problem
stringlengths 11
2.6k
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stringlengths 1
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stringclasses 10
values | domain
listlengths 1
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| llama8b_solve_rate
float64 0.11
0.89
|
|---|---|---|---|---|
Given the equations $a^{2}=16$ and $b^{3}=-27$, and the condition $|a-b|=a-b$, find the value of $a+b$. Express your answer as a single integer.
|
1
|
big_math
|
[
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.75
|
Given that the solutions of the quadratic equation in $x$, $a\left(x-h\right)^{2}+k=0$, are $x_{1}=-1$ and $x_{2}=3$, then the solutions of the equation $a\left(x-h-1\right)^{2}+k=0$ are ____.
|
x_1=0, x_2=4
|
cn_k12
|
[
"Mathematics -> Algebra -> Algebraic Expressions -> Equations and Inequalities"
] | 0.109375
|
What is the last digit of \(2^{2^{1989}} + 1\)? Express your answer as a single digit.
|
7
|
big_math
|
[
"Mathematics -> Number Theory -> Other"
] | 0.28125
|
A plane parallel to the base of a pyramid cuts the pyramid, resulting in a frustum whose upper and lower base areas have a ratio of 1:4. If the height of the cut pyramid is 3cm, then find the height of the frustum. Express your answer in centimeters (cm) as a whole number.
|
3
|
big_math
|
[
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.203125
|
If k is an integer and 0.000010101 x 10 ^ k is greater than 10000, what is the least possible value of k?
|
9
|
orca_math
|
[
"Mathematics -> Algebra -> Other"
] | 0.171875
|
Given a line and two points \(A\) and \(B\) on the same side of it. Find a point \(M\) on the line such that the sum \(MA + MB\) equals a given segment.
|
M
|
olympiads
|
[
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.3125
|
Given the complex number $z= \frac{1+i}{3-4i}$, find the value of $|\bar{z}|$. Express your answer as a fraction with a square root.
|
\frac{\sqrt{2}}{5}
|
big_math
|
[
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625
|
60 % antifreeze solution and another percentage of antifreeze solution were combined to give 39 gallons of 58 % antifreeze solution. The amount of 60% solution used was 26 gallons. What was the percentage of the second antifreeze solution?
|
54\%
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125
|
The function y = 2x is shifted down by 2 units. What is the resulting function expression? Express your answer as a function of x.
|
y = 2x - 2
|
big_math
|
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.765625
|
In triangle $ABC$, if $BC=3$, $AC= \sqrt {3}$, and $∠A= \frac {π}{3}$, then $∠B=$ ______.
|
∠B = \\frac{\\pi}{6}
|
cn_k12
|
[
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.28125
|
in a group of 28 junior high school students , 5 take french , 10 take spanish , and 4 take both languages . the students taking both french and spanish are not counted with the 7 taking french or the 10 taking spanish . how many students are not taking either french or spanish ?
|
17
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.796875
|
If 20% of a number equals a certain value, then 120% of that number is 1800. What is the value?
|
300
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625
|
Simplify first, then evaluate: $(\frac{x^2}{x-2}+\frac{4}{2-x})÷\frac{x^2+4x+4}{x}$, choose appropriate numbers from $-2$, $0$, $1$, $2$ to substitute and calculate.
|
\frac{1}{3}
|
cn_k12
|
[
"Mathematics -> Algebra -> Intermediate Algebra -> Rational Expressions"
] | 0.171875
|
There were 3 roses in the vase from other sources. Alyssa cut some roses from her garden and added them to the vase. If Alyssa wants the ratio of roses from her garden to roses from other sources in the vase to be 5:1, how many roses does she need to cut from her garden?
|
15
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875
|
Two congruent regular pentagons and a triangle are shown in the diagram. The angles marked \( x^\circ \) are equal. Find the measure of angle \( x \) in degrees.
|
36^\circ
|
big_math
|
[
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.359375
|
Let $i$ be the imaginary unit. If the complex number $\frac{{2+ai}}{{1-i}}$ is a pure imaginary number, then the real number $a=$____.
|
2
|
cn_k12
|
[
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.46875
|
In a factory, there are 86 workers, and on a certain day, some workers were present. The percentage that showed up for work is 83.7% (rounded to the nearest tenth). How many workers were present on that day?
