the sequence. Determine the largest prime factor that always divides S. | 101 | 0.5 |
Given a box containing $30$ red balls, $22$ green balls, $18$ yellow balls, $15$ blue balls, and $10$ black balls, determine the minimum number of balls that must be drawn from the box to guarantee that at least $12$ balls of a single color will be drawn. | 55 | 0.166667 |
Given a circle of radius $7$ inscribed in a rectangle, where the ratio of the length of the rectangle to its width is $3:1$, calculate the area of the rectangle. | 588 | 0.916667 |
Given trapezoid $ABCD$, $\overline{AD}$ is perpendicular to $\overline{DC}$, $AD = AB = 5$, and $DC = 10$. In addition, $E$ is on $\overline{DC}$ such that $DE = 4$. If $\overline{BE}$ is parallel to $\overline{AD}$, find the area of $\triangle ADE$. | 10 | 0.916667 |
The mean of three numbers is 20 more than the least of the numbers and 18 less than the greatest. The median of the three numbers is 9. Determine their sum. | 21 | 0.833333 |
Given spinners $P$, $Q$, and $R$ with values 1, 2, 3, 4; 2, 4, 6; and 1, 3, 5 respectively, what is the probability that the sum of the resulting numbers from spinning each is an odd number? | \frac{1}{2} | 0.75 |
For how many integers $n$ is $(n+i)^6$ an integer? | 1 | 0.666667 |
A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{4}$. Find the value of $d$ to the nearest tenth. | 0.3 | 0.166667 |
Find the number of ordered pairs $(a, b)$ of positive integers that satisfy the equation: $a\cdot b + 100 = 25\cdot \text{lcm}(a, b) + 15\cdot\text{gcd}(a,b)$. | 0 | 0.333333 |
Given that the car ran exclusively on battery for the first $75$ miles, then ran exclusively on gasoline for the rest of the journey, consuming gasoline at a rate of $0.05$ gallons per mile, and the car averaged $50$ miles per gallon for the entire trip, determine the total length of the trip in miles. | 125 | 0.833333 |
Evaluate the expression: $[x + (y-z)] - [(x+z) - y]$. | 2y - 2z | 0.75 |
For how many integers \( n \) in the set \(\{1, 2, 3, \ldots, 200\}\) is the units digit of \( n^3 \) greater than 5? | 80 | 0.75 |
Given a frog starting at point (0,0) and making jumps of length 4, determine the smallest possible number of jumps the frog must make to reach the point (6,2). | 2 | 0.583333 |
Given \( x = 1 + 3^p \) and \( y = 1 + 3^{-p} \), express y in terms of x. | \frac{x}{x-1} | 0.916667 |
Susie buys $5$ muffins and $2$ bananas. Calvin spends three times as much as Susie on $3$ muffins and $12$ bananas. Determine how many times a muffin is as expensive as a banana. | \frac{1}{2} | 0.75 |
Given g(x) = ax^2 - \sqrt{3} for some positive constant a, find the value of a if g(g(\sqrt{3})) = -\sqrt{3}. | \frac{\sqrt{3}}{3} | 0.833333 |
Jonas sets his watch correctly at 8:00 AM and notices that his watch reads 9:48 AM at the actual time of 10:00 AM. Assuming his watch loses time at a constant rate, calculate the actual time when his watch will first read 5:00 PM. | 6:00 PM | 0.25 |
If James has taken four tests, with scores of 82, 70, and 88, and wants to average 85 for a total of six tests, what is the lowest score he could earn on one of the remaining three tests? | 70 | 0.916667 |
How many pairs $(m,n)$ of integers satisfy the equation $m+n=mn-1$? | 4 | 0.75 |
Evaluate the expression \(\left(\frac{x^2+2}{x}\right)\left(\frac{y^2+2}{y}\right)+\left(\frac{x^2-2}{y}\right)\left(\frac{y^2-2}{x}\right)\), given \(xy \neq 0\). | 2xy + \frac{8}{xy} | 0.75 |
Liam read for 4 days at an average of 42 pages per day, and for 2 days at an average of 50 pages per day, then read 30 pages on the last day. What is the total number of pages in the book? | 298 | 0.166667 |
Given $x=\frac{-1+i\sqrt{3}}{2}$ and $y=\frac{-1-i\sqrt{3}}{2}$, where $i^2=-1$, calculate the value of $x^{15} + y^{15}$. | 2 | 0.916667 |
Given the equation $x^4y^2 - 10x^2y + 9=0$, determine the number of distinct ordered pairs $(x,y)$ where $x$ and $y$ have positive integral values. | 3 | 0.833333 |
A lemming starts at a corner of a square with side length 12 meters. It runs 7.8 meters along a diagonal towards the opposite corner, then turns 60 degrees to the right relative to its original direction and runs 3 meters. Find the average of the shortest distances from the lemming to each side of the square. | 6 | 0.75 |
