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A box contains chips, each of which is red, white, or blue. The number of blue chips is at least one third the number of white chips, and at most one fourth the number of red chips. The number which are white or blue is at least 70. Find the minimum number of red chips. | 72 | 0.833333 |
The integers from $-15$ to $9$, inclusive, are arranged to form a $5$-by-$5$ square such that the sum of the numbers in each row, each column, and each of the main diagonals is the same, calculate the value of this common sum. | -15 | 0.833333 |
Determine the number of pairs of regular polygons whose sides are of unit length and the ratio of their interior angles is $4:3$. | 4 | 0.333333 |
Given Miguel read an average of $48$ pages per day for the first four days, an average of $35$ pages per day for the next five days, $28$ pages per day for the subsequent four days, and read the remaining $19$ pages on the final day, calculate the total number of pages in Miguel’s book. | 498 | 0.833333 |
Given a small baseball league team consists of 18 players, each earning a minimum of $20,000, and the total budget for all players' salaries does not exceed $600,000, calculate the maximum possible salary for a single player. | 260,000 | 0.916667 |
How many sets of three or more consecutive positive integers have a sum of $18$? | 2 | 0.916667 |
Consider a sequence of four-digit integers where each integer has the property that the hundreds, tens, and units digits become, respectively, the thousands, hundreds, and tens digits of the next term, and the hundreds, tens, and units digits of the last term become, respectively, the thousands, hundreds, and tens digits of the first term. Find the largest prime factor that always divides the sum of all the terms in the sequence. | 101 | 0.75 |
Given the equation $x^4y^4 - 20x^2y^2 + 64 = 0$, find the number of distinct ordered pairs $(x, y)$ where $x$ and $y$ are positive integers. | 5 | 0.916667 |
Given the expression $(1+x^3)(1-x^4+x^5)$ evaluate the expression. | 1 + x^3 - x^4 + x^5 - x^7 + x^8 | 0.666667 |
A circle of radius $r$ is concentric with and outside a regular square of side length $2$. What is the radius $r$ such that the probability that two entire sides of the square are visible from a randomly chosen point on the circle is $1/2$?
A) $\sqrt{2}$
B) $2$
C) $2\sqrt{2}$
D) $4$
E) $4\sqrt{2}$ | 2 | 0.25 |
If \(x+6\) cows give \(x+9\) cans of milk in \(x+4\) days, determine how many days will it take \(x+4\) cows to give \(x+11\) cans of milk. | \frac{(x+11)(x+6)}{x+9} | 0.666667 |
Evaluate the expression: $\left(2\left(3\left(2\left(3\left(2\left(3 \times (2+1) \times 2\right)+2\right)\times 2\right)+2\right)\times 2\right)+2\right)$. | 5498 | 0.75 |
The greatest prime number that is a divisor of 65,536 is 2, and for 65,535, determine the sum of the digits of the greatest prime number that is a divisor of 65,535. | 14 | 0.75 |
Calculate the angle formed by the hands of a clock at 3:30, providing the larger of the two possible angles. | 285^\circ | 0.916667 |
A rectangular sheet of metal measures $12$ in. by $16$ in. Squares with side length $x$ inches are cut from each corner, and the sides are then folded to form an open box. Additionally, to ensure the structural integrity of the box, the side length $x$ must not exceed the width of the material divided by $5$. Determine the expression for the volume of the box. | 4x^3 - 56x^2 + 192x | 0.5 |
If the digit $5$ is placed after a two-digit number whose tens' digit is $t$ and units' digit is $u$, find the new number. | 100t + 10u + 5 | 0.416667 |
Given positive integers A, B, and C that are co-prime, satisfying the equation $A \log_{100} 5 + B \log_{100} 2 = C$, calculate the sum A + B + C. | 5 | 0.583333 |
Given that the average score of 20 students was 75, and the average score of the 21 students was 76, find Jessica's test score. | 96 | 0.833333 |
A woman buys a property for $150,000 with a goal to achieve a $7\%$ annual return on her investment. She sets aside $15\%$ of each month's rent for maintenance costs, and pays property taxes at $0.75\%$ of the property's value each year. Calculate the monthly rent she needs to charge to meet her financial goals. | 1139.71 | 0.166667 |