|
72
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125
|
Given vectors $\overrightarrow{a}=(-2,1)$ and $\overrightarrow{b}=(m,3)$, and the condition that $\overrightarrow{a}∥\overrightarrow{b}$, find the vector $\overrightarrow{a}-\overrightarrow{b}$. Express your answer as a vector in the form (x, y).
|
(4, -2)
|
big_math
|
[
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.40625
|
The number of students in each section of a school is 24. After admitting new students, three new sections were started. Now, the total number of sections is 16 and there are 21 students in each section. How many new students were admitted?
|
24
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125
|
Before James starts the strength phase training of his cycle, he has a powerlifting total of 2200 pounds at a bodyweight of 245 pounds. During the strength phase, he gains different percentages on each of his three powerlifting exercises: 12% on his squat, 10% on his bench press, and 18% on his deadlift. Additionally, he gains 8 pounds of body weight. Considering these changes, determine the new ratio of his lifting total to bodyweight after the strength phase. Assume that his initial squat, bench press, and deadlift are in the following proportion: 4x, 3x, and 5x.
|
9.91
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.34375
|
It is said that when asked how many students he had, the ancient Greek mathematician Pythagoras answered as follows: "Half of my students study mathematics, a quarter study nature, a seventh spend their time in silent contemplation, and the remaining part consists of 3 maidens." How many students did Pythagoras have?
|
28
|
olympiads
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.796875
|
Given sets $A=\{x|x^2-4x+3<0\}$ and $B=\{x|2<x<4\}$, calculate the intersection $A\cap B$. Express your answer as an interval, using parentheses to denote an open interval.
|
(2,3)
|
big_math
|
[
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.671875
|
How many points $(x, y)$, with $x$ and $y$ both integers, are on the line with equation $y=4x+3$ and inside the region bounded by $x=25, x=75, y=120$, and $y=250$?
|
32
|
omnimath
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875
|
In \\(∆ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively. Given that \\(8b=5c\\) and \\(C=2B\\), find \\(\\cos C\\).
|
\dfrac{7}{25}
|
cn_k12
|
[
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125
|
The insurance contract provides an insured amount of 500,000 rubles and a deductible of 1%. The actual damage amounted to 4,000 rubles. What will be the amount of insurance compensation?
|
0
|
olympiads
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.6875
|
What is the maximum possible number of equilateral triangles that can be formed on a plane by the intersection of six lines?
|
8
|
olympiads
|
[
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.1875
|
Given circle $C: x^{2}+y^{2}=4$, the equation of the line passing through point $P(1,1)$ with the shortest chord length cut by circle $C$ is ______.
|
x+y-2=0
|
cn_k12
|
[
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.390625
|
Given a geometric sequence $\{a_n\}$ with $a_n > 0$ and $q > 1$, where $a_3 + a_5 = 20$ and $a_2 \cdot a_6 = 64$, find the sum of the first six terms of the sequence, $S_6$. Express your answer as a single integer.
|
63
|
big_math
|
[
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625
|
Joy fosters dogs. The mom foster dog eats 1.5 cups of food, three times a day. The puppies each eat 1/2 cup of food, twice a day. There are 5 puppies. How much food will Joy need for a certain number of days if she needs 57 cups of food in total?
|
6
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125
|
Faye, her mom, and her two siblings were picking carrots from their garden. Faye picked 23, her mom picked 42, her brother picked 12, and her sister picked 18. If they found out that 60% of the carrots were good, how many bad carrots did they have?
|
38
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875
|
Find the smallest number $y$ such that $y^2 = 81$.
|
-9
|
math
|
[
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.5625
|
The orbital height of the Chinese space station is approximately 400,000 meters. Express this number in scientific notation in the format a × 10^b, where a is a number between 1 and 10, and b is an integer.
|
4 \times 10^5
|
big_math
|
[
"Mathematics -> Applied Mathematics -> Other"
] | 0.59375
|
Given two lines $l_1: \sqrt{3}x - y + 2 = 0$, $l_2: 3x + \sqrt{3}y - 5 = 0$, the angle between line $l_1$ and $l_2$ is \_\_\_\_\_\_.
|
\frac{\pi}{3}
|
cn_k12
|
[
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.796875
|
Compute: $$\lim_{n\to \infty } \frac {1-2n}{n+2}$$=\_\_\_\_\_\_.
