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Jean makes a mixed drink using $150$ grams of cranberry juice, $50$ grams of honey, and $300$ grams of water. There are $30$ calories in $100$ grams of cranberry juice, and $304$ calories in $100$ grams of honey. Water contains no calories. Calculate the total calories in $250$ grams of Jean's mixed drink. | 98.5 | 0.833333 |
The two concentric circles have the same center $O$. A chord $\overline{EF}$ of the larger circle is tangent to the smaller circle at point $G$. The distance from $O$ to $F$ is $12$ units, and the length of chord $\overline{EF}$ is $20$ units. Find the area between the two circles. | 100\pi | 0.833333 |
Given that the length of AB is 3 times the length of BD and the length of AC is 5 times the length of CD on line segment AD, express the length of BC as a fraction of the length of AD. | \frac{1}{12} | 0.833333 |
Given the sequence $x, 3x+3, 6x+6, \dots$ are in geometric progression, find the fourth term. | -24 | 0.916667 |
What is the maximum number of solid $3\text{-in} \times 1\text{-in} \times 1\text{-in}$ blocks that can be placed inside a $3\text{-in} \times 4\text{-in} \times 3\text{-in}$ box? | 12 | 0.583333 |
A group of friends collects $120 to buy flowers for their teacher. Each rose costs $4, and each daisy costs $3. They want to buy at least 5 roses. Find the number of different bouquets they could purchase for exactly $120. | 9 | 0.75 |
Starting with a display of "1," calculate the fewest number of keystrokes needed to reach "1458" using the keys [+1], [x2], and [x3]. | 7 | 0.666667 |
How many four-digit whole numbers are there such that the leftmost digit is an odd prime, the second digit is a multiple of 3, and all four digits are different? | 616 | 0.166667 |
Each of 8 balls is randomly and independently painted either black or white with equal probability. Find the probability that every ball is different in color from at least four of the other 7 balls. | \frac{35}{128} | 0.5 |
Two pitchers, each with a capacity of 800 mL, are partially filled with beverages. The first pitcher is $\frac{1}{4}$ full of orange juice, and the second pitcher is $\frac{3}{8}$ full of apple juice. Both pitchers are then filled to capacity with water and the contents are poured into a larger container. What fraction of the mixture in the large container is orange juice? | \frac{1}{8} | 0.916667 |
Determine how many positive integers $n$ satisfy the condition that $\frac{n}{40-n}$ is also a positive integer. | 7 | 0.833333 |
Let \( m \) be a positive integer and let the lines \( 17x+7y=1000 \) and \( y=mx+2 \) intersect in a point whose coordinates are integers. Find the possible value of \( m \). | 68 | 0.583333 |
Given a right rectangular prism with edge lengths $\log_{5}y, \log_{6}y,$ and $\log_{10}y$, find the value of $y$ such that the numerical value of its surface area equals its volume. | 90000 | 0.25 |
Given a pair of $8$-sided dice each with numbers from $1$ to $8$ is rolled once, the sum of the numbers rolled determines the diameter of a circle. Determine the probability that the numerical value of the area of the circle is less than the numerical value of the circumference. | \frac{3}{64} | 0.666667 |
Given that a contest began at noon one day and ended $1500$ minutes later, determine the time at which the contest ended. | 1:00 \text{ p.m.} | 0.916667 |
Given a square is displayed with 121 grid points, uniformly spaced, including those on the edges, and point $P$ is precisely at the center of the square, determine the probability that the line $PQ$ is a line of symmetry for the square if point $Q$ is selected at random from the remaining 120 points. | \frac{1}{3} | 0.083333 |
A triangle has one of its sides divided by the point of tangency of the inscribed circle into segments of $9$ and $5$ units. If the radius of the inscribed circle is $3$ units, calculate the length of the longest side of the triangle. | 14 | 0.666667 |
Given Ahn chooses a three-digit integer, subtracts it from $500$, and triples the result, find the largest number Ahn can get. | 1200 | 0.916667 |