If four times the larger of two numbers is three times the smaller and the difference between the numbers is 12, find the larger of the two numbers. | -36 | 0.25 |
The base of isosceles triangle $\triangle PQR$ is 30, and its area is 90. Find the length of one of the congruent sides. | 3\sqrt{29} | 0.75 |
Determine the number of positive integers less than $500$ that are divisible by neither $5$ nor $7$. | 343 | 0.916667 |
Determine the number of one-foot cubical blocks Maria uses to construct a fort with exterior dimensions of 15 feet in length, 12 feet in width, and 7 feet in height, given that the floor and the four walls are all 1.5 feet thick. | 666 | 0.166667 |
Samantha has moved to a new house which is 3 blocks east and 4 blocks north of the southeast corner of City Park. Her school is now 4 blocks west and 3 blocks south of the northwest corner of City Park. To get to school, she bikes to the southeast corner of City Park, walks a diagonal path across the park to the northwest corner, and then continues by bike to school. Determine the number of different routes she can take if her route remains as short as possible. | 1225 | 0.333333 |
Given the sum $2^{11}+7^{13}$, find the smallest prime number that divides the sum. | 3 | 0.916667 |
Let $R$ denote the set of students owning a rabbit and $G$ denote the set of students owning a guinea pig. Given that $|R|=35$ and $|G|=40$, and $|R|+|G|=50$, determine the number of students owning both a rabbit and a guinea pig. | 25 | 0.916667 |
A pyramid-like stack with a rectangular base containing $6$ apples by $9$ apples is constructed, with each apple above the first level fitting into a pocket formed by four apples below, until no more apples can be fit in a new layer. Determine the total number of apples in the completed stack. | 154 | 0.583333 |
Given $150^8$, determine the total number of positive integer divisors that are perfect squares or perfect cubes, or both. | 267 | 0.166667 |
Let \( r \) represent the result of tripling both the base and exponent of \( a^b \), where \( b \) is non-zero. If \( r \) equals the product of \( a^b \) by \( x^b \), determine the value of \( x \). | 27a^2 | 0.833333 |
A rectangular box has a total surface area of 118 square inches and the sum of the lengths of all its edges is 52 inches. Find the sum of the lengths in inches of all of its interior diagonals. | 4\sqrt{51} | 0.916667 |
Cagney can frost a cupcake every 15 seconds, and Lacey can frost a cupcake every 25 seconds. Lacey starts working 30 seconds after Cagney starts. Find the number of cupcakes that they can frost together in 10 minutes. | 62 | 0.75 |
What is the tens digit of $2023^{2024} - 2025^{2}$? | 1 | 0.916667 |
Given that $\log_b 243$ is a positive integer, how many positive integers $b$ satisfy this condition? | 2 | 0.833333 |
Given the base and height of a triangle are in an arithmetic progression, and the base is \(2a - d\) while the height is \(2a + d\), where \(d\) is the common difference and \(a\) is a positive number, determine the area of the triangle. | 2a^2 - \frac{d^2}{2} | 0.833333 |
In the $xy$-plane, the segment with endpoints $(-3,0)$ and $(27,0)$ is the diameter of a circle. A vertical line $x=k$ intersects the circle at two points, and one of the points has a $y$-coordinate of $12$. Find the value of $k$. | 21 | 0.666667 |
Given the numbers $2^{10} + 1$ and $2^{16} + 1$, find the number of perfect cubes between these two values, inclusive. | 30 | 0.833333 |
Given Brianna uses one fourth of her money to buy one half of the books she plans to purchase, and she spends an equal amount in the next transaction to buy the remaining books, determine the fraction of her money that Brianna will have left. | \frac{1}{2} | 0.833333 |
Given the expression $\left[ \sqrt [4]{\sqrt [8]{b^{16}}} \right]^6\left[ \sqrt [8]{\sqrt [4]{b^{16}}} \right]^6$, simplify the expression. | b^6 | 0.916667 |
Find the non-zero real value of $x$ that satisfies the equation $(9x)^{18} = (27x)^9$. | \frac{1}{3} | 0.833333 |
