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A box contains 30 red balls, 22 green balls, 18 yellow balls, 15 blue balls, 10 white balls, and 6 black balls. Calculate 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. | 88 | 0.166667 |
Given that the track has a width of 8 meters, the longer it takes Keiko to walk around the outer edge of the track compared to the inner edge is 48 seconds, and each straight section of the track has a length of 100 meters, calculate Keiko's speed in meters per second. | \frac{\pi}{3} | 0.583333 |
Given the numbers $5, 7, 10, 20,$ and $x$, find the sum of all real numbers $x$ for which the median of these numbers is equal to the mean of the numbers. | -7 | 0.833333 |
Given a number is called flippy if its digits alternate between two distinct digits from the set {4, 6}, calculate the number of four-digit flippy numbers that are divisible by 4. | 1 | 0.083333 |
A straight line passing through the point $(2,5)$ is perpendicular to the line $x-2y-8=0$. Find its equation. | y + 2x - 9 = 0 | 0.166667 |
Given the equation $(\log_b a)^3=\log_b(a^3)$, where $a$ is a positive real number and $b$ is an integer between $3$ and $300$, inclusive, determine the number of pairs $(a,b)$. | 894 | 0.25 |
Given that the new fort is designed to be $20$ feet long, $15$ feet wide, and $8$ feet high, with walls that are two feet thick and the floor one foot thick, calculate the total number of one-foot cubical blocks needed for the fort. | 1168 | 0.75 |
What is the value of $\dfrac{13!-12!}{10!}$? | 1584 | 0.916667 |
A cube with a volume of 1 cubic foot is divided into three slabs by making two cuts parallel to the top face of the cube. The first cut is made $\frac{1}{4}$ foot from the top, and the second cut is $\frac{1}{6}$ foot below the first cut. Determine the total surface area of the resulting solid assembly. | 10 | 0.083333 |
Three runners start simultaneously from the same point on a 600-meter circular track, running clockwise at constant speeds of 4.4 m/s, 4.9 m/s, and 5.1 m/s. Determine the time it takes for them to meet again somewhere on the track. | 6000 | 0.833333 |
If 1 pint of paint is needed to paint a statue 6 ft. high, calculate the number of pints it will take to paint 1080 statues similar to the original but only 2 ft. high. | 120 | 0.916667 |
The prime factorization of the denominator of the fraction $\frac{987654321}{2^{30} \cdot 5^3}$ is $2^{30} \cdot 5^3$. Therefore, to determine the minimum number of digits to the right of the decimal point, find the minimum number of factors of 2 and 5 in the numerator, 987654321. | 30 | 0.25 |
Given that $f(x+5)=4x^3 + 5x^2 + 9x + 6$ and $f(x)=ax^3 + bx^2 + cx + d$, find the value of $a+b+c+d$. | -206 | 0.333333 |
Determine how many perfect cubes exist between \(3^6 + 1\) and \(3^{12} + 1\), inclusive. | 72 | 0.583333 |
At Hilltop High, the ratio of sophomore students to freshman students involved in a yearly science fair is 7:4 and the ratio of junior students to sophomore students is 6:7. Determine the smallest number of students that could be participating in this fair from these three grades. | 17 | 0.75 |
Two circles with centers at points A and B are externally tangent and have radii 7 and 4, respectively. A line tangent to both circles intersects ray AB at point C. Calculate the length of segment BC. | \frac{44}{3} | 0.166667 |
Determine the number of minutes before Jack arrives at the park that Jill arrives at the park, given that they are 2 miles apart, Jill cycles at a constant speed of 12 miles per hour, and Jack jogs at a constant speed of 5 miles per hour. | 14 | 0.916667 |
Given the function $f(n) =\begin{cases}\log_{10}{n}, &\text{if }\log_{10}{n}\text{ is rational,}\\ 1, &\text{if }\log_{10}{n}\text{ is irrational.}\end{cases}$, calculate the value of $\sum_{n = 1}^{256}{f(n)}$. | 256 | 0.916667 |
Two 8-sided dice are rolled once. The sum of the numbers rolled determines the diameter of a circle, but only if this sum is at least 5. Find the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference. | 0 | 0.5 |