Circle A has a radius of 150. Circle B has an integer radius r < 150 and remains internally tangent to circle A as it rolls once around the circumference of circle A. Determine the number of possible values for r. | 11 | 0.5 |
What was the regular price of one tire, given that Sam paid 250 dollars for a set of four tires at the sale, with the fourth tire being offered for 10 dollars when three tires are bought at the regular price? | 80 | 0.833333 |
In a small town, Sarah usually bikes to her bookstore in 30 minutes. However, one windy day she decides to ride faster, biking 12 miles per hour faster than usual, getting her to the bookstore in 18 minutes. Determine the distance to the bookstore in miles. | 9 | 0.916667 |
If 1 yen is worth 0.0075 dollars, what percent is the value of Etienne's 5000 yen greater than or less than the value of Diana's 500 dollars? | 92.5\% | 0.833333 |
The point (c, d) in the xy-plane is first rotated clockwise by 180° around the point (2, -3) and then reflected about the line y = x. The image of (c, d) after these two transformations is at (5, -4). Find the value of d - c. | -19 | 0.916667 |
What is the tens digit of $2035^{2037} - 2039$? | 3 | 0.75 |
The even number 134 can be represented as the sum of two distinct prime numbers. Find the largest possible difference between these two prime numbers. | 128 | 0.833333 |
Given that $n$ is a positive integer, the sum of the positive factors of $n$ is represented as $\boxed{n}$, where $\boxed{5}$ equals $1 + 5 = 6$. Determine the value of $\boxed{\boxed{7}}$. | 15 | 0.25 |
The number of positive integers less than $500$ divisible by neither $5$ nor $7$. | 343 | 0.833333 |
A ferry boat begins shuttling tourists to an island every hour starting at 10 AM, with its last trip starting at 4 PM. On the 10 AM trip, there were 100 tourists on the ferry, and on each successive trip, the number of tourists decreased by 2 from the previous trip. Calculate the total number of tourists transported to the island that day. | 658 | 0.916667 |
Given the number $9,680$, find the number of its positive factors. | 30 | 0.916667 |
Anna selects a real number uniformly at random from the interval $[0, 3000]$, and Bob chooses a real number uniformly at random from the interval $[0, 6000]$. Determine the probability that Bob's number is greater than Anna's number. | \frac{3}{4} | 0.333333 |
What is the hundreds digit of $(18! + 14!)$? | 2 | 0.166667 |
Find the number of distinct points common to the curves $x^2 + 4y^2 = 4$, $4x^2 + y^2 = 4$, and $x^2 + y^2 = 1$. | 0 | 0.666667 |
The perimeter of a rectangular garden with dimensions 32 meters by 72 meters must be encircled by a fence, with fence posts placed every 8 meters, including the posts at each corner. Determine the fewest number of posts required. | 26 | 0.5 |
Given that Marie has 2500 coins consisting 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 greatest possible and least amounts of money that Marie can have. | 22473 | 0.416667 |
Point $O$ is the center of the regular octagon $ABCDEFGH$, and $Y$ is the midpoint of side $\overline{CD}$. Calculate the fraction of the area of the octagon that is shaded, where the shaded region includes triangles $\triangle DEO$, $\triangle EFO$, $\triangle FGO$, and half of $\triangle DCO$. | \frac{7}{16} | 0.5 |
Points E and F lie on segment GH. The length of segment GE is 3 times the length of segment EH, and the length of segment GF is 4 times the length of segment FH. Express the length of segment EF as a fraction of the length of segment GH. | \frac{1}{20} | 0.5 |
Given that liquid $Y$ forms a circular film $0.2$ cm thick when poured onto a water surface, find the radius, in centimeters, of the resulting circular film created when a rectangular box with dimensions $10$ cm by $5$ cm by $20$ cm is filled with liquid $Y$ and its contents are poured onto a large body of water. | \sqrt{\frac{5000}{\pi}} | 0.25 |
Given the expression $\frac{2^2+4^2+6^2}{1^2+3^2+5^2} - \frac{1^2+3^2+5^2}{2^2+4^2+6^2}$, simplify the expression. | \frac{1911}{1960} | 0.5 |
Maria drives 25 miles at an average speed of 40 miles per hour. Calculate the distance Maria needs to drive at 75 miles per hour to average 60 miles per hour for the entire trip. | 62.5 | 0.75 |