Let vertices $A, B, C$, and $D$ form a regular tetrahedron with each edge of length 1 unit. Define point $P$ on edge $AB$ such that $P = tA + (1-t)B$ for some $t$ in the range $0 \leq t \leq 1$ and point $Q$ on edge $CD$ such that $Q = sC + (1-s)D$ for some $s$ in the range $0 \leq s \leq 1$. Determine the minimum possible distance between $P$ and $Q$. | \frac{\sqrt{2}}{2} | 0.5 |
In her youth, Lina could cycle 20 miles in 2 hours, and now she cycles 12 miles in 3 hours. Calculate the difference in her cycling time per mile now compared to her youth. | 9 | 0.416667 |
Given $\log_4{5}=p$ and $\log_5{7}=q$, calculate $\log_{10}{7}$ in terms of $p$ and $q$. | \frac{2pq}{2p+1} | 0.083333 |
Given that in triangle PQR, PS is the altitude to QR, and the angle ∠QRP is 30°, find the length of PR in terms of PS. | 2PS | 0.75 |
A contractor estimated that one of his two bricklayers would take $8$ hours to build a certain wall and the other $12$ hours. However, he knew from experience that when they worked together, their combined output fell by $12$ bricks per hour. Being in a hurry, he put both men on the job and found that it took them exactly 6 hours to build the wall. Determine the number of bricks in the wall. | 288 | 0.916667 |
Given that two fifths of Jamie's marbles are blue, one third of her marbles are red, and four of them are green, determine the smallest number of yellow marbles Jamie could have. | 0 | 0.25 |
Emily cycles at a constant rate of 15 miles per hour, and Leo runs at a constant rate of 10 miles per hour. If Emily overtakes Leo when he is 0.75 miles ahead of her, and she can view him in her mirror until he is 0.6 miles behind her, calculate the time in minutes it takes for her to see him. | 16.2 | 0.416667 |
Given a right circular cone and a sphere have volumes related such that the volume of the cone is one-third that of the sphere, and the radius of the base of the cone is twice the radius of the sphere, determine the ratio of the altitude of the cone to the radius of its base. | \frac{1}{6} | 0.833333 |
Given that $3z$ varies inversely as the square of $w$ and directly as $v$, and given that when $z = 15$ and $w = 2$, $v = 4$, find $z$ when $w = 4$ and $v = 8$. | 7.5 | 0.833333 |
Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. Calculate the probability that the sum of the die rolls is odd. | \frac{7}{16} | 0.5 |
The side length of a cube is 2 units. Four of the eight vertices of this cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the tetrahedron to the surface area of the cube. | \frac{\sqrt{3}}{3} | 0.75 |
What is the largest quotient that can be obtained using two numbers from the set $\{ -30, -4, 0, 3, 5, 10 \}$? | 7.5 | 0.25 |
Given the 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 speed in meters per second. | \frac{16\pi}{48} = \frac{\pi}{3} | 0.75 |
Given a positive integer n that has 72 divisors and 5n has 90 divisors, calculate the greatest integer k such that 5^k divides n. | 3 | 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 |
Given points $P$ and $Q$ are $8$ units apart in a plane, determine the number of lines containing $P$ and $Q$ that are $4$ units from $P$ and $6$ units from $Q$. | 2 | 0.583333 |
Determine the number of ways a student can schedule four mathematics courses — algebra, geometry, number theory, and statistics — on an 8-period day, given that no two mathematics courses can be scheduled in consecutive periods. | 120 | 0.833333 |
Two candles of the same height are lighted at the same time. The first is consumed in $5$ hours and the second in $4$ hours. Assuming that each candle burns at a constant rate, determine in how many hours after being lighted was the first candle three times the height of the second. | \frac{40}{11} | 0.916667 |
While Jessica and Shawn are fishing 2 miles from the shore, their boat begins to take on water at a constant rate of 12 gallons per minute. The boat will sink if it accumulates more than 50 gallons of water. Jessica starts rowing towards the shore at a constant rate of 3 miles per hour while Shawn bails water out of the boat. Determine the slowest rate, in gallons per minute, at which Shawn can bail if they are to reach the shore without sinking. | 10.75 | 0.916667 |
Three years ago, Tom was twice as old as his sister Sara, and five years before that, Tom was three times as old as Sara.
