id stringlengths 12 82 | source stringclasses 26
values | question stringlengths 8 1.61k | steps listlengths 3 48 | answer stringlengths 1 634 |
|---|---|---|---|---|
dbb6e346-17d8-5107-804c-31ef3a2b27e6 | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following operations is correct?
A: $a^{5}\div a^{2}=a^{3}$
B: $a^{3}+a^{3}=a^{6}$
C: $(a^{3})^{2}=a^{5}$
D: $\sqrt{{a^2}}=a$ | [
"Okay, let's see. The question is asking which of the operations A through D is correct. I need to check each one step by step. Let me start by recalling some exponent rules.",
"Starting with option A: a⁵ ÷ a² = a³. Hmm, when you divide exponents with the same base, you subtract the exponents, right? So, a^(5-2) ... | Only option A is correct as \(a^{5} \div a^{2} = a^{3}\) follows exponent rules; options B, C, and D are incorrect due to violations of exponent properties or lack of domain restrictions. |
73a5fc6f-76bd-5dfc-8127-63a53d41820b | open-r1/OpenR1-Math-220k/cn_k12 | The opposite of $23$ is ( )
A: $-23$
B: $23$
C: $-\frac{1}{23}$
D: $\frac{1}{23}$ | [
"Okay, let's see. The problem is asking for the opposite of 23, and the options are A: -23, B: 23, C: -1/23, D: 1/23. Hmm, opposite. So, what does opposite mean in math? I think it could mean the additive inverse or maybe the multiplicative inverse. Let me recall. The additive inverse of a number is what you add to... | The opposite of 23 is its additive inverse, \(-23\), as \(23 + (-23) = 0\). The correct answer is \boxed{A}. |
bbe81a48-0158-55fe-8422-0012630deb9e | open-r1/OpenR1-Math-220k/olympiads | 2. Jenny had to add 26 to a certain number. Instead she subtracted 26 and obtained -14 . What number should she have obtained?
A 28
B 32
C 36
D 38
E 42 | [
"Okay, so the problem says that Jenny had to add 26 to a certain number, but instead she subtracted 26 and got -14. We need to find out what number she should have obtained if she did it correctly. The answer choices are A 28, B 32, C 36, D 38, E 42. Let me try to figure this out step by step.",
"First, let's par... | Jenny subtracted 26 from a number \( x \) instead of adding it, getting -14, which means \( x = 12 \). The correct result is \( 12 + 26 = 38 \), corresponding to option D. |
c1521f61-54b9-59ab-a19f-6004c18bfb07 | open-r1/OpenR1-Math-220k/olympiads | ## Task 8
The area of a new playground is square and measures $1600 \mathrm{~m}^{2}$.
How long is one side of the playground?
How many meters of fence are needed for three sides? | [
"Okay, so I need to figure out the length of one side of a square playground that has an area of 1600 square meters. Then, I also need to determine how many meters of fence are required for three sides of this playground. Let me start with the first question.",
"Since the playground is square, all four sides are ... | The side length of the square playground is 40 meters, and the total length of fencing required for three sides is 120 meters. |
e1cc1f33-ea60-585b-9ef2-1cfda7707233 | open-r1/OpenR1-Math-220k/olympiads | ## Task 5
Mr. Müller has manufactured 16 chair legs.
How many chairs can he build? | [
"Okay, so Mr. Müller made 16 chair legs. The question is asking how many chairs he can build with those legs. Hmm, let's see. I need to figure out how many legs each chair requires.",
"From what I know, most chairs have four legs each. Like, a standard chair, right? You know, one at each corner. But wait, maybe s... | Mr. Müller can build 4 chairs by dividing the 16 available legs by the 4 legs required per chair. |
20d46f66-8581-54e1-b6dd-868a9eebd039 | open-r1/OpenR1-Math-220k/olympiads | 29. The numbers $1, 2, 3, 4, 5, 6, 7, 8, 9, 0$ that we use for counting are called Arabic numerals, and they have become the universal symbols for numbers used around the world. Arabic numerals were actually invented by the ancient $\qquad$.
