problem
stringlengths 18
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stringlengths 1
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0.92
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|---|---|---|
Find the ordered pair $(x,y)$ if
\begin{align*}
x + y &= (7-x) + (7-y),\\
x - y &= (x-2) + (y-2).
\end{align*}
|
(5,2)
| 0.916667 |
Find the largest value of $c$ such that $4$ is in the range of $f(x) = x^2 + 5x + c$.
|
10.25
| 0.083333 |
Compute $\sqrt{128}\cdot\sqrt{50}\cdot \sqrt[3]{27}$.
|
240
| 0.916667 |
If the two roots of the quadratic $9x^2 + 5x + m$ are $\frac{-5 \pm i\sqrt{371}}{18}$, what is $m$?
|
11
| 0.916667 |
Evaluate $\log_{\sqrt{12}} (1728\sqrt{12})$.
|
7
| 0.916667 |
What multiple of 18 is closest to 2500?
|
2502
| 0.75 |
A convex polyhedron $Q$ has $30$ vertices, $72$ edges, and $44$ faces consisting of $30$ triangular and $14$ quadrilateral faces. Calculate the number of space diagonals in $Q$.
|
335
| 0.833333 |
Chandra has four bowls of different colors (red, blue, yellow, green) and five glasses, with an extra glass having no matching bowl color (purple). If she randomly chooses a bowl and a glass, how many different color pairings are possible considering that one glass color has no pairing dish color?
|
20
| 0.083333 |
If $x$, $y$, and $z$ are positive with $xy=30$, $xz=60$, and $yz=90$, what is the value of $x+y+z$?
|
x + y + z = 11\sqrt{5}
| 0.25 |
Convert $\rm{B}2\rm{F}_{16}$ to a base 10 integer.
|
2863
| 0.916667 |
How many positive integers less than 300 are divisible by 4, 5, and 6?
|
4
| 0.416667 |
What is the smallest positive four-digit integer equivalent to 3 mod 8?
|
1003
| 0.916667 |
Each box in the new grid diagram forms part of a larger layout composed of squares. Determine the total number of different-sized squares that can be traced using the lines in this expanded grid:
[asy]
unitsize(0.2 inch);
for (int i=0; i <= 6; ++i) {
draw((i,0)--(i,6));
}
for (int j=0; j <= 6; ++j) {
draw((0,j)--(6,j));
}
[/asy]
|
91
| 0.25 |
A line is parameterized by a parameter $t$, so that the vector on the line at $t = 1$ is $\begin{pmatrix} 2 \\ 7 \end{pmatrix},$ and the vector on the line at $t = 4$ is $\begin{pmatrix} 8 \\ -5 \end{pmatrix}.$ Find the vector on the line at $t = 5.$
|
\begin{pmatrix} 10 \\ -9 \end{pmatrix}
| 0.583333 |
Jayden borrowed money from his friend to buy a bicycle. His friend agreed to let him repay the debt by gardening, with the payment structure as follows: first hour of gardening earns $\$2$, second hour $\$3$, third hour $\$4$, fourth hour $\$5$, fifth hour $\$6$, sixth hour $\$7$, and then it resets. If Jayden repays his debt by gardening for 47 hours, how much money did he borrow?
|
\$209
| 0.916667 |
Three of the following test scores are Alex's and the other three are Jamie's: 80, 85, 90, 95, 100, 105. Alex's mean score is 85. What is Jamie's mean score?
|
100
| 0.916667 |
Convert the point \((\rho,\theta,\phi) = \left( 15, \frac{5 \pi}{4}, \frac{\pi}{4} \right)\) in spherical coordinates to rectangular coordinates.
|
\left(-\frac{15}{2}, -\frac{15}{2}, \frac{15\sqrt{2}}{2}\right)
| 0.5 |
Olivia's Omelette Oasis offers a range of omelettes that include various fillings: cheese, ham, mushrooms, peppers, onions, tomatoes, spinach, and olives. A customer can choose omelette egg base ranging from one to four eggs, and any combination of fillings. How many different kinds of omelettes can be ordered?
|
1024
| 0.25 |
The graphs of the function $f(x) = 2x + b$ and its inverse function $f^{-1}(x)$ intersect at the point $(-4, a)$. Given that $b$ and $a$ are both integers, what is the value of $a$?