The difference between the sum of the first 1000 even numbers, starting from 0, and the sum of the first 1000 odd counting numbers. | -1000 | 0.25 |
For what value of $x$ does $2^{3x} \cdot 8^{x-1} = 32^{2x}$ | -\frac{3}{4} | 0.916667 |
What is the value of $2468 + 8642 + 6824 + 4286$? | 22220 | 0.916667 |
If $\frac{b}{a} = 3$, $\frac{c}{b} = 2$, and $\frac{d}{c} = 4$, calculate the ratio of $a + c$ to $b + d$. | \frac{7}{27} | 0.916667 |
A driver travels for 3 hours at a speed of 50 miles per hour. The car has a fuel efficiency of 25 miles per gallon, earns $0.60 per mile, and gasoline costs $2.50 per gallon. Calculate her net rate of pay per hour after considering gasoline expenses. | 25 | 0.833333 |
Cagney can frost a cupcake every 25 seconds and Lacey can frost a cupcake every 20 seconds. If they both take a 30-second break after every 3 minutes of work, how many cupcakes can they frost together in 10 minutes? | 48 | 0.25 |
If a total of 28 conference games were played during the 2017 season, determine the number of teams that were members of the BIG N conference. | 8 | 0.916667 |
With all three valves open, the tank fills in 0.5 hours, with only valves A and C open it takes 1 hour, and with only valves B and C open it takes 2 hours. Determine the number of hours required with only valves A and B open. | 0.4 | 0.666667 |
Ted's grandfather used his treadmill on 4 days this week with distances of 3 miles on Monday, Wednesday, and Friday, and 4 miles on Sunday. The times taken to travel these distances were 0.5 hours, 0.75 hours, and 0.6 hours on Monday, Wednesday, and Friday, respectively, and 4/3 hours on Sunday. If Grandfather had always walked at 5 miles per hour, calculate the time saved in minutes. | 35 | 0.166667 |
Let $x$ be the smallest positive real number such that $\cos(x) = \cos(3x)$, where $x$ is measured in radians. Express $x$ in radians, rounded to two decimal places. | 1.57 | 0.833333 |
If two factors of $3x^3 - mx + n$ are $x-3$ and $x+4$, determine the value of $|3m-2n|$. | 45 | 0.833333 |
Given a real number $x$, simplify the expression $\sqrt{x^4 - x^2}$. | |x|\sqrt{x^2-1} | 0.916667 |
Sam drove $150$ miles in $135$ minutes. His average speed during the first $45$ minutes was $50$ mph, and his average speed during the second $45$ minutes was $60$ mph. Calculate the average speed, in mph, during the last $45$ minutes. | 90 | 0.916667 |
A $3\times 3$ block of calendar dates is shown. First, the order of the numbers in the first and the third rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums? | 0 | 0.916667 |
Given that $\frac{x}{3} = y^2$ and $\frac{x}{6} = 3y$, solve for the value of $x$. | 108 | 0.916667 |
Given the quadratic polynomial $ax^2+bx+c$, identify the value that is incorrect among the outcomes $2107, 2250, 2402, 2574, 2738, 2920, 3094, 3286$. | 2574 | 0.833333 |
Given that Mrs. Wanda Wake drives to work at 7:00 AM, if she drives at an average speed of 30 miles per hour, she will be late by 2 minutes, and if she drives at an average speed of 50 miles per hour, she will be early by 2 minutes. Find the required average speed for Mrs. Wake to get to work exactly on time. | 37.5 | 0.916667 |
Given that there are $400$ adults in total in City Z, with $370$ adults owning bikes and $75$ adults owning scooters, determine the number of bike owners who do not own a scooter. | 325 | 0.833333 |
In 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. At the end of the tournament, the teams are ranked by the number of games won, calculate the maximum number of teams that could be tied for the most wins. | 7 | 0.25 |
Given that John begins the first task at 8:00 AM and finishes the third task at 11:30 AM, determine the time when he finishes the fourth task. | 12:40 \text{ PM} | 0.5 |
A merchant buys goods at a 30% discount off the list price and wants to mark up the goods. He plans to offer a 20% discount on his marked price but still aims to achieve a 30% profit on the selling price. What percentage of the list price should he set as the marked price? | 125\% | 0.75 |
The fifth term in the expansion of $(\frac{b}{x}-x^2b)^7$ when simplified is $-35b^7x^5$. | -35b^7x^5 | 0.916667 |