Let $t$ and $s$ represent Tom's current age and Sara's current age, respectively. Determine the number of years until the ratio of their ages will be 3:2. | 7 | 0.166667 |
Three eighths of a gallon of paint is used on the first day. One quarter of the remaining paint is used on the second day. Determine the fraction of the original amount of paint available to use on the third day. | \frac{15}{32} | 0.916667 |
A digital watch displays hours and minutes in a 24-hour format. Calculate the largest possible sum of the digits in this display. | 24 | 0.083333 |
The ratio of the width to the length of a rectangle is $3$ : $5$. If the rectangle has a diagonal of length $10d$, find the constant $k$ such that the area of the rectangle can be written as $kd^2$. | \frac{750}{17} | 0.583333 |
Let $r$ be the number that results when both the base and the exponent of $a^b$ are doubled, where $a,b>0$. If $r$ equals the product of $a^b$ and $x^{2b}$ where $x>0$, find $x$. | 2 \sqrt{a} | 0.416667 |
Given quadrilateral $\Box FRDS$ with $\triangle FDR$ being a right-angled triangle at point $D$, with side lengths $FD = 3$ inches, $DR = 4$ inches, $FR = 5$ inches, and $FS = 8$ inches, and $\angle RFS = \angle FDR$, find the length of RS. | \sqrt{89} | 0.25 |
Evaluate the sum $\frac{4}{3} + \frac{7}{6} + \frac{13}{12} + \frac{25}{24} + \frac{49}{48} + \frac{97}{96} - 8$. | -\frac{43}{32} | 0.666667 |
What is the largest sum of two 3-digit numbers that can be obtained by placing each of the six digits `1, 2, 3, 7, 8, 9` in one of the six boxes in this addition problem? | 1803 | 0.833333 |
A sequence of diagrams consists of equilateral triangles, each subsequently divided into smaller equilateral triangles. In the first diagram, there are no shaded triangles. In the second diagram, 1 out of 4 small triangles are shaded. In the third diagram, 4 out of 9 small triangles are shaded. If this pattern of shading continues, determine the fraction of the small triangles in the sixth diagram that would be shaded. | \frac{25}{36} | 0.25 |
Given that Tom had two containers, and the first was $\frac{3}{5}$ full of oil and the second was empty, then he poured all the oil from the first container into the second container, after which the second container was $\frac{2}{3}$ full of oil. What is the ratio of the volume of the first container to the volume of the second container? | \frac{10}{9} | 0.916667 |
A point P is located 15 inches away from the center of a circle. A secant from P intersects the circle at points Q and R such that the external segment PQ is 11 inches and segment QR is 5 inches. Find the radius of the circle. | 7 | 0.833333 |
Given Alice has $4q + 3$ quarters and Bob has $2q + 8$ quarters. Find the difference in their money in nickels. | 10q - 25 | 0.75 |
Given that \( \text{rem} \left(\frac{5}{7}, \frac{3}{4}\right) \) must be calculated, determine the value of the remainder. | \frac{5}{7} | 0.5 |
The sum of six integers is $20$. What is the maximum number of the six integers that can be larger than $15$? | 5 | 0.083333 |
Consider the system of equations where $3x - 4y - 2z = 0$ and $x + 4y - 10z = 0$, with the condition $z \neq 0$. Calculate the numerical value of $\frac{x^2 + 4xy}{y^2 + z^2}$. | \frac{96}{13} | 0.916667 |
Find the area of the irregular quadrilateral formed by the vertices at points (2,1), (4,3), (7,1), and (4,6). | 7.5 | 0.75 |
The taxi fare in Metropolis City is $3.00 for the first $\frac{3}{4}$ mile and additional mileage charged at the rate of $0.25 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate the total distance you can ride for $15. | 4.35 | 0.5 |
Let $x$, $x$, and $x$ represent the ages of Liam's twin sisters and Liam. If the product of their ages is 72, then find the sum of their ages. | 14 | 0.083333 |
Let $x = -2023$. Find the value of $\bigg| \, ||x|-x|-|x| \, \bigg| - x$. | 4046 | 0.916667 |
Al is diagnosed with algebritis and has been prescribed to take one green pill and one pink pill each day for 18 days. Each green pill costs $2 more than each pink pill, and the total cost for all the pills over the 18 days comes to $\textdollar 738$. Given this information, calculate the cost of one green pill. | 21.5 | 0.916667 |
Approximately how many sheets of the type of paper would be in a stack 10 cm high? | 1000 | 0.583333 |
A half-sector of a circle with a radius of 6 inches, along with its interior, is rolled up to form the lateral surface area of a right circular cone by taping together along the two radii. Calculate the volume of the cone in cubic inches. | 9\pi\sqrt{3} | 0.916667 |
Consider a 6-digit palindrome $m$ formed as $\overline{abccba}$ and chosen uniformly at random. Every digit ($a$, $b$, $c$) can be any digit from 0 to 9, except $a$, which cannot be 0 because $m$ is a 6-digit number. What is the probability that both $m$ and $m+11$ are palindromes?