Given that the median of the numbers $3, 5, 7, 23,$ and $x$ is equal to the mean of those five numbers, calculate the sum of all real numbers $x$. | -13 | 0.583333 |
Given that four $\Delta$'s and two $\diamondsuit$'s balance twelve $\bullet$'s, and two $\Delta$'s balance a $\diamondsuit$ and three $\bullet$'s, calculate the number of $\bullet$'s that balance three $\diamondsuit$'s in this balance. | 4.5 | 0.333333 |
Determine the number of sets of two or more consecutive positive integers whose sum is 120. | 3 | 0.833333 |
It takes Clea 75 seconds to walk down an escalator when it is not moving, and 30 seconds when it is moving. Determine the time it would take Clea to ride the escalator down when she is not walking. | 50 | 0.916667 |
Given Hammie and his triplet siblings' weights are 120, 4, 7, and 10 pounds respectively, calculate the difference between the mean and the median of their weights. | 26.75 | 0.916667 |
If it costs two cents for each plastic digit used to number each locker and it costs $294.94 to label all lockers up to a certain number, calculate the highest locker number labeled. | 3963 | 0.333333 |
The product of two positive numbers is 16. The reciprocal of one of these numbers is 3 times the reciprocal of the other number. Find the sum of the two numbers. | \frac{16\sqrt{3}}{3} | 0.916667 |
Using a calculator with only [+1] and [x2] keys and starting with the display "1", calculate the fewest number of keystrokes needed to reach "500". | 13 | 0.25 |
Determine the percentage by which the price was increased and then decreased, given that the resulting price is 75% of the original price. | 50\% | 0.916667 |
Moe, Loki, Nick, and Ott are friends. Thor initially had no money, while the other friends did have money. Moe gave Thor one-sixth of his money, Loki gave Thor one-fifth of his money, Nick gave Thor one-fourth of his money, and Ott gave Thor one-third of his money. Each friend gave Thor $2. Determine the fractional part of the group's total money that Thor now has. | \frac{2}{9} | 0.833333 |
A bag contains four pieces of paper, each labeled with one of the prime digits 2, 3, 5, or 7. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. Calculate the probability that this three-digit number is a multiple of 3. | \frac{1}{2} | 0.916667 |
Let $x$ and $y$ be three-digit positive integers with a mean of $505$. Find the maximum value of the difference $x - y$. | 810 | 0.25 |
Given the fraction $\frac{987654321}{2^{30}\cdot 5^6}$, calculate the minimum number of digits to the right of the decimal point needed to express it as a decimal. | 30 | 0.5 |
Set $A$ has $30$ elements, and set $B$ has $25$ elements. Set $C$ has $10$ elements and is a subset of both $A$ and $B$. Determine the smallest possible number of elements in $A \cup B$. | 30 | 0.25 |
The centers of two circles are $50$ inches apart. One circle has a radius of $7$ inches and the other has a radius of $10$ inches. Determine the length of the common internal tangent. | \sqrt{2211}\text{ inches} | 0.333333 |
A sphere with center \(O\) has a radius of \(8\). An isosceles triangle with sides \(17, 17,\) and \(16\) is situated in space such that each of its sides is tangent to the sphere. Determine the distance between \(O\) and the plane determined by the triangle. | 6.4 | 0.083333 |
Given that Chelsea is ahead by 60 points halfway through a 120-shot archery contest, with each shot scoring 10, 8, 5, 3, or 0 points and Chelsea scoring at least 5 points on every shot, determine the smallest number of bullseyes (10 points) Chelsea needs to shoot in her next n attempts to ensure victory, assuming her opponent can score a maximum of 10 points on each remaining shot. | 49 | 0.166667 |
In a right triangle, the sides adjacent to the right angle are $a$ and $b$, with the hypotenuse being $c$. A perpendicular from the right angle vertex divides the hypotenuse $c$ into two segments $r$ and $s$, where $r$ is adjacent to $a$ and $s$ is adjacent to $b$. Given that the ratio of $a$ to $b$ is $2 : 5$, find the ratio of $r$ to $s$. | \frac{4}{25} | 0.75 |