A. Indians
B. China
C. Babylon
D. Arabs | [
"Okay, let's see. The question is about the origin of Arabic numerals. The options are Indians, China, Babylon, and Arabs. Hmm, I remember that Arabic numerals are the numbers we use today, like 1, 2, 3, etc. But the name might be misleading because sometimes things are named after the people who popularized them r... | \boxed{A} |
9503b700-334e-5379-a645-f4dc5285db6a | open-r1/OpenR1-Math-220k/cn_k12 | Calculate: $(-2m^{3})^{2}$ is equal to ( ).
A: $-2m^{6}$
B: $4m^{6}$
C: $4m^{5}$
D: $-4m^{6}$ | [
"Okay, so I need to calculate (-2m³) squared and choose the correct answer from the options given. Let me think about how to approach this.",
"First, when you square a term that has both a coefficient and a variable raised to a power, you need to square each part separately. That is, both the coefficient (-2) and... | Squaring \((-2m^3)\) gives \((-2)^2 \cdot (m^3)^2 = 4m^6\), corresponding to option B. |
0d3d603c-8ffe-5a1b-aef2-fc2fa55c74bd | open-r1/OpenR1-Math-220k/cn_k12 | The domain of the function $f\left(x\right)=\sqrt{x+2}+\dfrac{1}{x-1}$ is ( ).
A: $\left(-\infty ,1\right)\cup \left(1,+\infty \right)$
B: $ \left[-2,+\infty \right)$
C: $ \left[-2,1\right)\cup \left(1,+\infty \right)$
D: $ \left(1,+\infty \right)$ | [
"Okay, let's see. I need to find the domain of the function f(x) = sqrt(x + 2) + 1/(x - 1). Hmm, the domain is all the real numbers x for which the function is defined. Since this function has two parts: a square root and a rational expression, I need to consider the domains of each part separately and then find th... | The domain of \( f(x) = \sqrt{x+2} + \frac{1}{x-1} \) is \( [-2, 1) \cup (1, \infty) \), corresponding to option C. |
fd26fccd-e9b4-59a5-a1ab-ad21c6c4c1bb | open-r1/OpenR1-Math-220k/cn_k12 | The medicine is on sale everywhere. You can get it at \_\_\_ chemist’s.
A. each B. some C. certain D. any | [
"Okay, let's tackle this question. The sentence is: \"The medicine is on sale everywhere. You can get it at _____ chemist’s.\" The options are A. each, B. some, C. certain, D. any.",
"First, I need to figure out which determiner fits best in the blank. The first sentence says the medicine is available everywhere,... | The medicine is available everywhere, meaning it can be obtained at any chemist's without restriction. Option D ("any") correctly conveys this universal availability, while other options imply limited access. |
e6f0869a-29ad-546a-a41b-51833a93b4fa | open-r1/OpenR1-Math-220k/cn_k12 | In the process of deriving a contradiction using the method of proof by contradiction, which of the following should be used as conditions? ( )
① The judgment opposite to the conclusion, i.e., the assumption
② The conditions of the original proposition
③ Axioms, theorems, definitions, etc.
④ The original concl... | [
"Okay, let's see. The question is about proof by contradiction and which conditions should be used in deriving a contradiction. The options are combinations of ①, ②, ③, ④. Let me recall how proof by contradiction works.",
"So, in a proof by contradiction, you start by assuming the opposite of what you want to pro... | In a proof by contradiction, we assume the opposite of the conclusion (①), use the original conditions (②), and apply axioms, theorems, and definitions (③) to derive a contradiction, making ①, ②, and ③ the correct conditions. The original conclusion (④) is not used as a condition but is what we aim to prove. |
62a2d7f9-4094-5161-bcf2-026e799de7c6 | open-r1/OpenR1-Math-220k/cn_k12 | Given that $p$ is a true statement, and $q$ is a false statement, then ( )
A: The statement $p \land q$ is true.
B: The statement $p \lor q$ is false.
C: The statement $\lnot p$ is true.
