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Calculate the sum $E(1)+E(2)+E(3)+\cdots+E(500)$, where $E(n)$ denotes the sum of the even digits of $n$. For example, $E(5681) = 6 + 8 = 14$.
A) 2000
B) 2200
C) 2400
D) 2500
E) 2600 | 2600 | 0.083333 |
What is the greatest number of consecutive integers, including both negative and non-negative integers, whose sum is $120$? | 240 | 0.083333 |
Given a magician's hat contains 4 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 4 of the reds are drawn or until both green chips are drawn, calculate the probability that all 4 red chips are drawn before both green chips are drawn. | \frac{1}{3} | 0.083333 |
Given the four-digit number $2055$, calculate the number of different four-digit numbers that can be formed by rearranging its digits. | 9 | 0.75 |
Given that every 7-digit whole number can be a telephone number, except those that begin with numbers 0, 1, or 2, or end with 9, calculate the fraction of telephone numbers that begin with 7 and end with 5. | \frac{1}{63} | 0.416667 |
A computer can perform 15,000 additions per second, calculate the total number of additions it can perform in 2 hours and 30 minutes. | 135,000,000 | 0.916667 |
Let $A$, $B$ and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\triangle ABC$ is a right triangle with area $2016$. Find the sum of the digits of the $y$-coordinate of $C$. | 26 | 0.25 |
The number of positive integers less than $1200$ divisible by neither $6$ nor $8$ must be calculated. | 900 | 0.916667 |
Given a fair coin is tossed 4 times, calculate the probability of getting at least two consecutive heads. | \frac{1}{2} | 0.25 |
If the radius of a circle is increased by $k$ units (where $k > 0$), determine the ratio of the new circumference to the original diameter. | \pi \left(1 + \frac{k}{r}\right) | 0.666667 |
Given circle $O$, point $C$ is on the opposite side of diameter $\overline{AB}$ from point $A$, and point $D$ is on the same side as point $A$. Given $\angle AOC = 40^{\circ}$, and $\angle DOB = 60^{\circ}$, calculate the ratio of the area of the smaller sector $COD$ to the area of the circle. | \frac{4}{9} | 0.25 |
Suppose a square piece of paper with side length 4 units is folded in half diagonally, then cut along the fold, producing two right-angled triangles. Calculate the ratio of the perimeter of one of the triangles to the perimeter of the original square. | \frac{1}{2} + \frac{\sqrt{2}}{4} | 0.333333 |
Given $f(x) = x^2 + 5x + 6$, consider the set $S = \{0, 1, 2, \dots, 30\}$. Find the number of integers $s$ in $S$ such that $f(s)$ is divisible by $5$. | 12 | 0.833333 |
What is the value of the expression $2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2}}}$? | \frac{29}{12} | 0.25 |
Given that each vertex of a regular tetrahedron is to be labeled with an integer $1$ through $4$, with each integer being used exactly once, such that the sum of the numbers on the vertices of each triangular face is the same, determine the number of distinct arrangements possible. | 0 | 0.333333 |
Given 2 gallons of paint are required for the room's complete coverage, determine the fraction of the original amount of paint available to use on the third day, given that one quarter of the paint is used on the first day and one third of the remaining paint is used on the second day. | \frac{1}{2} | 0.916667 |
Given that Brianna is planning to use some of her earnings from a weekend job to purchase several music albums at equal prices, if she spends one quarter of her money to buy one quarter of the albums, determine the fraction of her money that she will have left after purchasing all the albums. | 0 | 0.75 |
Let $g(x) = 3x^2 + 4x + 5$. Calculate $g(x + h) - g(x)$. | h(6x + 3h + 4) | 0.5 |
Given a quadrilateral with vertices at the points (2,1), (0,7), (5,5), and (6,9), calculate the area enclosed by this quadrilateral. | 9 | 0.583333 |
Given the podcast series with a total content of 500 minutes, evenly distributed throughout the week with each day featuring up to 65 minutes of content, calculate the number of minutes of content that each day will feature and the number of days it will take to release the entire series. | 8 | 0.333333 |
What is the greatest number of consecutive integers whose sum is $36$? | 72 | 0.25 |
Given that the new track has a width of 8 meters, and it takes Keiko 48 seconds longer to walk around the outside edge of the track than around the inside edge, determine Keiko's new speed in meters per second. | \frac{\pi}{3} | 0.916667 |
What is the area of the polygon with vertices at $(2,1)$, $(4,3)$, $(6,1)$, and $(4,6)$? | 6 | 0.583333 |
Lucas wakes up at 6:00 a.m., walks to the bus stop at 6:45 a.m., attends 7 classes each lasting 45 minutes, gets 40 minutes for lunch, and spends an additional 1.5 hours at school for extracurricular activities. He catches the bus home and arrives at 3:30 p.m. Calculate the number of minutes Lucas spends on the bus. | 80 | 0.666667 |
A triangle has a fixed base $AB$ that is $6$ inches long. The median from $A$ to side $BC$ is $4$ inches long and must be at a 60-degree angle with the base. Determine the locus of vertex $C$.