Parallelogram ABCD has an area of 80 square meters. Points E and G are the midpoints of sides AB and CD respectively. Point F is the intersection of line segment EG and diagonal BD. Find the area of triangle BFG in square meters. | 10 | 0.583333 |
Given the set $\{-10, -7, -5, 0, 4, 6, 9\}$, find the minimum possible product of three different numbers from this set. | -540 | 0.333333 |
An automobile travels $2a/5$ feet in $2r$ seconds. If this rate is maintained for $5$ minutes, how many yards does it travel in $5$ minutes? | \frac{20a}{r} | 0.916667 |
Evaluate the expression $\sqrt{7+4\sqrt{3}} - \sqrt{7-4\sqrt{3}}$. | 2\sqrt{3} | 0.916667 |
Eight points on a circle are given. Four of the chords joining pairs of the eight points are selected at random. Find the probability that the four chords form a convex quadrilateral. | \frac{2}{585} | 0.333333 |
A store owner purchased $1800$ pencils at $0.15$ each. If he sells them for $0.30$ each, how many of them must he sell to make a profit of exactly $150.00$? | 1400 | 0.916667 |
Given the lighthouse is 60 meters tall and the conical light container has a volume of 150,000 liters, determine the height of the model lighthouse if the volume of the light container for the model is 0.15 liters. | 0.6 | 0.583333 |
Given a number ten times as large as y is decreased by three, then find one third of the result equals what expression? | \frac{10y}{3} - 1 | 0.416667 |
The smallest possible even five-digit number is formed using the digits 1, 2, 3, 5, and 8. Determine the digit that must be in the tens place. | 5 | 0.166667 |
Determine the greatest number of consecutive integers whose sum is $36$. | 72 | 0.166667 |
Jo and Blair take turns counting numbers, starting from $3$. Jo begins the sequence, and each subsequent number said by Blair or Jo is two more than the last number said by the other person. Find the $20^\text{th}$ number said. | 41 | 0.666667 |
At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis, and by $6{:}00$ PM, the temperature in Minneapolis has fallen by $7$ degrees while the temperature in St. Louis has risen by $4$ degrees. The temperatures in the two cities differ by $1$ degree at that time. What is the product of all possible values of $N?$ | 120 | 0.916667 |
Given Professor Lee has a collection of ten different books lined up on a shelf: three Arabic, three German, and four Spanish, calculate the number of arrangements of the ten books on the shelf so that the Arabic books are always together and the German books are always together. | 25920 | 0.25 |
Determine the number of perfect cubic divisors in the product $1! \cdot 2! \cdot 3! \cdot \ldots \cdot 6!$. | 10 | 0.416667 |
Given a $25$-quart radiator is filled with water, and the described process of removing and replacing quarts of the mixture with pure antifreeze is repeated five times, calculate the fractional part of the final mixture that is water. | \frac{1024}{3125} | 0.333333 |
For how many positive integer values of $n$ are both $\frac{n}{4}$ and $4n$ three-digit whole numbers? | 0 | 0.916667 |
Given $\theta$ is an acute angle, and $\cos 2\theta = a$, then find $\sin\theta \cos\theta$. | \frac{\sqrt{1-a^2}}{2} | 0.833333 |
Determine the number of distinct points common to the curves $x^2 + 9y^2 = 9$ and $9x^2 + y^2 = 9$. | 4 | 0.916667 |
A square hallway is covered entirely with tiles in a repeating pattern. Each corner of the hallway displays a similar arrangement of tiles. What fraction of the tiled hallway is made of darker tiles if the pattern repeats every $8 \times 8$ tiles, and in each top-left $4 \times 4$ square of these tiles there are 9 dark tiles? | \frac{9}{16} | 0.833333 |
If $a \star b = \frac{a \times b}{a+b-3}$ for $a,b$ positive integers where $a+b>3$, calculate the value of $6 \star 9$. | \frac{9}{2} | 0.25 |
Given a group of 60 girls, each of whom is either blonde or brunette and has either blue eyes or brown eyes, and if there are 20 blue-eyed blondes, 35 brunettes, and 22 brown-eyed girls, calculate the number of brown-eyed brunettes. | 17 | 0.833333 |
Given that (x,y) are real numbers, what is the least possible value of (xy+1)^2 + (x+y+1)^2 ? | 0 | 0.25 |