|
-4
| 0.25 |
If $\log (xy^2) = 2$ and $\log (x^3y) = 3$, what is $\log (xy)$?
|
\frac{7}{5}
| 0.916667 |
Determine the range of the function
\[ g(x) = \frac{3(x + 5)(x - 4)}{x + 5}. \]
|
(-\infty, -27) \cup (-27, \infty)
| 0.916667 |
The probability it will rain on Friday is $30\%$, the probability it will rain on Saturday is $50\%$, and the probability it will rain on Sunday is $40\%$. Assuming the probability that it rains on any day is independent of the other days except that if it rains on Saturday, the probability it will rain on Sunday increases to $70\%$. What is the probability it will rain on all three days, expressed as a percent?
|
10.5\%
| 0.916667 |
Find all solutions to the inequality \[\frac{x^2}{(x+2)^2} \ge 0.\]
|
(-\infty, -2) \cup (-2, \infty)
| 0.916667 |
In a sports conference, ten sports star players are seated. The players are from four teams: Cubs (3 players), Red Sox (3 players), Yankees (2 players), and Dodgers (2 players). If teammates insist on sitting together, how many ways can the ten sports stars be seated in a row?
|
3456
| 0.833333 |
If a polynomial of degree $15$ is divided by a polynomial $d(x)$, resulting in a quotient of degree $8$ and a remainder of $2x^3 + 5x^2 + x + 7$, what is $\deg d(x)$?
|
7
| 0.916667 |
Find the sum of the coefficients in the polynomial $3(x^8 - x^5 + 2x^3 - 6) - 5(x^4 + 3x^2) + 2(x^6 - 5)$.
|
-40
| 0.916667 |
The matrix $\mathbf{B} = \begin{pmatrix} 4 & 5 \\ 7 & e \end{pmatrix}$ satisfies
\[\mathbf{B}^{-1} = m \mathbf{B}\]for some constant $m.$ Find the ordered pair $(e,m).$
|
(-4, \frac{1}{51})
| 0.833333 |
Factor the following expression: $294b^3 + 63b^2 - 21b$.
|
21b(14b^2 + 3b - 1)
| 0.666667 |
Define \( h(x) = 7x - 6 \). If \( h(x) = f^{-1}(x) - 5 \) and \( f^{-1}(x) \) is the inverse of the function \( f(x) = cx + d \), find \( 7c + 7d \).
|
2
| 0.916667 |
Determine the area enclosed by the graph of $|2x| + |5y| = 20$.
|
80
| 0.666667 |
In a bag, there are 7 blue chips, 5 yellow chips, and 4 red chips. One chip is randomly drawn from the bag and then placed back into the bag before a second chip is drawn. What is the probability that the two selected chips are of different colors? Express your answer as a common fraction.
|
\frac{83}{128}
| 0.833333 |
What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand of a 12-hour clock at 9:35?
|
77.5^\circ
| 0.916667 |
Expand $(2x^{15} - 4x^7 + 3x^{-3} - 9) \cdot (7x^3)$.
|
14x^{18} - 28x^{10} - 63x^3 + 21
| 0.25 |
The arithmetic mean of these five expressions is 26. What is the value of $x$? $$x + 10 \hspace{.5cm} 17 \hspace{.5cm} 2x \hspace{.5cm} 15 \hspace{.5cm} 2x + 6$$
|
x = 16.4
| 0.916667 |
Calculate the sum of the squares of the coefficients when the expression $3(x^3 - 4x^2 + x) - 5(x^3 + 2x^2 - 5x + 3)$ is fully simplified.
|
1497
| 0.333333 |
Determine the value of `p` for the quadratic equation $5x^2 + 7x + p$ when the roots are given as $\frac{-7 \pm i\sqrt{231}}{10}$.
|
14
| 0.833333 |
Two circles, one of radius 7 inches, the other of radius 3 inches, are tangent at point P. Two bugs start crawling at the same time from point P, one crawling along the larger circle at $4\pi$ inches per minute, the other crawling along the smaller circle at $3\pi$ inches per minute. How many minutes is it before their next meeting at point P?