A) A circle with center A and radius 6 inches
B) A circle with center A and radius 7 inches
C) An ellipse with A as focus
D) A straight line perpendicular to AB at A | B) A circle with center A and radius 7 inches | 0.166667 |
Given a sequence where each term increases by 1 from the previous term and the first term is $x_1 = 3$, find the sum of the first $n$ terms of the sequence where $n$ is a positive integer such that $n \leq 10$. | \frac{n(n + 5)}{2} | 0.833333 |
Between $5^{5} - 1$ and $5^{10} + 1$, inclusive, calculate the number of perfect cubes. | 199 | 0.166667 |
Samia jogged to her friend's house at an average speed of 8 kilometers per hour, and then walked the remaining two-thirds of the way at 4 kilometers per hour. If the entire trip took her 105 minutes to complete, calculate the distance in kilometers that Samia walked. | 5.6 | 0.916667 |
Calculate the product of \(\frac{4}{3} \times \frac{5}{4} \times \frac{6}{5} \times \cdots \times \frac{2010}{2009}\). | 670 | 0.75 |
For how many positive integer values of N is the expression $\dfrac{48}{N+3}$ an integer? | 7 | 0.666667 |
A vehicle is equipped with five tires (four standard tires and one enhanced durability spare tire) is driven for a total of 45,000 miles. The spare tire can endure 20% more mileage than the regular tires. If the rotation policy ensures each tire is used the maximum amount without exceeding its durability, how many miles was each type of tire used?
A) 30,000 miles, 36,000 miles
B) 34,615 miles, 41,538 miles
C) 37,500 miles, 45,000 miles
D) 40,000 miles, 48,000 miles | B) 34,615 miles, 41,538 miles | 0.083333 |
Given that the number is $180^8$, calculate the number of positive integer divisors that are perfect squares or perfect cubes (or both). | 495 | 0.333333 |
Three jars each contain the same number of marbles, and each marble is either blue or green. In Jar $1$ the ratio of blue to green marbles is $7:3$, in Jar $2$ it is $6:4$, and in Jar $3$ it is $5:5$. There are a total of $240$ green marbles across all jars. Calculate the difference in the number of blue marbles in Jar $1$ and Jar $3$. | 40 | 0.916667 |
The maximum number of points of intersection of the graphs of two different third degree polynomial functions $y=f(x)$ and $y=g(x)$, each with leading coefficient 2, must be determined. | 2 | 0.583333 |
A divisor of $10!$ that is a multiple of 12 is chosen at random. What is the probability that this divisor is also a perfect square? Express your answer as a fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. | \frac{2}{21} | 0.666667 |
Given that Sangho's video had a score of 140, and 75% of the total votes were like votes, determine the total number of votes cast on Sangho's video. | 280 | 0.416667 |
Given that Maria's video had a score of $120$, with $75\%$ of the votes being likes, determine the total number of votes cast on her video. | 240 | 0.583333 |
Given a quadrilateral with vertices $P(k,a)$, $Q(a,k)$, $R(-k,-a)$, and $S(-a,-k)$, where $k$, $a$, and $b$ are integers with $k > a > 0$, find the value of $k+a$, given that the area of the quadrilateral $PQRS$ is $32$. | 8 | 0.666667 |
Given that $x \not= 0$ or $3$ and $y \not= 0$ or $4$, solve the equation $\frac{3}{x} + \frac{2}{y} = \frac{5}{6}$ and find $x$ in terms of $y$. | \frac{18y}{5y - 12} | 0.666667 |
Casper started with some candies. On the first day, he ate $\frac{1}{4}$ of his candies then gave $3$ candies to his brother. On the second day, he ate $\frac{1}{5}$ of the remaining candies and gave $5$ candies to his sister. On the third day, he gave $7$ candies to his friend and finally ate $\frac{1}{6}$ of what was left, leaving him with $10$ candies. Determine the number of candies that Casper had at the beginning. | 44 | 0.833333 |