|
-2
|
cn_k12
|
[
"Mathematics -> Precalculus -> Limits"
] | 0.75
|
What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?
|
180
|
math
|
[
"Mathematics -> Number Theory -> Other"
] | 0.21875
|
Willie came back from Europe with 70 euros. Normally a certain amount of euros is worth 1 dollar, but the money exchange at the airport will only give Willie 5/7ths of the official exchange rate. Willie gets 10 dollars. How many euros are worth 1 dollar at the official exchange rate?
|
5
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125
|
In the arithmetic sequence {a_n}, a_3 and a_15 are the two roots of the equation x^2 - 6x + 8 = 0. Find the sum of a_7 + a_8 + a_9 + a_10 + a_11. Express your answer as a single integer.
|
15
|
big_math
|
[
"Mathematics -> Algebra -> Sequences and Series"
] | 0.21875
|
Find the maximum value of the function y = -x - 9/x + 18 for x > 0. Express your answer as a single number.
|
12
|
big_math
|
[
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.859375
|
Frank is practicing a new dance move. It starts with him take 5 steps back, and then 10 steps forward, and then 2 steps back, and then double that amount forward. How many steps forward is Frank from his original starting point?
|
7
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125
|
What is the coefficient of \( x^{3} \) in the binomial expansion of \( (2x+1)^{5} \)?
|
80
|
cn_k12
|
[
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.859375
|
In front of the theater box office, there are $2n$ people lined up to buy tickets. Each ticket costs 5 yuan. Among these people, $n$ people each have a 5 yuan bill, and the other $n$ people each have a 10 yuan bill. The box office does not have any change to give. How many ways can the people line up such that everyone can buy a ticket without encountering difficulties in providing change?
|
\frac{(2n)!}{n!(n+1)!}
|
olympiads
|
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.34375
|
Given that $\{a_n\}$ is an arithmetic sequence with the first term $2$ and a non-zero common difference, if $a_1$, $a_3$, $a_6$ form a geometric sequence, calculate the sum of the first $9$ terms of $\{a_n\}$. Express your answer as a single integer.
|
36
|
big_math
|
[
"Mathematics -> Algebra -> Sequences and Series"
] | 0.25
|
Given $|a| = 1$, then $a^{2014} =$ _____
|
1
|
cn_k12
|
[
"Mathematics -> Algebra -> Complex Numbers"
] | 0.875
|
If the sum of the coefficients in the expansion of $(4\sqrt{x} + \frac{1}{x})^n$ is 125, then the constant term in the expansion is ______.
|
48
|
cn_k12
|
[
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.125
|
In $\triangle ABC$, it is known that $AB=2$, $BC=5 \sqrt {3}$, $\cos B= \frac {4}{5}$, then the area of $\triangle ABC$ is _______.
|
3 \sqrt {3}
|
cn_k12
|
[
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.609375
|
Point A(-3,-5) is moved 4 units up and then 3 units to the left to point B. Find the coordinates of point B. Express your answer as an ordered pair (x, y).
|
(-6, -1)
|
big_math
|
[
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.421875
|
Bill is golfing with 3 friends and can either buy generic golf tees that are packaged by the dozen or the higher quality aero flight tees that come by a certain amount. The minimum number of packages of aero flight tees Bill must purchase to ensure that he has at least 20 golf tees for each member of his foursome, if he will buy no more than 2 packages of the generic golf tees, is 28. How many aero flight tees are in one package?
|
2
|
orca_math
|
[
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.609375
|
The equation of a circle is given as (x+2)^2 + y^2 = 5. Find the equation of the circle that is symmetrical to this circle with respect to the origin (0,0). Express your answer in the standard form of a circle equation.
|
(x-2)^2 + y^2 = 5
|
big_math
|
[
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.515625
|
If a / b = 4 / 3, and the ratio of (3a + 2b) to a certain expression involving a and b is 3, what is that expression?
|
2b
|
orca_math
|
[
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.71875
|
Given a geometric sequence $\{a_n\}$ satisfying $a_1=2$, $q=2$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, and $S_n=126$, find the value of $n$. Express your answer as a whole number.