Evaluate \((x^x)^{(x^x)}\) at \(x = 3\). | 27^{27} | 0.833333 |
A woman buys a house for $20,000 and rents it out. She sets aside $15%$ of each month's rent for maintenance, pays $400 a year in taxes, and wants to realize a $6%$ return on her investment. Calculate the monthly rent she needs to charge. | 156.86 | 0.75 |
A triangle has vertices at $(0,0)$, $(3,3)$, and $(8m,0)$, where $m$ is a nonzero constant. The line $y = mx$ divides the triangle into two regions of equal area. Determine the sum of all possible values of $m$. | -\frac{3}{8} | 0.083333 |
Given that the ceiling is 3.0 meters above the ground, Bob is 1.8 meters tall and can reach 50 centimeters above the top of his head, and the light fixture is 15 centimeters below the ceiling, calculate the height of the box in centimeters. | 55 | 0.5 |
Given two integers, their sum is 30. After adding two more integers to the first two, the sum is 47. Finally, after adding two more integers to the sum of the previous four, the sum is 65. Determine the minimum number of odd integers among the six integers. | 1 | 0.583333 |
Find the difference of the roots of the quadratic equation $3x^2 + 4x - 15 = 0$. | \frac{14}{3} | 0.916667 |
Determine the number of positive integer divisors of $255^8$ that are either perfect squares or perfect cubes (or both). | 144 | 0.666667 |
If $M$ is $40 \%$ of $Q$, $Q$ is $30 \%$ of $P$, and $N$ is $60 \%$ of $2P$, calculate the value of $\frac {M}{N}$. | \frac{1}{10} | 0.75 |
Given Orvin went to the store with just enough money to buy 40 balloons at the regular price, and he noticed a promotion: buy 1 balloon at the regular price and get the next one at 1/2 off. What is the greatest number of balloons he could purchase? | 53 | 0.833333 |
Given a skewed six-sided die is structured so that rolling an odd number is twice as likely as rolling an even number, calculate the probability that, after rolling the die twice, the sum of the numbers rolled is odd. | \frac{4}{9} | 0.166667 |
Given the equation $x^{4} + x + y^2 = 2y + 3$, determine the number of ordered pairs of integers $(x, y)$ that satisfy the equation. | 4 | 0.916667 |
For values of \( x \) between $0$ and $5$, evaluate the maximum and minimum values of the expression
$$\frac{x^2 - 4x + 5}{2x - 4}.$$
A) Minimum value of -1 and maximum value of 1
B) Minimum value of 1 and maximum value of -1
C) Maximum value only of 0
D) No extremum values within the domain | A) Minimum value of -1 and maximum value of 1 | 0.083333 |
Given that eight teams play each other three times, with a team earning $3$ points for a win and $2$ points for a draw, determine the maximum possible number of total points that each of the four teams could have earned. | 54 | 0.083333 |
Given the equation $x^{2024} + y^2 = 2y + 1$, calculate the number of ordered pairs of integers $(x, y)$. | 4 | 0.833333 |
For how many integers \( x \) does a triangle with side lengths \( 13, 15 \) and \( x \) have all its angles acute? | 12 | 0.666667 |
Given Chloe chooses a real number uniformly at random from the interval $[0, 100]$ and Laurent chooses a real number uniformly at random from the interval $[0, 200]$, find the probability that Laurent's number is greater than Chloe's number. | \frac{3}{4} | 0.333333 |
Calculate the area of a quadrilateral with vertices at \((2,1)\), \((4,3)\), \((7,1)\), and \((4,6)\). | 7.5 | 0.916667 |
A mixture of $50$ liters of paint is composed of $20\%$ red tint, $40\%$ yellow tint, and $40\%$ water. Six liters of red tint are added to the original mixture. What is the percent of red tint in the new mixture? | 28.57\% | 0.833333 |
Given that 40% of the birds are sparrows, 20% are pigeons, 15% are parrots, and 25% are crows in Oakwood Park, calculate the percentage of the birds that are not pigeons which are sparrows. | 50\% | 0.666667 |
Shelby drives her car at a speed of $40$ miles per hour if it is not raining, and $25$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $20$ miles in $36$ minutes. Determine the number of minutes she drove in the rain. | 16 | 0.75 |