Hadi has a $5 \times 7$ index card. If he shortens the length of one side of this card by $2$ inches, the card would have an area of $21$ square inches. Calculate the area of the card if instead he shortens the length of the other side by $2$ inches. | 25 | 0.5 |
Given a pentagonal grid with a base of 10 toothpicks and four other sides made of 8 toothpicks each, each vertex joint consumes an additional toothpick for structural support. Calculate the total number of toothpicks needed. | 47 | 0.583333 |
Calculate the sum $\triangle$ for two triangles, where the first triangle has numbers 2, 3, and 5, and the second triangle has numbers 3, 4, and 6. | 23 | 0.75 |
Given Ben's test scores $95, 85, 75, 65,$ and $90$, and his goal to increase his average by at least $5$ points and score higher than his lowest score of $65$ with his next test, calculate the minimum test score he would need to achieve both goals. | 112 | 0.916667 |
Evaluate the sum $2345 + 3452 + 4523 + 5234$ and then subtract $1234$ from the result. | 14320 | 0.833333 |
Given points $A, B, C,$ and $D$ on a line with $AB=2$, $BC=1$, and $CD=3$, and points $E, F,$ and $G$ on a second line, parallel to the first, with $EF=1$ and $FG=2$, and the perpendicular distance between the two lines is a fixed, positive value $h$, determine the number of possible distinct areas that can be formed using any three of these seven points as vertices. | 5 | 0.333333 |
Given that Mary is 30% older than Sally, and Sally is 50% younger than Danielle, and the sum of their ages is 45 years, determine Mary's age on her next birthday. | 14 | 0.333333 |
Given the expressions $(6+16+26+36+46)$ and $(14+24+34+44+54)$, evaluate their sum. | 300 | 0.916667 |
How many primes less than $100$ have $3$ as the ones digit? | 7 | 0.666667 |
A regular hexagon has a point at each of its six vertices, and each side of the hexagon measures one unit. If two of the 6 points are chosen at random, calculate the probability that the two points are one unit apart. | \frac{2}{5} | 0.916667 |
Evaluate the ratio $\frac{2^{3002} \cdot 3^{3005}}{6^{3003}}$. | \frac{9}{2} | 0.916667 |
If the shopper saves $\textdollar{3.75}$ when buying a coat on sale and the final price paid for the coat was $\textdollar{50}$, determine the percentage that the shopper saved. | 7\% | 0.416667 |
Determine the number of digits in the expression $(8^4 \cdot 4^{12}) / 2^8$. | 9 | 0.833333 |
In a distant kingdom, four apples can be traded for three bottles of juice, and each bottle of juice can be traded for five slices of cake. Calculate the value in slices of cake that one apple is worth. | 3.75 | 0.5 |
Evaluate the value of $\frac{(2210-2137)^2 + (2137-2028)^2}{64}$. | 268.90625 | 0.416667 |
Find the sum of the squares of all real numbers satisfying the equation $x^{64} - 16^{16} = 0$. | 8 | 0.916667 |
What is the sum of the two smallest prime factors of $280$? | 7 | 0.666667 |
What is the sum of the mean, median, and mode of the numbers $1, 2, 2, 4, 5, 5, 5, 7$? | 13.375 | 0.583333 |
Let $M$ be the greatest four-digit number whose digits have a product of $24$. Calculate the sum of the digits of $M$. | 13 | 0.75 |
Brianna uses one quarter of her money to buy one half of the CDs she wants. What fraction of her money will she have left after she buys all the CDs? | \frac{1}{2} | 0.833333 |