In four consecutive soccer matches of a season, a player scored $18$, $12$, $15$, and $14$ goals, respectively. His goals-per-game average was higher after these four games compared to the previous three games. If his total number of goals after seven matches is at least $100$, determine the minimum number of goals he must have scored in the three previous matches. | 41 | 0.416667 |
Susie buys 5 muffins and 4 bananas. Calvin spends three times as much as Susie buying 3 muffins and 20 bananas. Determine the ratio of the price of one muffin to the price of one banana. | \frac{2}{3} | 0.833333 |
In $\triangle ABC$, $AB = 6$, $BC = 8$, $AC = 10$, and $D$ is on $\overline{AC}$ with $BD = 6$. Calculate the ratio of $AD:DC$. | \frac{18}{7} | 0.666667 |
Given that the base of the number system is changed to six, count the twentieth number in this base. | 32 | 0.083333 |
For how many positive integer values of $N$ is the expression $\dfrac{49}{N+3}$ an integer, and this resulting integer is odd? | 2 | 0.833333 |
Lily is riding her bicycle at a constant rate of 15 miles per hour and Leo jogs at a constant rate of 9 miles per hour. If Lily initially sees Leo 0.75 miles in front of her and later sees him 0.75 miles behind her, determine the duration of time, in minutes, that she can see Leo. | 15 | 0.916667 |
A sphere with center O has a radius of 5. A triangle with sides of length 13, 13, and 10 is situated in space so that each of its sides is tangent to the sphere. Find the distance between O and the plane determined by the triangle. | \frac{5\sqrt{5}}{3} | 0.583333 |
Given $S$ is the set of the 1000 smallest positive multiples of $5$, and $T$ is the set of the 1000 smallest positive multiples of $9$, determine the number of elements common to both sets $S$ and $T$. | 111 | 0.5 |
Given a rhombus with diagonals such that one diagonal is three times the length of the other, express the side of the rhombus in terms of $K$, where $K$ is the area of the rhombus in square inches. | \sqrt{\frac{5K}{3}} | 0.5 |
A rectangular metal plate measuring \(10\) cm by \(8\) cm has a circular piece of maximum size cut out, followed by cutting a rectangular piece of maximum size from the circular piece. Calculate the total metal wasted in this process. | 48 | 0.25 |
Let M be the third smallest positive integer that is divisible by every positive integer less than 9. What is the sum of the digits of M? | 9 | 0.916667 |
A wooden cube with side length $n$ units is painted green 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 green. What is $n$? | 3 | 0.666667 |
Consider the set $\{2,3,4,5,6,7,8,9,10\}$. Find the number of subsets of three different numbers, one of which is $7$, that sum to $18$. | 3 | 0.916667 |
A 7' × 11' table sits in the corner of a square room. The table is to be rotated so that the side formerly 7' now lies along what was previously the end side of the longer dimension. Determine the smallest integer value of the side S of the room needed to accommodate this move. | 14 | 0.916667 |
Given that the average daily running time for third graders is $14$ minutes, fourth graders is $17$ minutes, and fifth graders is $12$ minutes, and there are three times as many third graders as fifth graders, and one and a half times as many fourth graders as fifth graders, find the average number of minutes run per day by all students. | \frac{159}{11} | 0.416667 |
Javier the painter initially had enough paint for 40 identically sized rooms. On his way to the painting site, he accidentally loses 4 cans of paint, leaving him with enough paint for only 30 rooms. Determine the number of cans he used for these 30 rooms. | 12 | 0.916667 |
Given Sarah earns $2$ dollars for doing her tasks, $4$ dollars for doing them well, or $6$ dollars for doing them exceptionally over a period of $15$ days, and she accumulated a total of $70$ dollars, determine the number of days on which Sarah did her tasks exceptionally well. | 10 | 0.416667 |
Given a ferry starts transporting tourists to an island at 9 AM and continues every hour until the last trip at 4 PM, and the number of tourists on each trip decreases by 2 compared to the previous trip, starting with 120 tourists on the first trip, calculate the total number of tourists transported to the island that day. | 904 | 0.916667 |