|
6
|
big_math
|
[
"Mathematics -> Algebra -> Sequences and Series"
] | 0.84375
|
A certain percentage of the population of an ant colony is red, and of these 45% are females. If 46.75% of the total ant population are male red ants, what percentage of the ant population is red?
|
85\%
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875
|
Let the set $M=\{(x,y)|x^2+y^2\leq 4\}$, and $N=\{(x,y)|(x-1)^2+(y-1)^2\leq r^2\}$ where $r>0$. When $M\cap N=N$, the range of the real number $r$ is \_\_\_\_\_\_.
|
(0, 2-\sqrt{2}])
|
cn_k12
|
[
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625
|
Kylie makes 10 beaded necklaces on Monday, 2 beaded necklaces on Tuesday, 5 beaded bracelets on Wednesday, and 3 beaded earrings on Thursday. On Friday, she makes 4 beaded anklets and 6 beaded rings. 20 beads are needed to make one beaded necklace, 10 beads are needed to make one beaded bracelet, 5 beads are needed to make one beaded earring, 8 beads are required to make one beaded anklet, and 7 beads are required to make one beaded ring. How many beads does Kylie use in total to make her jewelry over the week?
|
379
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875
|
A store sells luxury watches listed for rs. 25000 each. If the store offers a 15% discount for the first purchase, a 20% loyalty discount for the second purchase, and an additional 10% holiday discount on the third purchase, how much will the third purchase cost?
|
15300
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125
|
If it takes a machine some time to produce one item, the machine will produce 150 items in 2 hours. How long does it take for the machine to produce one item?
|
0.8
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875
|
In triangle $ABC$, with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$ respectively, given that $a=3$, $C=120^{\circ}$, and the area of triangle $ABC$, $S= \frac {15 \sqrt {3}}{4}$, find the value of $c$. Express your answer as a single number.
|
7
|
big_math
|
[
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.578125
|
Given \(\vec{a}=\{6,4\}\), \(\vec{b}=\{0,2\}\), \(\vec{c}=\vec{a}+m \vec{b}\), and \(|\vec{c}|=10\), find the real number \( m \).
|
2 \text{ or } -6
|
olympiads
|
[
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.703125
|
Are the vectors \(c_{1}\) and \(c_{2}\), constructed from vectors \(a\) and \(b\), collinear?
\(a = \{2, -1, 4\}\)
\(b = \{3, -7, -6\}\)
\(c_{1} = 2a - 3b\)
\(c_{2} = 3a - 2b\)
|
The vectors are not collinear.
|
olympiads
|
[
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.703125
|
Given a geometric sequence {a_n} that satisfies a_1 = 1/4 and a_3a_5 = 4(a_4-1), find the value of a_2. Express your answer as a fraction.
|
\frac{1}{2}
|
big_math
|
[
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.671875
|
There are some boys and 13 girls in a class. If 1 boy is added to the class, the percentage of the class that are girls is 52%. How many boys were originally in the class?
|
11
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875
|
Zach wants to ride the Ferris wheel, the roller coaster, and the log ride. The roller coaster costs 7 tickets, and the log ride costs 1 ticket. Zach has 1 ticket and needs to buy 9 more tickets. How many tickets does the Ferris wheel cost?
|
2
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.734375
|
Given that $0 < x < \frac{\pi}{2}$, and $\tan (x- \frac{\pi}{4})=-\frac{1}{7}$, find the value of $\sin x+\cos x$.
|
\frac{7}{5}
|
cn_k12
|
[
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.234375
|
A boat having a certain length and breadth of 3 m is floating on a lake. The boat sinks by 1 cm when a man gets on it. The mass of the man is 210 kg. What is the length of the boat?
|
7
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.4375
|
Xiao Li participated in a strange math exam with a total of 100 questions. Each correct answer earns 1 point, each wrong answer deducts 3 points, and each unanswered question deducts 2 points. Given that Xiao Li scored a total of 50 points, what is the maximum number of questions Xiao Li could have answered correctly?
|
87
|
olympiads
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125
|
The increasing interval of the function $y=x-2\sin x$ on $[0,\pi]$ is ______.
|
[\frac{\pi}{3},\pi]
|
cn_k12
|
[
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.4375
|
A sample of size $45$ is drawn from high school students in a certain school using stratified sampling. There are $600$ students in the second year of high school, and $15$ students are selected. The total number of high school students in the school is ______.