|
14
| 0.833333 |
Given the table below, calculate the median salary among the 67 employees of a different company:
\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
CEO&1&$\$150{,}000$\\\hline
Senior Vice-President&4&$\$105{,}000$\\\hline
Manager&15&$\$80{,}000$\\\hline
Team Leader&8&$\$60{,}000$\\\hline
Office Assistant&39&$\$28{,}000$\\\hline
\end{tabular}
|
\$28,000
| 0.75 |
Compute the product $(3x^2 - 4y^3)(9x^4 + 12x^2y^3 + 16y^6)$.
|
27x^6 - 64y^9
| 0.416667 |
I run at a constant speed, and it takes me 30 minutes to run to the park from my house. The park is 4 miles away. There is a library exactly midway between my house and the park. How many minutes will it take me to run from my house to the library and then to the park?
|
30 \text{ minutes}
| 0.916667 |
Calculate the value of $7^4 + 4(7^3) + 6(7^2) + 4(7) + 1$.
|
4096
| 0.916667 |
If the parabola defined by $y = bx^2 + 4$ is tangent to the line $y = 2x + 2$, then calculate the constant $b$.
|
\frac{1}{2}
| 0.916667 |
A person has $440.55$ in their wallet. They purchase goods costing $122.25$. Calculate the remaining money in the wallet. After this, calculate the amount this person would have if they received interest annually at a rate of 3% on their remaining money over a period of 1 year.
|
327.85
| 0.166667 |
In parallelogram $EFGH$, the measure of angle $EFG$ is twice the measure of angle $FGH$. Determine the measure of angle $EHG$.
|
120^\circ
| 0.5 |
If \(a + 3b = 27\) and \(5a + 2b = 40\), what is the value of \(a + b\)?
|
\frac{161}{13}
| 0.166667 |
A running track is now composed of three concentric circles. The circumferences of the inner and middle circles differ by $20\pi$ feet, and the circumferences of the middle and outer circles differ by $30\pi$ feet. What is the total width of the track from the innermost to the outermost circle?
|
25 \text{ feet}
| 0.916667 |
The hyperbolas
\[\frac{y^2}{49} - \frac{x^2}{25} = 1\]
and
\[\frac{x^2}{T} - \frac{y^2}{18} = 1\]
have the same asymptotes. Find \(T.\)
|
\frac{450}{49}
| 0.833333 |
What is $(-1)^1+(-1)^2+\cdots+(-1)^{2011}$ ?
|
-1
| 0.916667 |
Fifty slips are placed into a hat, each bearing a number from 1 to 10, with each number entered on five slips. Five slips are drawn from the hat at random and without replacement. Let \( p \) be the probability that all five slips bear the same number. Let \( q \) be the probability that three of the slips bear a number \( a \) and the other two bear a number \( b \neq a \). What is the value of \( q/p \)?
|
450
| 0.25 |
Given that $f(x) = x^2 - 3x + 7$ and $g(x) = 2x + 4$, find the value of $f(g(5)) - g(f(5))$.
|
123
| 0.833333 |
If $a$, $b$, and $c$ are digits and $0.abc$ can be expressed as $\frac{1}{y}$ where $y$ is an integer such that $0<y\le10$, then what is the largest possible value of $a+b+c$?
|
8
| 0.333333 |
A tailor needs to make a pair of pants for Jessica, who provides her waist size in inches. Given that there are $12$ inches in a foot and $25.4$ centimeters in a foot, determine Jessica's waist size in centimeters if she states her waist is $28$ inches. Round your answer to the nearest tenth.
|
59.3\ \text{cm}
| 0.333333 |
Let $g(x) = 3x^4 - 22x^3 + 47x^2 - 44x + 24$. Find the value of $g(5)$ using the Remainder Theorem.
|
104
| 0.916667 |
Determine the count of positive integers less than 150 that satisfy the congruence $x + 20 \equiv 70 \pmod{45}$.
|
4
| 0.75 |
How many four-character license plates can be formed if the plate must start with an even digit, followed by a consonant, then a vowel, and end with a consonant? Assume Y is considered a vowel.
|
12,000
| 0.333333 |
Find the smallest, positive five-digit multiple of $18.$
|
10008
| 0.833333 |
The numbers $x$ and $y$ are inversely proportional. If the sum of $x$ and $y$ is now 60, and $x$ is three times $y$, find the value of $y$ when $x=-12$.