Given \(\frac{8^x}{2^{x+y}} = 16\) and \(\frac{16^{x+y}}{4^{5y}} = 1024\), find the value of \(xy\). | -\frac{7}{8} | 0.916667 |
What is the radius of a circle inscribed in a rhombus with diagonals of length $14$ and $30$? | \frac{105\sqrt{274}}{274} | 0.416667 |
Shea and Ara were once the same height. Since then, Shea has grown 30%, while Ara has grown 5 inches less than Shea. Shea is now 65 inches tall. Calculate Ara's current height. | 60 | 0.916667 |
The graphs of $y = -|x-a|^2 + b$ and $y = |x-c|^2 + d$ intersect at points $(1,8)$ and $(9,4)$. Find $a+c$. | 10 | 0.416667 |
Given $x \otimes y = x^2 - 2y$, calculate the value of $k \otimes (k \otimes k)$. | -k^2 + 4k | 0.833333 |
The notation $N!$ represents the product of integers $1$ through $N$. Consider the expression $102! + 103! + 104! + 105!$. Determine the largest integer $n$ for which $5^n$ is a factor of this sum. | 24 | 0.833333 |
Evaluate the expression $\left(\left((5+2)^{-1} - \frac{1}{2}\right)^{-1} + 2\right)^{-1} + 2$. | \frac{3}{4} | 0.5 |
In ∆ABC, AB = 6, BC = 8, AC = 10, and D is on AC with BD = 6. Find the ratio of AD:DC. | \frac{18}{7} | 0.416667 |
Let $m = {2021}^2 + {3}^{2021}$. What is the units digit of $m^2 + 3^m$? | 7 | 0.833333 |
Given that one of every 400 people has a particular disease and a blood test has a 5% false positive rate, determine the probability that a person randomly selected from this population and who tests positive actually has the disease. | \frac{20}{419} | 0.916667 |
Given Sarah has a six-ounce cup of coffee and an identical cup containing three ounces of milk. She pours one-third of the coffee into the milk, mixes it thoroughly, and then pours half of this mixed content back into the coffee cup. What fraction of the liquid in the coffee cup is now milk. | \frac{3}{13} | 0.333333 |
Points $A = (2,8)$, $B = (0,2)$, $C = (6,4)$, and $D=(a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$. The quadrilateral formed by joining the midpoints of $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$ is a square. Find the sum of the coordinates of point $D$. | 10 | 0.25 |
What is the least possible value of $(xy-2)^2 + (x-y)^2$ for real numbers $x$ and $y$? | 0 | 0.5 |
Given that Spinner A has the numbers 1, 4, 7, 9 and Spinner B has the numbers 2, 3, 5, 8, calculate the probability that the sum of the numbers on which the spinners land is a prime number. | \frac{1}{4} | 0.75 |
Consider a regular polygon with $n$ sides, $n>4$, where each side is bent inward at an angle $\theta$ such that $\theta = \frac{360^\circ}{2n}$. Calculate the total angle formed by all the bends around the entire shape. | 180^\circ | 0.833333 |
Given the equation $x^2 + y^2 = m^2$ is tangent to the line $x - y = m$, find the possible value(s) of $m$. | m = 0 | 0.75 |
Find the number of pairs (x, y) of integers that satisfy the equation x^3 + 4x^2 + x = 18y^3 + 18y^2 + 6y + 3. | 0 | 0.833333 |
The printing machine starts operating at $\text{9:00}\ {\small\text{AM}}$. By $\text{12:00}\ {\small\text{PM}}$, it has completed half of the day's printing job. Find the time when the printing machine will finish the entire job. | 3:00 PM | 0.916667 |
Calculate $300,000$ times $300,000$ times $3$. | 270,000,000,000 | 0.75 |
Samantha gets up at 7:00 a.m., catches the school bus at 8:00 a.m. and arrives home at 5:30 p.m. Given that she has 7 classes that last 45 minutes each, has 45 minutes for lunch, and participates in a 1.5 hour chess club after class, calculate the total number of minutes she spends on the bus. | 120 | 0.166667 |