The probability that when all 8 dice are rolled, the sum of the numbers on the top faces is 12. What other sum occurs with the same probability as this? | 44 | 0.666667 |
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 along the two radii shown. Calculate the volume of the cone. | 9\pi \sqrt{3} | 0.833333 |
Given that Maria's pedometer flips 50 times from $99999$ to $00000$ and reads $25000$ on December 31, and she covers $1500$ steps per mile, calculate the total distance Maria walked during the year. | 3350 | 0.333333 |
A company has 28 employees, among which 16 have brand A computers and 12 have brand B computers. The technician can install cables between brand A and brand B computers, ensuring no redundant connections between the same pairs. Determine the maximum possible number of cables the technician could use. | 192 | 0.75 |
Given that $x$ is a positive real number, find an equivalent expression for $\sqrt[4]{x\sqrt[3]{x}}$. | x^{1/3} | 0.916667 |
Segments $AD=15$, $BE=9$, $CF=30$ are drawn from vertices $A$, $B$, and $C$ of triangle $ABC$, each perpendicular to a straight line $RS$, which does not intersect the triangle. Points $D$, $E$, and $F$ are the intersection points of $RS$ with the perpendiculars. If $x$ is the length of the perpendicular segment $GH$ drawn to $RS$ from the intersection point $G$ of the medians of the triangle, find the length of $x$. | 18 | 0.833333 |
The price of a ticket to a concert is \( x \) dollars, where \( x \) is an even whole number. A group of students from one class buys tickets costing a total of \( \$72 \), and another group from a different class buys tickets costing a total of \( \$108 \). Determine the number of possible values for \( x \). | 6 | 0.833333 |
During a mathematics competition preparation period, John aims to score a perfect "100" on at least $85\%$ of his $40$ practice tests. He has scored a "100" on $30$ of the first $36$ tests. Calculate the maximum number of the remaining tests on which he can score less than "100". | 0 | 0.916667 |
Suppose $3 + \frac{1}{2 + \frac{1}{3 + \frac{3}{4+x}}} = \frac{225}{68}$. Determine the value of $x$. | -\frac{102}{19} | 0.333333 |
Given that $M = 58^3 + 3 \cdot 58^2 + 3 \cdot 58 + 1$, determine the number of positive integers that are factors of $M$. | 4 | 0.916667 |
A list of $3042$ positive integers has a unique mode, which occurs exactly $15$ times. Calculate the least number of distinct values that can occur in the list. | 218 | 0.333333 |
Calculate the probability that a randomly chosen divisor of $24!$ is odd. | \frac{1}{23} | 0.5 |
Given that spinner S has numbers 1, 4, 3, spinner T has numbers 2, 4, 6, and spinner U has numbers 2, 3, 5, determine the probability that the sum of the numbers obtained from spinning each of the three spinners is an even number. | \frac{5}{9} | 0.25 |
Professors Alpha, Beta, Gamma, and Delta choose their chairs so that each professor will be between two students. Given that there are 13 chairs in total, determine the number of ways these four professors can occupy their chairs. | 1680 | 0.083333 |
Alice has $30$ apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least four apples? | 190 | 0.916667 |
Given that a recipe that makes $4$ servings requires $3$ squares of chocolate, $1/2$ cup of sugar, $2$ cups of water, and $3$ cups of milk, if Emily maintains the same ratio of ingredients, calculate the greatest number of servings of the dessert she can make with $9$ squares of chocolate, $3$ cups of sugar, plenty of water, and $10$ cups of milk. | 12 | 0.666667 |
The relationship between the arithmetic mean and the geometric mean of the segments $a$, $b$, and $c$ can be accurately represented. | \frac{a+b+c}{3} \geq \sqrt[3]{abc} | 0.916667 |
When $\left(1 - \frac{1}{a}\right)^8$ is expanded, calculate the sum of the last three coefficients. | 21 | 0.916667 |
Given that \( x \) and \( y \) are nonzero real numbers and \(\frac{4x+y}{x-4y} = 3\), calculate the value of \(\frac{x+4y}{4x-y}\). | \frac{9}{53} | 0.916667 |
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