|
-56.25
| 0.333333 |
What is the value of $x + y$ if the sequence $3, 9, 15, \ldots, x, y, 45$ is an arithmetic sequence?
|
72
| 0.583333 |
What is the probability that the same odd number will be facing up on each of four standard eight-sided dice that are tossed simultaneously? Express your answer as a common fraction.
|
\frac{1}{1024}
| 0.916667 |
Determine the greatest common divisor (GCD) of the numbers 4410 and 10800.
|
90
| 0.75 |
Bryan now has some 5 cent stamps and some 6 cent stamps. What is the least number of stamps he can combine to make a total of 60 cents?
|
10
| 0.916667 |
In the right triangle $XYZ$, we know $\angle X = \angle Z$ and $XZ = 12$. What is the perimeter of $\triangle XYZ$?
|
12\sqrt{2} + 12
| 0.166667 |
At a school cafeteria, Sam wants to buy a lunch consisting of one main course, one beverage, and one snack. The table below lists Sam's options available in the cafeteria. How many different lunch combinations can Sam choose from?
\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Courses} & \textbf{Beverages} & \textbf{Snacks} \\ \hline
Burger & Water & Apple \\ \hline
Pasta & Soda & Banana \\ \hline
Salad & Juice & \\ \hline
Tacos & & \\ \hline
\end{tabular}
|
24
| 0.75 |
Ten people decide to attend a basketball game, but five of them are only 1/3 sure that they will stay for the entire time (the other five are certain they'll stay the whole time). What is the probability that at the end, at least 9 people stayed the entire time?
|
\frac{11}{243}
| 0.833333 |
A triangle in a Cartesian coordinate plane has vertices at (3, -3), (8, 4), and (3, 4). Calculate the area of this triangle.
|
17.5 \text{ square units}
| 0.5 |
Climbing the first flight of stairs takes Jimmy 25 seconds, and each following flight takes 10 seconds more than the preceding one. How many total seconds does it take to climb the first seven flights of stairs?
|
385
| 0.916667 |
Find the integer $n$, $0 \le n \le 11$, that satisfies \[n \equiv -5033 \pmod{12}.\]
|
7
| 0.916667 |
If the two roots of the quadratic $5x^2 - 2x + m$ are $\frac{2 \pm i \sqrt{78}}{10}$, what is $m$?
|
4.1
| 0.25 |
Find the number of integers $n$ that satisfy
\[25 < n^2 < 144.\]
|
12
| 0.833333 |
What is the sum of all integer solutions to $|n| < |n-5| < 5$?
|
3
| 0.416667 |
Calculate $(-6)^5 \div 6^2 + 4^3 - 7^2$ and express it as an integer.
|
-201
| 0.916667 |
What is the 150th digit to the right of the decimal point in the decimal representation of $\frac{16}{81}$?
|
0
| 0.416667 |
Evaluate $\cfrac{\left\lceil\cfrac{19}{6}-\left\lceil\cfrac{34}{21}\right\rceil\right\rceil}{\left\lceil\cfrac{34}{6}+\left\lceil\cfrac{6\cdot19}{34}\right\rceil\right\rceil}$
|
\frac{1}{5}
| 0.916667 |
In a pet shelter housing 100 dogs, 20 dogs enjoy apples, 70 dogs enjoy chicken, and 10 dogs enjoy cheese. 7 dogs enjoy both apples and chicken, 3 dogs enjoy both apples and cheese, and 5 dogs enjoy both chicken and cheese. 2 dogs enjoy all three. How many dogs enjoy none of these foods?
|
13
| 0.916667 |
What is the smallest positive integer that leaves a remainder of 6 when divided by 8 and a remainder of 5 when divided by 9?
|
14
| 0.916667 |
What is the coefficient of $x^2$ when $3x^3 - 4x^2 - 2x - 3$ is multiplied by $2x^2 + 3x - 4$ and the like terms are combined?
|
4
| 0.75 |
Kadin makes a larger snowman by stacking four snowballs with radii of 2 inches, 4 inches, 5 inches, and 6 inches. Assuming all snowballs are perfect spheres, calculate the total volume of snow used only for the snowballs with radii greater than 3 inches. Express your answer in terms of $\pi$.