A) $\frac{800}{900}$
B) $\frac{9}{10}$
C) $\frac{7}{9}$
D) $\frac{8}{9}$
E) $\frac{9}{9}$ | \frac{8}{9} | 0.666667 |
In a physics lab experiment, a certain fixed natural constant $k$ is measured as $3.56897$ with an accuracy of $\pm 0.00145$. The researcher aims to declare a value of $k$ where every reported digit is significant. The announced value must precisely reflect $k$ when it is rounded to that number of digits. What is the most accurate value that the researcher can announce for $k$? | 3.57 | 0.083333 |
Evaluate the limit as $x \to \infty$ of the expression $\log_4{(8x-7)} - \log_4{(3x+2)$. | \log_4 \left(\frac{8}{3}\right) | 0.333333 |
Given that a line $x = k$ intersects the graph of $y = \log_2 x$ and the graph of $y = \log_2 (x + 6)$ and the distance between the points of intersection is $1$, determine the sum of the integers $a$ and $b$ in the expression $k = a + \sqrt{b}$. | 6 | 0.916667 |
Given that three spinners X, Y, and Z are spun, determine the probability that the sum of the numbers shown is an even number, where spinner X lands on 2, 5, or 7, spinner Y lands on 2, 4, or 6, and spinner Z lands on 1, 2, 3, or 4. | \frac{1}{2} | 0.166667 |
Given $945=a_1+a_2\times2!+a_3\times3!+a_4\times4!+a_5\times5!+\ldots$ where $0 \le a_k \le k$ for all $k$, determine the value of $a_4$. | 4 | 0.666667 |
Alan, Beth, and Chris went on a camping trip and decided to share the expenses equally. Alan paid $110, Beth paid $140, and Chris paid $190. To equalize the cost, Alan gave Chris $a$ dollars and Beth gave Chris $b$ dollars after realizing they had forgotten to split the cost of a $60 picnic they also enjoyed. What is $a-b$? | 30 | 0.666667 |
Given a set of test scores is $[55, 68, 72, 72, 78, 81, 81, 83, 95, 100]$, median score $Q_2 = 79.5$, first quartile $Q_1 = 72$, and third quartile $Q_3 = 83$, determine the number of outliers present in this data set. | 2 | 0.583333 |
Determine the number of positive integer divisors of $202^8$ that are either perfect squares, perfect cubes, or both. | 30 | 0.166667 |
The ratio of the length to the width of a rectangle is $3:2$. If the diagonal of the rectangle is $\sqrt{13}x$, express the area in terms of $kd^2$, where $d$ is the diagonal. Find the value of $k$. | \frac{6}{13} | 0.916667 |
In a school drama club of 150 students, each student can either write, direct, or produce. Every student has at least one skill but no student can do all three. There are 60 students who cannot write, 90 students who cannot direct, and 40 students who cannot produce. Determine the number of students who have exactly two of these skills. | 110 | 0.833333 |
Given that for every dollar David spent on sandwiches, Ben spent $50$ cents more, and David paid $15.00$ less than Ben, calculate the total amount they spent on sandwiches together. | 75.00 | 0.916667 |
Given that $(-\frac{1}{343})^{-2/3}$, simplify the expression. | 49 | 0.916667 |
Given that a square $S_1$ has an area of $25$, the area of the square $S_3$ constructed by bisecting the sides of $S_2$ is formed by the points of bisection of $S_2$. | 6.25 | 0.25 |
If John earned a 'B' on $32$ of the first $40$ tests and needs to earn a 'B' on at least $75\%$ of his $60$ tests, determine the maximum number of remaining tests for which he can earn a grade lower than a 'B'. | 7 | 0.916667 |
Given a number 180, find the sum of its two smallest prime factors. | 5 | 0.666667 |
Yesterday, Han drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Jan drove 3 hours longer than Ian at an average speed 15 miles per hour faster than Ian. Han drove 120 miles more than Ian. Calculate the difference in the distances driven by Jan and Ian. | 195 | 0.916667 |
Three positive consecutive integers start with $a+1$. Find the average of the next three consecutive integers that start with this new average. | a + 3 | 0.416667 |
Given the price of a product increased by 30% in January, decreased by 30% in February, increased by 40% in March, and then decreased by y% in April, find the value of y, to the nearest integer, if the price of the product at the end of April is the same as it was at the beginning of January. | 22 | 0.583333 |
If the legs of a right triangle are in the ratio $3:4$, find the ratio of the areas of the two triangles created by dropping an altitude from the right-angle vertex to the hypotenuse. | \frac{9}{16} | 0.416667 |
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