D: The statement $\lnot q$ is true. | [
"Okay, let's see. I need to figure out which of the options A, B, C, or D is correct given that p is true and q is false. Hmm, okay, let me start by recalling what each logical connective means.",
"First, p is true, q is false. So, let's go through each option one by one.",
"Option A: The statement p ∧ q is tru... | Given \( p \) is true and \( q \) is false, \( p \land q \) is false, \( p \lor q \) is true, \( \lnot p \) is false, and \( \lnot q \) is true, so the correct answer is \(\boxed{D}\). |
715a26a6-4b8b-5a6a-b60b-68d743806463 | open-r1/OpenR1-Math-220k/olympiads | 1. Given the set $M$ satisfies $M \subseteq\{0,1,2,3,4\}$ and $M \subseteq\{0,2,4,8\}$, the maximum number of elements in set $M$ is ( ).
A. 1
B. 2
C. 3
D. 6 | [
"Okay, so I need to figure out the maximum number of elements that set M can have. The problem says that M is a subset of both {0,1,2,3,4} and {0,2,4,8}. Hmm, subsets... right, so M has to be contained within both of these sets. That means M can only have elements that are common to both sets, right? Because if it'... | The intersection of the sets \(\{0,1,2,3,4\}\) and \(\{0,2,4,8\}\) is \(\{0,2,4\}\), so the maximum number of elements in \(M\) is 3, corresponding to option C. |
704b5f6a-7b5f-5a74-908f-803f5dd1d0d8 | open-r1/OpenR1-Math-220k/olympiads | ## Task 1
In every 100 m preliminary race, 8 athletes start. The two best runners will participate in the final. How many athletes participated in the preliminaries if 8 of them competed in the final. | [
"Okay, let's see. So the problem is about figuring out how many athletes participated in the preliminaries if 8 of them ended up competing in the final. Hmm. Let me read the question again to make sure I understand it correctly.",
"\"In every 100 m preliminary race, 8 athletes start. The two best runners will par... | With 8 finalists and 2 qualifying from each preliminary race, there were 4 preliminary races, each with 8 athletes, resulting in 32 athletes total. |
fad768a4-ba62-5b23-a177-2bcb11b8c4b8 | open-r1/OpenR1-Math-220k/cn_k12 | If ¬(p∧q) is a false proposition, then ( )
A: p is true, q is false
B: p is false, q is false
C: p is true, q is true
D: p is false, q is true | [
"Okay, let's see. The problem says that ¬(p ∧ q) is a false proposition. Then we need to figure out which of the options A to D is correct. Hmm, let me recall some logic basics. So, ¬ is the negation symbol, and ∧ is the conjunction, meaning \"and\". So ¬(p ∧ q) being false means that the negation of \"p and q\" is... | Since ¬(p ∧ q) is false, p ∧ q must be true, which requires both p and q to be true, making option C correct. |
185c641b-c732-59d2-b7a1-1b6cbcbea878 | open-r1/OpenR1-Math-220k/olympiads | 3. In the Caribbean, loggerhead turtles lay three million eggs in twenty thousand nests. On average, how many eggs are in each nest?