Alice is baking two batches of cookies, one batch requiring $3 \frac{1}{2}$ cups of sugar. Her measuring cup can hold $\frac{1}{3}$ cup of sugar. Determine the total number of times she must fill her cup to measure the sugar needed for both batches. | 21 | 0.916667 |
Given that a semipro baseball league has 25 players per team and each player's minimum salary is $15,000, while the total salary cap for each team is $875,000, calculate the maximum possible salary for a single player. | 515,000 | 0.416667 |
Given that in a bag of marbles, $\frac{2}{3}$ of the marbles are blue and the rest are red, find the new fraction of the marbles that will be red if the number of red marbles is tripled while the number of blue marbles stays the same. | \frac{3}{5} | 0.916667 |
Given two circles with centers 50 inches apart, where the radius of the smaller circle is 7 inches and the radius of the larger circle is 10 inches, calculate the length of the common internal tangent between these two circles. | \sqrt{2211} \text{ inches} | 0.75 |
A number, $n$, is added to the set $\{4, 8, 12, 15\}$ to make the mean of the set of five numbers equal to its median. Find the smallest possible value of $n$. | 1 | 0.916667 |
Jenny and Joe shared a pizza that was cut into 12 slices. Jenny prefers a plain pizza, but Joe wanted mushrooms on four of the slices. A whole plain pizza costs $12. Each slice with mushrooms adds an additional $0.50 to the cost of that slice. Joe ate all the mushroom slices and three of the plain slices. Jenny ate the remaining plain slices. Each paid for what they had eaten. What is the difference, in dollars, that Joe paid compared to Jenny? | 4 | 0.666667 |
In parallelogram $\text{EFGH}$, point $\text{J}$ is on $\text{EH}$ such that $\text{EJ} + \text{JH} = \text{EH} = 12$ and $\text{JH} = 8$. If the height of parallelogram $\text{EFGH}$ from $\text{FG}$ to $\text{EH}$ is 10, find the area of the shaded region $\text{FJGH}$. | 100 | 0.916667 |
The sum of five integers is $3$. Calculate the maximum number of these integers that can be larger than $26$. | 4 | 0.166667 |
Given real numbers $x$ and $y$, find the least possible value of $(x^2y-1)^2 + (x^2+y)^2$. | 1 | 0.833333 |
Given two pipes, one with an inside diameter of $2$ inches and a length of $2$ meters, and the other with an inside diameter of $8$ inches and a length of $1$ meter, determine the number of circular pipes with an inside diameter of $2$ inches and a length of $2$ meters that will carry the same amount of water as one pipe with an inside diameter of $8$ inches and a length of $1$ meter. | 8 | 0.333333 |
Given that $f(x+6) + f(x-6) = f(x)$ for all real $x$, determine the least positive period $p$ for these functions. | 36 | 0.583333 |
Consider a line segment whose endpoints are (5, 10) and (68, 178), determine the number of lattice points on this line segment, including both endpoints. | 22 | 0.916667 |
Given that Eva is using 24 fence posts to fence her rectangular vegetable patch, with each post spaced 6 yards apart, including the corners, and the longer side of the patch has three times as many posts as the shorter side, calculate the area of Eva’s vegetable patch. | 576 | 0.583333 |
Evaluate the expression $\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{4}}}}$. | \frac{14}{19} | 0.916667 |
Find the area of quadrilateral $EFGH$, given that $m\angle F = m \angle G = 135^{\circ}$, $EF=4$, $FG=6$, and $GH=8$. | 18\sqrt{2} | 0.833333 |
Given the annual incomes of $500$ families ranging from $12,000$ dollars to $150,000$ dollars, where the largest income was incorrectly entered as $1,500,000$ dollars, calculate the difference between the mean of the incorrect data and the mean of the actual data. | 2700 | 0.5 |
Evaluate the expression $\frac{2^{3001} \cdot 3^{3003}}{6^{3002}}$. | \frac{3}{2} | 0.833333 |
Lucas, Emma, and Noah collected shells at the beach. Lucas found four times as many shells as Emma, and Emma found twice as many shells as Noah. Lucas decides to share some of his shells with Emma and Noah so that all three will have the same number of shells. What fraction of his shells should Lucas give to Emma? | \frac{5}{24} | 0.583333 |