|
540\pi
| 0.916667 |
How many two-digit numbers are there in which the tens digit is greater than the ones digit and the ones digit is odd?
|
20
| 0.5 |
A bin contains 10 black balls and 10 white balls. Four balls are drawn at random. What is the probability of drawing exactly 2 black balls and 2 white balls?
|
\frac{135}{323}
| 0.666667 |
If $n = 2^{12} \cdot 3^{15} \cdot 5^{9}$, how many of the natural-number factors of $n$ are multiples of 360?
|
1260
| 0.833333 |
What is the fourth power of the cube of the third smallest prime number?
|
244140625
| 0.916667 |
What is the first year after 2010 for which the sum of the digits is 8?
|
2015
| 0.916667 |
The sum of the first 1000 terms of a geometric sequence is 500. The sum of the first 2000 terms is 950. Find the sum of the first 3000 terms.
|
1355
| 0.666667 |
Semicircles of diameter 3 inches are lined up as shown. What is the area, in square inches, of the shaded region in an 18-inch length of this pattern? Express your answer in terms of \(\pi\).
|
\frac{27}{4}\pi
| 0.166667 |
If $\frac{7}{12}$ is expressed in decimal form, what digit is in the 101st place to the right of the decimal point?
|
3
| 0.916667 |
Four concentric circles are drawn with radii of 2, 4, 6, and 8. The inner circle is painted white, the ring around it is black, the next ring is white, and the outer ring is black. What is the ratio of the black area to the white area? Express your answer as a common fraction.
|
\frac{5}{3}
| 0.916667 |
An infinite geometric series has a first term of $18$ and a second term of $6.$ A second infinite geometric series has the same first term of $18,$ a second term of $6+n,$ and a sum of five times that of the first series. Find the value of $n.$
|
9.6
| 0.5 |
Let $p$ and $q$ be the solutions to the equation $3x^2 - 5x - 8 = 0$. Compute the value of $(5p^3 - 5q^3)(p - q)^{-1}$.
|
\frac{245}{9}
| 0.75 |
Marvin has forgotten his bike lock code. The code is a sequence of four numbers: the first number is between 1 and 25 and is a prime number, the second number is a multiple of 4 between 1 and 20, the third number is a multiple of 5 between 1 and 30, and the fourth number is an even number between 1 and 12. How many different codes could Marvin's combination be?
|
1620
| 0.75 |
Compute \(\sqrt{(20)(19)(18)(17) + 1}\).
|
341
| 0.666667 |
Calculate the value of \( n \) for which the sum $$2 + 33 + 444 + 5555 + 66666 + 777777 + 8888888 + 99999999$$ is congruent to \( n \) modulo 9, where \( 0 \leq n < 9 \). Additionally, find out if \( n \) is even or odd.
|
6
| 0.75 |
Find the distance from the point $(0,3,-1)$ to the line passing through $(1,-2,0)$ and $(2,-5,3)$.
|
2\sqrt{2}
| 0.666667 |
What is the value of $102^{4} - 4 \cdot 102^{3} + 6 \cdot 102^2 - 4 \cdot 102 + 1$?
|
104060401
| 0.666667 |
The greatest common divisor of 24 and some number between 70 and 90 is 6. What is the smallest such number?
|
78
| 0.833333 |
The ratio of irises to roses in Carter's garden is 3:7. He currently has 42 roses. He plans to add 35 more roses and needs to add enough irises to maintain the same ratio. How many irises will he need in total after this addition?
|
33
| 0.416667 |
Calculate $\sqrt{50} \cdot \sqrt{18} \cdot \sqrt{8}$.
|
60\sqrt{2}
| 0.916667 |
What is the base-10 integer 624 when expressed in base 7?
|
1551_7
| 0.833333 |
Calculate the area, in square units, of a triangle with vertices at $(1,1)$, $(1,8)$, and $(10,15)$. Use the determinant method for a solution.
|
31.5
| 0.333333 |
Given that $\frac{a}{45-a}+\frac{b}{85-b}+\frac{c}{75-c}=9$, evaluate $\frac{9}{45-a}+\frac{17}{85-b}+\frac{15}{75-c}$.
|
2.4
| 0.083333 |
What is the smallest integer value of $x$ for which $3x^2 - 4 < 20$?
|
-2
| 0.916667 |
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