A 15
B 150
C 1500
D 15000
E 150000 | [
"Okay, so I need to figure out the average number of eggs per nest for these loggerhead turtles in the Caribbean. The problem says there are three million eggs laid in twenty thousand nests. Hmm, averages... right, average is total divided by the number of items. So, if there are 3,000,000 eggs and 20,000 nests, I ... | The average number of eggs per nest is 150, calculated by dividing 3,000,000 eggs by 20,000 nests, which corresponds to option B. |
d66b5d06-0300-56f9-95a4-fca6c2cb9f0d | open-r1/OpenR1-Math-220k/cn_k12 | To make the fraction $\frac{3}{{x-2}}$ meaningful, the value of $x$ should satisfy ______. | [
"Okay, so I need to figure out what value of x makes the fraction 3 divided by (x minus 2) meaningful. Hmm, let's start by recalling what makes a fraction undefined or meaningless. From what I remember, a fraction becomes undefined when its denominator is zero because you can't divide by zero. So, in this case, the... | The fraction \(\frac{3}{x-2}\) is undefined when \(x = 2\), so \(x\) must satisfy \(\boxed{x \neq 2}\). |
10b6d5a6-4d1e-504c-ab78-f691e9d46e29 | open-r1/OpenR1-Math-220k/olympiads | 77. $\int \frac{\sin 2 x}{\cos x} d x$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
77. $\int \frac{\sin 2 x}{\cos x} d x$. | [
"Okay, so I have to solve the integral of sin(2x) divided by cos(x) with respect to x. Let me think about how to approach this.",
"First, I remember that sin(2x) is a trigonometric identity. Maybe I can use that to simplify the expression. The double angle formula for sine is sin(2x) = 2 sinx cosx. Let me write t... | Using the double-angle identity $\sin 2x = 2 \sin x \cos x$, the integral simplifies to $\int 2 \sin x \, dx = -2 \cos x + C$. |
ad0fa471-e751-594e-bc52-00a7ecd2eee3 | open-r1/OpenR1-Math-220k/cn_k12 | Given the set $P = \{x \,|\, x^2 - x - 2 = 0\}$, and the set $T = \{x \,|\, -1 < x \leq 2\}$, find $P \cap T$. | [
"Okay, so I need to find the intersection of sets P and T. Let me start by understanding each set individually.",
"First, set P is defined as all x such that x squared minus x minus 2 equals zero. That's a quadratic equation. Maybe I should solve for x to find the elements of P. Let me write that down: x² - x - 2... | The intersection \( P \cap T \) is \(\{2\}\) since \( P = \{-1, 2\} \) and only \( 2 \) is in the interval \( (-1, 2] \). |
33db7ec2-b91f-59cb-9701-c28cb6058062 | open-r1/OpenR1-Math-220k/cn_k12 | The program obtained by using the bisection method to solve the equation $x^2-2=0$ is ( )
A: Organizational structure diagram
B: Process flow diagram
C: Knowledge structure diagram
D: Program flowchart | [
"Okay, let's see. The question is about which diagram represents the program that uses the bisection method to solve the equation x² - 2 = 0. The options are organizational structure, process flow, knowledge structure, or program flowchart.",
"Hmm, first, I need to remember what each of these diagrams represents.... | A program flowchart (Option D) is the appropriate choice for representing the bisection method algorithm as it visually outlines the control flow, sequences, and logical decisions within the program. Other options are unrelated or typically apply to non-programming contexts. |
504f1c8e-ef9d-5fc3-9d6c-6e59fd3a367e | open-r1/OpenR1-Math-220k/cn_k12 | If the fraction $\frac{x}{2-x}$ is meaningful, then the value of $x$ cannot be:
A: $1$
B: $2$
C: $3$
D: $0$ | [
"Okay, so I have this problem here: If the fraction x over (2 minus x) is meaningful, then the value of x cannot be which of the options A, B, C, or D. The options are 1, 2, 3, and 0. Hmm, let me think about this step by step.",
"First, I remember that for a fraction to be meaningful or defined, the denominator c... | The fraction \( \frac{x}{2-x} \) is undefined when \( x = 2 \), which corresponds to option B. |
9f181d73-ebb5-5415-8d73-b2fecb2ab994 | open-r1/OpenR1-Math-220k/cn_k12 | Among the following descriptions, the one that can determine a point's location is ______.