Square $ABCD$ has its vertex $A$ at the origin of the coordinate system, and side length $AB = 2$. Vertex $E$ of isosceles triangle $\triangle ABE$, where $AE=BE$, is inside the square. A circle is inscribed in $\triangle ABE$ tangent to side $AB$ at point $G$, exactly at the midpoint of $AB$. Find the area of $\triangle ABF$, where $F$ is the point of intersection of diagonal $BD$ of the square and line segment $AE$. | 1 | 0.666667 |
Calculate the number of integer values of $n$ such that $3200 \cdot \left(\frac{4}{5}\right)^n$ remains an integer. | 6 | 0.75 |
Given seven positive consecutive integers starting with $c$, find the average of seven consecutive integers starting with $d$, where $d$ is the average of the first set of integers. | c + 6 | 0.666667 |
Given a positive number $x$ that satisfies the inequality $\sqrt{x} > 3x$, determine the inequality that must be true. | 0 < x < \frac{1}{9} | 0.916667 |
What is the expression $\frac{x-2}{4-z} \cdot \frac{y-3}{2-x} \cdot \frac{z-4}{3-y}$ in its simplest form? | -1 | 0.75 |
At $5:50$ o'clock, calculate the angle between the hour and minute hands of a clock. | 125^{\circ} | 0.916667 |
Given that circle $O$ contains points $C$ and $D$ on the same side of diameter $\overline{AB}$, $\angle AOC = 40^{\circ}$, and $\angle DOB = 60^{\circ}$, calculate the ratio of the area of the smaller sector $COD$ to the area of the circle. | \frac{2}{9} | 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 |
Jeremy wakes up at 6:00 a.m., catches the school bus at 7:00 a.m., has 7 classes that last 45 minutes each, enjoys 45 minutes for lunch, and spends an additional 2.25 hours (which includes 15 minutes for miscellaneous activities) at school. He takes the bus home and arrives at 5:00 p.m. Calculate the total number of minutes he spends on the bus. | 105 | 0.25 |
Given $a,b>0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by=8$ has area 8, determine the value of $ab$. | 4 | 0.5 |
Jack drove 150 miles in 2.5 hours. His average speed during the first hour was 50 mph. After a 15-minute stop, he resumed travel for another hour at an average speed of 55 mph. Calculate his average speed, in mph, during the last 30 minutes. | 90 | 0.916667 |
The interior of a quadrilateral is bounded by the graphs of $(x+by)^2 = 9b^2$ and $(bx-y)^2 = 4b^2$, where $b$ is a positive real number. Determine the area of this region in terms of $b$, valid for all $b > 0$. | \frac{24b^2}{1 + b^2} | 0.083333 |
Let $s$ be the result of tripling both the base and exponent of $c^d$, where $d$ is a non-zero integer. If $s$ equals the product of $c^d$ by $y^d$, determine the value of $y$. | 27c^2 | 0.833333 |
Given that John divides a circle into 15 sectors with central angles that form an arithmetic sequence, calculate the degree measure of the smallest possible sector angle. | 3 | 0.333333 |
Let the roots of $px^2 + qx + r = 0$ be $u$ and $v$. Determine the quadratic equation with roots $p^2u - q$ and $p^2v - q$. | x^2 + (pq + 2q)x + (p^3r + pq^2 + q^2) = 0 | 0.166667 |
Suppose that $\frac{2}{3}$ of $15$ bananas are worth as much as $12$ oranges, calculate the number of oranges that are worth as much as $\frac{1}{4}$ of $20$ bananas. | 6 | 0.916667 |
Find how many 3-digit whole numbers have a digit-sum of 27 and are even. | 0 | 0.833333 |
Given a bag of popping corn that contains $\frac{3}{4}$ white kernels and $\frac{1}{4}$ yellow kernels, only $\frac{1}{3}$ of the white kernels will pop, while $\frac{3}{4}$ of the yellow ones will pop, calculate the probability that the kernel selected was white. | \frac{4}{7} | 0.5 |
Determine how many integers $n$ exist such that $(n+i)^6$ is an integer. | 1 | 0.416667 |
The sum other than $11$ which occurs with the same probability when all 8 dice are rolled is equal to what value. | 45 | 0.75 |
Four runners start running simultaneously from the same point on a 600-meter circular track. They each run clockwise around the course maintaining constant speeds of 5.0, 5.5, 6.0, and 6.5 meters per second. Calculate the total time until the runners are together again somewhere on the circular course. | 1200 | 0.333333 |
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