|
1800
|
cn_k12
|
[
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.8125
|
Convert the binary number $1001101_{(2)}$ to decimal. Express your answer as a single integer.
|
77
|
big_math
|
[
"Mathematics -> Number Theory -> Other"
] | 0.859375
|
Marcus spends 20 minutes giving his dog a bath and half as long blow-drying her. Then he takes her for a walk along a 3-mile trail. If Marcus walks at 6 miles per hour, how much time does he spend with his dog total?
|
60
|
openmath
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75
|
Rocco stores his coins in piles of 10 coins each. He has 4 piles of quarters, 6 piles of dimes, 9 piles of nickels, and 5 piles of pennies. How much money does Rocco have?
|
21
|
openmath
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.625
|
Let the set A consist of positive integers that include only the prime factors 2, 3, or 5. If the sum of the reciprocals of all elements in set A can be expressed as m/n, where m, n are coprime positive integers, find the value of m + n.
|
19
|
big_math
|
[
"Mathematics -> Number Theory -> Other"
] | 0.609375
|
Happy Cattle Ranch is home to 200 cows. Every year, the mother cows have enough calves that the number of cows the rancher owns rises by a certain ratio of the number of cows living on the ranch. If the rancher does not buy or sell any cows, the rancher will own 450 cows on Happy Cattle Ranch in two years. What is the ratio of the number of new calves born each year to the number of cows living on the ranch?
|
0.5,
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.734375
|
On Valentine's Day, each boy in a school gave each girl a valentine. It turned out that the number of valentines was 46 more than the total number of students. How many valentines were given?
|
96
|
olympiads
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125
|
Deepak and his wife start from the same point on a jogging track and walk in opposite directions at 20 km/hr and 13 km/hr respectively. They meet for the first time in 33 minutes. What is the circumference of the jogging track?
|
18.15
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.640625
|
Tom and Michael have some car robots, and Bob has 9 times more than that. Bob has a total of 81 car robots. How many car robots do Tom and Michael have?
|
9
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125
|
Two planes can divide space into at most ______ parts.
|
4
|
cn_k12
|
[
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.28125
|
The selling price of an item is first increased by 15%, then reduced by 20%, and the final selling price is 46 yuan. What was the original price of this item in yuan?
|
50
|
olympiads
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.734375
|
Translate the graph of the function $y=\sin 2x$ by shifting it $\frac{\pi}{8}$ units to the left. Determine the corresponding function expression of the transformed graph.
|
y = \sin\left(2x + \frac{\pi}{4}\right)
|
cn_k12
|
[
"Mathematics -> Precalculus -> Functions -> Other"
] | 0.765625
|
Determine the value of \(a\) such that in the equation
$$
x^{2}+a x+2 a=0
$$
one root is four times the other root.
|
\frac{25}{2}
|
olympiads
|
[
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.78125
|
Compute $\tan 75^\circ.$
|
2 + \sqrt{3}
|
math
|
[
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.8125
|
Given the vector $\overrightarrow{a} = (\cos\theta, \sin\theta)$ and the vector $\overrightarrow{b} = (1, 3)$, and knowing that $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $\frac{\sin\theta + \cos\theta}{\sin\theta - \cos\theta}$.
|
2
|
cn_k12
|
[
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other"
] | 0.765625
|
Given two complex numbers $z_1 = 2+3i$ and $z_2 = a-2+i$, if $|z_1 - z_2| < |z_1|$, find the range of values for the real number $a$.
|
(1, 7)
|
cn_k12
|
[
"Mathematics -> Algebra -> Complex Numbers"
] | 0.140625
|
There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$ -way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of $n$ .
|
4
|
aops_forum
|
[
"Mathematics -> Discrete Mathematics -> Graph Theory"
] | 0.203125
|
A basketball player practices some hours a day Monday through Friday. He must practice 11 hours combined on Saturday and Sunday if he wants to practice an average (arithmetic mean) of 3 hours a day for the 7 day week. How many hours does he practice each day from Monday to Friday?
|
2
|
orca_math
|
[
"Applied Mathematics -> Math Word Problems"
] | 0.84375
|
25 people are arranged in a 5×5 matrix. Now, select 3 people under the condition that these 3 people are not in the same row or column. How many different ways are there to make this selection?