Suppose that $f(x+2)=5x^2 + 2x + 6$ and $f(x)=ax^2 + bx + c$. Find the value of $a + 2b + 3c$. | 35 | 0.916667 |
Given that all the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are arranged in a $3\times3$ grid such that if two numbers are consecutive, they occupy squares that share an edge, and the numbers in the four corners add up to $20$, determine the number in the center of the grid. | 5 | 0.75 |
The sides of a triangle are $40$, $90$, and $100$ units. Given that an altitude is dropped upon the side of length $100$, calculate the length of the larger segment cut off on this side. | 82.5 | 0.166667 |
Calculate the number of terms in the expansion of $[(a+3b)^{3}(a-3b)^{3}]^{2}$ when fully simplified. | 7 | 0.75 |
When a certain biased die is rolled, an even number is twice as likely to appear as an odd number. The die is rolled twice. Calculate the probability that the sum of the numbers rolled is odd. | \frac{4}{9} | 0.083333 |
$\log 216$ equals $3 \log 2 + 3 \log 3$. | 3 \log 2 + 3 \log 3 | 0.75 |
Given a cyclic quadrilateral $ABCD$ inscribed in a circle, with side $AB$ extended beyond $B$ to point $E$. If $\angle BAD = 75^{\circ}$ and $\angle ADC = 83^{\circ}$, find $\angle EBC$. | 83^{\circ} | 0.166667 |
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 $[0, 2000]$, find the probability that Laurent's number is greater than twice Chloe's number. | \frac{1}{2} | 0.25 |
Alice has 4 sisters and 6 brothers. Given that Alice's sister Angela has S sisters and B brothers, calculate the product of S and B. | 24 | 0.5 |
Given Evan and his two older twin sisters' ages are three positive integers whose product is 162, calculate the sum of their ages. | 20 | 0.166667 |
Given that the product of Kiana's older brother, Kiana, and Kiana's younger sister's ages is 72, find the sum of their three ages. | 13 | 0.166667 |
Given that $x$ and $y$ are distinct nonzero real numbers such that $x + \frac{3}{x} = y + \frac{3}{y}$, find the value of $xy$. | 3 | 0.916667 |
A circle with a radius of 8 is inscribed in a rectangle. The ratio of the length of the rectangle to its width is 3:1. Calculate the area of the rectangle. | 768 | 0.833333 |
Cagney can frost a cupcake every 20 seconds, Lacey can frost a cupcake every 30 seconds, and Jamie can frost a cupcake every 15 seconds. Working together, calculate the number of cupcakes they can frost in 10 minutes. | 90 | 0.916667 |
John works on three tasks sequentially. The second task takes twice as long as the first, and the third task takes the same amount of time as the first. He starts the first task at 9:00 AM and completes the second task by 11:30 AM. Determine the completion time of the third task. | 12:20 \text{ PM} | 0.916667 |
Given that Mark drove the first one-fourth of his trip on a country road, the next 30 miles on a highway, and the remaining one-sixth on a city street, calculate the total distance of Mark's trip in miles. | \frac{360}{7} | 0.333333 |
How many positive odd multiples of $5$ less than $5000$ are perfect squares? | 7 | 0.75 |
Determine the smallest value of n for which the total cost, after a 5% sales tax, is exactly n dollars. | 21 | 0.75 |
A woman was born in the nineteenth century and was $x$ years old in the year $x^2$. Find the birth year of the woman. | 1892 | 0.333333 |
Given that LeRoy, Bernardo, and Camilla paid $A$, $B$, and $C$ dollars, respectively, with $A < B < C$, calculate the total number of dollars LeRoy must give to Bernardo and Camilla so that they share the costs equally. | \frac{B + C - 2A}{3} | 0.416667 |
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