Given the line y = (1/2)x + 3 and a line K that is parallel to the given line but 5 units away from it, find the equation of line K. | y = \dfrac{1}{2}x + \left(3 - \dfrac{5\sqrt{5}}{2}\right) | 0.25 |
Given that 20% of the students scored 65 points, 40% scored 75 points, 25% scored 85 points, and the rest scored 95 points, calculate the difference between the mean and median score of the students' scores on this test. | 3.5 | 0.916667 |
If $2^{2010} - 2^{2009} - 2^{2008} + 2^{2007} - 2^{2006} = m \cdot 2^{2006}$, calculate the value of $m$. | 5 | 0.833333 |
Given that 20% of students in a class are juniors and the remaining 80% are seniors, the average score of the class on the test was 84, and the average score of seniors was 82, determine the score of each junior. | 92 | 0.916667 |
Consider the quadratic function $f(x) = px^2 + qx + r$ where $p > 0$ and the discriminant $q^2 - 4pr < 0$. Calculate the least value of this function. | \frac{4pr - q^2}{4p} | 0.083333 |
Consider the sequence where the n-th term is given by $(-1)^n \cdot n$. Determine the average of the first $200$ terms of the sequence. | 0.5 | 0.416667 |
Samantha has \(3q + 2\) quarters, while Bob has \(2q + 8\) quarters. Determine the difference in their amount of money in dimes. | 2.5q - 15 | 0.666667 |
Let’s consider Nubia has an $8 \times 10$ index card. She decides to reduce the length of one side of this card by $2$ inches, resulting in an area of $64$ square inches. Determine the area of the card in square inches if instead she reduces the length of the other side by $2$ inches. | 60 | 0.916667 |
Points $P$ and $Q$ are 12 units apart, points $Q$ and $R$ are 7 units apart, and points $R$ and $S$ are 5 units apart. If $P$ and $S$ are as close as possible, find the distance between them. | 0 | 0.5 |
If $\log_k x \cdot \log_5 (k^2) = 3$, calculate the value of $x$. | 5\sqrt{5} | 0.916667 |
What is the greatest number of consecutive integers whose sum is $136$? | 272 | 0.166667 |
Evaluate \((x^x)^{(x^x)}\) at \(x = 3\). | 27^{27} | 0.833333 |
Given that the product $\dfrac{5}{3}\cdot \dfrac{6}{5}\cdot \dfrac{7}{6}\cdot \dfrac{8}{7}\cdot \ldots\cdot \dfrac{a}{b} = 16$, calculate the sum of $a$ and $b$. | 95 | 0.583333 |
Let $M = 72^5 + 5\cdot72^4 + 10\cdot72^3 + 10\cdot72^2 + 5\cdot72 + 1$. Find the number of positive integers that are factors of $M$. | 6 | 0.833333 |
A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle in the center of the floor with the sides of the rectangle parallel to the floor's sides. The unpainted part of the floor forms a border of width $2$ feet around the painted rectangle and occupies two-thirds of the area of the entire floor. Determine the number of possible ordered pairs $(a, b)$. | 3 | 0.666667 |
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$. Find the probability that $\lfloor\log_3x\rfloor=\lfloor\log_3y\rfloor$. | \frac{1}{2} | 0.416667 |
What is the maximum number of balls of clay of radius $3$ that can completely fit inside a cube of side length $9$ assuming the balls can be reshaped but not compressed before they are packed in the cube? | 6 | 0.666667 |
The area of the triangle formed by the lines $y = 7$, $y = 2x + 3$, and $y = -2x + 3$. | 8 | 0.916667 |
Given that $1024 = 2^{10}$, find the number of positive integers $b$ for which $\log_b 1024$ is a positive integer. | 4 | 0.916667 |
Points $G$, $H$, $I$, and $J$ are collinear on a line with $GH = HI = IJ = 2$. Points $K$ and $L$ rest on another line, parallel to the first, and distanced such that $KL = 2$. Find the number of distinct possible values for the area of the triangle formed by any three of the six points. | 3 | 0.416667 |
The base of a new rectangle is twice the product of the smaller side and the larger side of a given rectangle, while the altitude of the new rectangle is half the product of the diagonal and the smaller side of the given rectangle. Calculate the area of the new rectangle. | a^2b \sqrt{a^2 + b^2} | 0.5 |