|
600
|
cn_k12
|
[
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.125
|
If three non-identical real numbers \( a, b, c \) satisfy \( a^{3}+b^{3}+c^{3}=3abc \), then \( a+b+c = \)
|
0
|
olympiads
|
[
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.734375
|
Given sin(2α) = 1/4, calculate cos²(α - π/4). Express your answer as a simplified fraction.
|
\frac{5}{8}
|
big_math
|
[
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.296875
|
Given an even function $f(x)$ defined on $\mathbb{R}$ that is monotonically decreasing on $[0,+\infty)$, a number $x$ is randomly selected from $[-4,4]$. The probability of the event "the inequality $f(x-1) \geqslant f(1)$" occurring is ______.
|
\dfrac{1}{4}
|
cn_k12
|
[
"Mathematics -> Applied Mathematics -> Probability"
] | 0.109375
|
Given $\frac{{\cos(\frac{π}{2}-α)-3\cosα}}{\sinα-\cos(π+α)}=2$, calculate the value of $\tan \alpha$. Express your answer as a single numerical value.
|
-5
|
big_math
|
[
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.34375
|
Given that the derivative of the function $f(x)$ is $f′(x)$, and they satisfy the relationship $f(x)=x^{2}+3xf′(2)$, find the value of $f′(2)$. Express your answer as a single number.
|
-2
|
big_math
|
[
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.5625
|
Given the parabola equation y = 4x^2, find the equation of its directrix. Express your answer in the format of the equation of a line (e.g., y =... or x =...).
|
y = -\frac{1}{16}
|
big_math
|
[
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.796875
|
Given a complex number z = 1 - i, calculate the value of 1/z^2 - conjugate(z). Express your answer as a complex number in the form a + bi.
|
-1 - \frac{1}{2}i
|
big_math
|
[
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.265625
|
10 men and 15 women together can complete a work in 6 days. The efficiency of men to complete the work decreases by 5% every day while the efficiency of women to complete the work increases by 3% every day. It takes 100 days for one man alone to complete the same work at his initial efficiency. How many days will be required for one woman alone to complete the same work at her initial efficiency?
|
225
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375
|
There are 6 identical computers ready to be distributed to 5 primary schools, among which schools A and B each want at least 2 computers, while the other schools may receive none. Find the total number of different distribution schemes. Express your answer as a single integer.
|
15
|
big_math
|
[
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.109375
|
Given the set $ M = \{ x \mid -2 \leq x < 2 \} $ and the set $ N = \{ x \mid x^2 - 2x - 3 \geq 0 \} $, find $ M \cap N $.
|
M \\cap N = [-2, -1]
|
cn_k12
|
[
"Mathematics -> Algebra -> Other"
] | 0.171875
|
There was a string that was 70 centimeters (cm) long. Of these, 27 centimeters (cm) was given to Minyoung, and 7/9 of the remaining was used to make (A). How many centimeters (cm) is the length of the remaining string?
|
9.56
|
orca_math
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625
|
Find the exact value of cos(2018π/3). Express your answer as a fraction.
|
-\frac{1}{2}
|
big_math
|
[
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.703125
|
Given the sets $M = \{x | x^2 - 2x < 0\}$ and $N = \{x | x > 1\}$, determine the set $M \cap \lnot_R N$. Express your answer in interval notation.
|
(0, 1]
|
big_math
|
[
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.453125
|
Given that the standard deviation of sample data $x_{1}$, $x_{2}$, $\ldots $, $x_{10}$ is $8$, find the standard deviation of the data after applying the transformation $3x_{i} - 1$ to each data point $x_{i}$. Express your answer as a numerical value.
|
24
|
big_math
|
[
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.796875
|
In $\triangle ABC$, if $|\overrightarrow{AB} + \overrightarrow{AC}| = |\overrightarrow{AB} - \overrightarrow{AC}|$ and $|\overrightarrow{AB}| = 2$, then $\overrightarrow{AB} \cdot \overrightarrow{BC} =$ _____
|
-4
|
cn_k12
|
[
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.34375
|
John decides to replace all his VHS with DVDs. He can trade in his VHS for $2 each and then buy the DVDs for $10 each. If he has 100 movies, how much does it cost to replace his movies?
|
800
|
gsm8k
|
[
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375
|
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