① East longitude 118°, north latitude 40°
② Baoji City, Wenhua East Road
③ 60° northeast
④ Oscar Cinema, Hall 1, Row 3. | [
"Okay, so I need to figure out which of these four options can determine a point's location. Let me go through each one step by step.",
"First, option ① is \"East longitude 118°, north latitude 40°.\" Hmm, coordinates using longitude and latitude. I remember that longitude and latitude are the standard ways to pi... | The correct answer is ①, as East longitude 118° and north latitude 40° provide precise geographic coordinates that uniquely determine a point on Earth, while other options lack specificity. |
8be917a8-6544-58fc-b0e1-57fef7b5f4f3 | open-r1/OpenR1-Math-220k/cn_k12 | Given that variables $x$ and $y$ are linearly related, and the regression equation is $y = 3 - 2x$, then the relationship between variables $x$ and $y$ is ( )
A: Positive linear correlation
B: Negative linear correlation
C: Linear correlation
D: The positive or negative correlation cannot be determined from the regr... | [
"Okay, let's see here. The problem states that variables x and y are linearly related, and the regression equation is y = 3 - 2x. We need to determine the type of correlation between x and y from the given options. The choices are positive, negative, linear, or undetermined. Hmm.",
"First, I remember that in a re... | The regression equation \( y = 3 - 2x \) has a negative slope of \(-2\), indicating a negative linear correlation between \( x \) and \( y \). The correct answer is option B. |
364969f4-6c71-5c18-b261-c5052dc7acf9 | open-r1/OpenR1-Math-220k/cn_k12 | Given that a normal vector of plane $\alpha$ is $\overrightarrow{n\_1}=(1,2,-2)$, and a normal vector of plane $\beta$ is $\overrightarrow{n\_2}=(-2,-4,k)$, find the value of $k$ if $\alpha$ is parallel to $\beta$. | [
"Okay, let's see. I need to find the value of k such that the two planes α and β are parallel. The normal vector of plane α is n₁ = (1, 2, -2), and the normal vector of plane β is n₂ = (-2, -4, k). Hmm, if two planes are parallel, their normal vectors must also be parallel, right? Because if the planes are parallel... | For the planes to be parallel, their normal vectors must be scalar multiples, giving \( k = 4 \) when \(\lambda = -2\). |
cd253f38-7cbe-546d-b133-698db2f0e2a1 | open-r1/OpenR1-Math-220k/olympiads | You have exactly $\$ 4.40$ ( 440 ) in quarters ( $25 \phi$ coins), dimes ( $10 \phi$ coins), and nickels ( 5 coins). You have the same number of each type of coin. How many dimes do you have?
(A) 20
(B) 11
(C) 10
(D) 12
(E) 4 | [
"Okay, let's see. The problem says I have exactly $4.40, which is 440 cents. The coins are quarters, dimes, and nickels. Each type of coin has the same number. So, if I let the number of each coin be, say, N, then I have N quarters, N dimes, and N nickels.",
"Hmm, right. So the total value from quarters would be ... | The number of each type of coin is 11, and the number of dimes is option B. |
de391b61-b536-5a6a-9c7d-86ceb97a4912 | open-r1/OpenR1-Math-220k/olympiads | $20 \%$ of 20 is equal to
(A) 400
(B) 100
(C) 5
(D) 2
(E) 4 | [
"Okay, let's see. The problem is asking for 20% of 20, and then gives five options: 400, 100, 5, 2, and 4. Hmm, percentages can sometimes be tricky, but I think I remember that \"of\" usually means multiplication in math problems. So, maybe I need to convert 20% into a decimal and then multiply it by 20.",
"First... | 20% of 20 is 4, which corresponds to option E. |
54f33688-5f6c-509a-bf2a-c64f4734587c | open-r1/OpenR1-Math-220k/cn_k12 | If the set $A=\{x|x^2-2x+m=0\}=\emptyset$, then the range of the real number $m$ is ( )
A: $(-\infty, -1)$
B: $(-\infty, 1)$
C: $(1, +\infty)$
D: $[1, +\infty)$ | [
"Okay, let's see. The problem says that set A is empty. Set A is defined as all x such that x² - 2x + m = 0. So, if A is empty, that means there are no real solutions to the equation x² - 2x + m = 0. Therefore, the quadratic equation must have no real roots.",
"I remember that for a quadratic equation ax² + bx + ... | For the set \( A = \{x \mid x^2 - 2x + m = 0\} \) to be empty, the discriminant \( D = 4 - 4m \) must be negative, yielding \( m > 1 \). Thus, the range of \( m \) is \( (1, +\infty) \), corresponding to option C. |
63ddae99-6f6b-5f82-8f74-76993abcaf87 | open-r1/OpenR1-Math-220k/cn_k12 | A school plans to purchase 400 workbooks, with each workbook costing x yuan. The school should pay a total of __ yuan. | [
"Okay, let's see. The problem says that a school plans to purchase 400 workbooks, and each workbook costs x yuan. The question is asking how much the school should pay in total. Hmm, that seems straightforward. Let me think step by step.",
"First, I need to find the total cost. If each workbook costs x yuan and t... | The school should pay a total of \boxed{400x} yuan for 400 workbooks costing \( x \) yuan each. |
26615747-61e8-5eb8-a076-ed70fc81a8d3 | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following real numbers is an irrational number?