The number of significant digits in the measurement of the side of a square whose computed area is $2.4896$ square inches to the nearest ten-thousandth of a square inch. | 5 | 0.416667 |
Determine the number of lattice points on the line segment whose endpoints are $(15, 35)$ and $(75, 515)$. | 61 | 0.916667 |
Let $n$ be the smallest positive integer such that $n$ is divisible by $30$, $n^2$ is a perfect fourth power, and $n^4$ is a perfect cube. Determine the number of digits of $n$. | 9 | 0.833333 |
Given a box containing 30 red balls, 25 green balls, 20 yellow balls, 15 blue balls, 10 white balls, and 5 black balls, determine the minimum number of balls that must be drawn without replacement to guarantee that at least 18 balls of a single color will be drawn. | 82 | 0.333333 |
Given the hexagons grow by adding subsequent layers of hexagonal bands of dots, with each new layer having a side length equal to the number of the layer, calculate how many dots are in the hexagon that adds the fifth layer, assuming the first hexagon has only 1 dot. | 61 | 0.833333 |
Determine the volume of the original cube given that one dimension is increased by $3$, another is decreased by $2$, and the third is left unchanged, and the volume of the resulting rectangular solid is $6$ more than that of the original cube. | (3 + \sqrt{15})^3 | 0.333333 |
At Pine Lake Summer Camp, $70\%$ of the children play soccer, $50\%$ of the children swim, and $30\%$ of the soccer players swim. Find the percent of the non-swimmers that play soccer. | 98\% | 0.916667 |
For how many integers $x$ does a triangle with side lengths $12, 24$ and $x$ have all its angles acute? | 6 | 0.833333 |
A point is randomly chosen within a square with side length 4040 units. The probability that the point is within d units of a lattice point is 3/5. Calculate d to the nearest tenth. | 0.4 | 0.25 |
Given \(1+3+5+\cdots+2023+2025-2-4-6-\cdots-2022-2024\), evaluate the value of the expression. | 1013 | 0.833333 |
Given that \(x\) is a perfect square, find an expression that represents the second larger perfect square after \(x\). | x + 4\sqrt{x} + 4 | 0.833333 |
A gumball machine contains $10$ red, $12$ white, $9$ blue, and $11$ green gumballs. Calculate the least number of gumballs a person must buy to be sure of getting four gumballs of the same color. | 13 | 0.833333 |
Evaluate the expression: \[
\frac{12-11+10-9+8-7+6-5+4-3+2-1}{2-3+4-5+6-7+8-9+10-11+12} | \frac{6}{7} | 0.5 |
Given Chloe chooses a real number uniformly at random from the interval $[0,1000]$ and Laurent chooses a real number uniformly at random from the interval $[0,3000]$, find the probability that Laurent's number is greater than Chloe's number. | \frac{5}{6} | 0.333333 |
An iterative average of the numbers 6, 7, and 8 is computed as follows: Arrange the three numbers in some order. Firstly, find the mean of the first two numbers, and then compute the mean of that result with the third number. Find the difference between the largest and smallest possible values that can be obtained using this procedure. | \frac{1}{2} | 0.916667 |
What is the largest number of solid \(1\text{-in} \times 3\text{-in} \times 2\text{-in}\) blocks that can fit in a \(4\text{-in} \times 3\text{-in} \times 5\text{-in}\) box? | 10 | 0.416667 |
Determine the value of $r$ if the line $x - 2y = r$ is tangent to the parabola $y = x^2 - r$. | -\frac{1}{8} | 0.75 |
Consider a large semicircle with diameter $D$ and $N$ congruent semicircles fitting exactly on its diameter. Let $A$ be the combined area of these $N$ small semicircles, and $B$ be the area of the large semicircle that is not covered by the small semicircles. Given that the ratio $A:B$ is $1:10$, determine the value of $N$. | 11 | 0.833333 |
Given that Jeff, Maria, and Lee paid $90, $150, and $210 respectively, find j - m where Jeff gave Lee $j dollars and Maria gave Lee $m dollars to settle the debts such that everyone paid equally. | 60 | 0.666667 |
A truck travels $\dfrac{2b}{7}$ feet every $2t$ seconds. There are $3$ feet in a yard. Given this information, calculate the distance the truck travels in $4$ minutes. | \frac{80b}{7t} | 0.416667 |
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