A: $0.7$
B: $\frac{1}{2}$
C: $\sqrt{2}$
D: $\sqrt{9}$ | [
"Okay, let's see. The question is asking which of the given options is an irrational number. The choices are A: 0.7, B: 1/2, C: sqrt(2), and D: sqrt(9). Hmm, I need to remember what makes a number irrational. Irrational numbers can't be expressed as a fraction of two integers, right? Their decimal expansions are no... | Among the options, \(\sqrt{2}\) is irrational as it cannot be expressed as a fraction of two integers, while the others are rational. Thus, the correct answer is option C. |
42c312d4-855c-5a3a-8558-2701db151124 | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following calculations is correct?
A: $a^{2}\cdot a^{6}=a^{8}$
B: $a^{6}\div a^{3}=a^{2}$
C: $2a^{2}+a^{6}=3a^{8}$
D: $(-3a)^{2}=-9a^{2}$ | [
"Okay, so I need to figure out which of these calculations is correct. There are four options: A, B, C, and D. Let me go through each one step by step.",
"Starting with option A: \\(a^{2} \\cdot a^{6} = a^{8}\\). Hmm, when you multiply exponents with the same base, you add the exponents. So \\(a^{2} \\cdot a^{6}\... | Option A is correct as \(a^{2} \cdot a^{6} = a^{8}\), while options B, C, and D are incorrect due to errors in exponent rules and operations. The correct answer is \(\boxed{A}\). |
c8b7524f-4e39-5aca-8f04-17b68f13b40d | open-r1/OpenR1-Math-220k/olympiads | 5. (3 points) Wang Lei was born on National Day. If three times his age minus 8 is exactly the total number of days in the month he was born, then Wang Lei is $\qquad$ years old this year. | [
"Okay, let's see. The problem is about Wang Lei's age. He was born on National Day, which in China is October 1st, right? So the month he was born in is October. The problem says that three times his age minus 8 equals the total number of days in the month he was born. So first, I need to figure out how many days a... | Wang Lei is 13 years old, as determined by solving the equation \(3x - 8 = 31\) where \(x\) represents his age. |
0760720c-57b3-55dd-94f4-7032e24d04ec | open-r1/OpenR1-Math-220k/olympiads | ## Task 1
Out of 33 pioneers in a pioneer group, 8 pioneers were able to win medals at the pioneer sports festival because they took first, second, or third place. All other pioneers received certificates for participation.
How many pioneers in this group received certificates? | [
"Okay, let's see here. The problem is about pioneers in a group, some of whom won medals and others got certificates. The total number of pioneers is 33, and out of these, 8 won medals. The question is asking how many received certificates instead.",
"Hmm, so if there are 33 pioneers in total and 8 of them got me... | By subtracting the 8 medal winners from the 33 total pioneers, we find that 25 pioneers received certificates. |
5f767954-6eab-59bc-a723-5dbf10fa5048 | open-r1/OpenR1-Math-220k/cn_k12 | Variance is a measure of the magnitude of fluctuations in a set of data. For a set of data $x_{1}$, $x_{2}$, $x_{3}$, $\ldots$, $x_{n}$, the variance can be calculated using the following formula: $S^{2}=\frac{1}{n}[(x_{1}-3)^{2}+(x_{2}-3)^{2}+(x_{3}-3)^{2}+\ldots +(x_{n}-3)^{2}]$, where "3" represents the ( ).
A: min... | [
"Okay, so I need to figure out what the \"3\" represents in the variance formula given here. The formula is S squared equals 1 over n times the sum of each data point minus 3 squared. Let me recall what variance is. Variance measures how spread out the data is from a central value. Typically, the formula for varian... | In the variance formula \( S^2 = \frac{1}{n}[(x_1 - 3)^2 + (x_2 - 3)^2 + \ldots + (x_n - 3)^2] \), the number 3 represents the mean of the data set, as variance is defined as the average squared deviation from the mean. The correct answer is B: mean. |
cb6f7e20-b7bd-5cd0-ac52-3787e0571332 | open-r1/OpenR1-Math-220k/cn_k12 | Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(x,-2)$, if $\overrightarrow{a} \parallel \overrightarrow{b}$, then the real number $x=$ ______. | [
"Okay, so I need to find the value of x such that the vectors a and b are parallel. Let me recall what it means for two vectors to be parallel. If two vectors are parallel, one must be a scalar multiple of the other. That means there exists some scalar k where each component of vector b is k times the corresponding... | For vectors \(\overrightarrow{a} = (1, 2)\) and \(\overrightarrow{b} = (x, -2)\) to be parallel, \(x = -1\) satisfies both the scalar multiple condition and the zero cross product condition. |
025a4e5e-57c4-5c37-a8e5-6922025c8d85 | open-r1/OpenR1-Math-220k/cn_k12 | Translate the graph of the function $y=x^{2}$ up by one unit to obtain the graph of which function?
A: $y=x^{2}-1$
B: $y=\left(x+1\right)^{2}$
C: $y=x^{2}+1$
D: $y=\left(x-1\right)^{2}$ | [
"Okay, so I need to figure out which function represents the graph of y = x² shifted up by one unit. Let me recall how function transformations work.",
"When you translate a graph vertically, it affects the y-values of the function. If you want to shift a graph up by a certain number of units, you add that number... | The correct function for shifting \( y = x^2 \) up by one unit is \( y = x^2 + 1 \), which corresponds to option C. |
d0a01921-400f-54cb-b44c-ecb7b4c5aa70 | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following equations is correct?
A: $0-\left(-3\right)=3$
B: $(-14)-5=-9$
C: $(-3)-\left(-3\right)=-6$
D: $|5-3|=-\left(5-3\right)$ | [
"Okay, let's see. I need to figure out which of these equations is correct. There are four options: A, B, C, and D. Let me go through each one step by step.",
"Starting with option A: 0 - (-3) = 3. Hmm, subtracting a negative is the same as adding the positive, right? So, 0 - (-3) becomes 0 + 3, which is 3. That ... | Option A is correct as \(0 - (-3) = 3\), while options B, C, and D are incorrect. The correct answer is \(\boxed{A}\). |
b6042d44-599e-5c4c-a29d-f680c498afd2 | open-r1/OpenR1-Math-220k/olympiads | A square garden has an area of 9 square metres. The perimeter of the garden is $N$ metres. What is the value of $N$ ?
## | [
"Okay, so I need to find the perimeter of a square garden that has an area of 9 square meters. Hmm, let me think. First, I remember that the area of a square is calculated by squaring the length of one of its sides. So, if the area is 9, then each side must be the square root of 9. Let me write that down.",
"Area... | The side length of the square garden is 3 meters, and the perimeter is 4 times the side length, resulting in a value of 12 meters. |
ef699f3b-94a7-533f-be7a-543d56d67b66 | open-r1/OpenR1-Math-220k/olympiads | ## Task 4
From class 1a, 3 Pioneers receive certificates for good performance. An equal number of Pioneers from class 1b also receive certificates.
How many Pioneers will be awarded? | [
"Okay, so I need to figure out how many Pioneers will be awarded certificates in total from classes 1a and 1b. Let me start by reading the problem again carefully.",
"From class 1a, 3 Pioneers receive certificates for good performance. Then it says an equal number of Pioneers from class 1b also receive certificat... | 3 Pioneers from class 1a and 3 from class 1b receive certificates, totaling 6 Pioneers. |
4409aeda-7fc0-54a5-b576-8752b01058b6 | open-r1/OpenR1-Math-220k/cn_k12 | Evaluate the integral $$∫_{ 0 }^{ 1 }(e^{x}+2x)dx$$.
Options:
A: 1
B: e-1
C: e+1
D: e | [
"Okay, so I need to evaluate the integral from 0 to 1 of (e^x + 2x) dx. Let me think about how to approach this. The integral is a definite integral, so once I find the antiderivative, I can plug in the upper and lower limits and subtract.",
"First, let me recall that the integral of a sum is the sum of the integ... | The integral \( \int_{0}^{1} (e^x + 2x) \, dx \) evaluates to \( e \), corresponding to Option D. |
2531fe3d-6706-5699-ba3d-52cce81c87aa | open-r1/OpenR1-Math-220k/cn_k12 | From 10 basketballs, one is randomly selected to test its quality. The sampling method that should be used is ____. | [
"Okay, let's see. The problem is about selecting one basketball out of ten to test its quality. The question is asking which sampling method should be used. Hmm, I need to recall different types of sampling methods. Let me think.",
"First, there's simple random sampling. That's where each member of the population... | The appropriate sampling method for randomly selecting one basketball from ten with equal probability is simple random sampling, which ensures unbiased selection without subgroups or systematic patterns. |
5e2e0759-26a8-5dfd-b615-4633c4976216 | open-r1/OpenR1-Math-220k/olympiads | 274. $\int_{0}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}}$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
274. $\int_{0}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}}$.
The above text is already in a mathematical format and does not require transl... | [
"Okay, let's tackle this integral problem. The question is to compute the definite integral from 0 to 1/2 of 1 over the square root of (1 minus x squared) dx. So, written out, it's ∫₀^(1/2) dx/√(1 - x²). Hmm, I remember that integrals involving 1 over √(1 - x²) are related to inverse trigonometric functions. Specif... | The integral of 1 over the square root of 1 minus x squared from 0 to 1/2 equals arcsin(1/2) minus arcsin(0), which simplifies to π/6. |
e3f9ed39-d9e2-5504-921e-790061178992 | open-r1/OpenR1-Math-220k/cn_k12 | Given the set $M=\{-1, 1\}$, the number of sets $N$ that satisfy $N \subseteq M$ is ( )
A: 1
B: 2
C: 3
D: 4 | [
"Alright, let's tackle this problem step by step. So, the question is about finding the number of subsets of the set M = {-1, 1}. The options given are A:1, B:2, C:3, D:4. Hmm, okay. Let me recall what a subset is. A subset N of M is a set where every element of N is also an element of M. That means N can have some... | The set \( M = \{-1, 1\} \) has 4 subsets, which are \( \emptyset \), \( \{-1\} \), \( \{1\} \), and \( \{-1, 1\} \), corresponding to option D. |
f27b451d-3381-5f04-825b-93a1a9ad696d | open-r1/OpenR1-Math-220k/cn_k12 | Sampling methods include ( )
A: Random sampling, systematic sampling, and stratified sampling
B: Random number method, drawing lots, and stratified sampling method
C: Simple random sampling, stratified sampling, and systematic sampling
D: Systematic sampling, stratified sampling, and random number method | [
"Okay, let's see. The question is about sampling methods and the options are A, B, C, D. I need to figure out which one is correct.",
"First, I remember that there are several basic sampling methods. Let me try to recall them. The main ones are simple random sampling, stratified sampling, cluster sampling, and sy... | Option C correctly identifies the three primary sampling methods: simple random sampling, stratified sampling, and systematic sampling. The other options either use imprecise terminology or include techniques rather than methods. |
5fb1f1e4-b69c-56e5-befb-bf906447ae95 | open-r1/OpenR1-Math-220k/olympiads | 2. Xiao Pang and his father played a dart game together, each of them threw 5 times, father scored 48 points, Xiao Pang's score was 8 points less than half of his father's, Xiao Pang scored $\qquad$ points. | [
"Okay, so let's see. The problem is about Xiao Pang and his father playing a dart game. Each of them threw 5 times. The father scored 48 points. Xiao Pang's score was 8 points less than half of his father's. We need to find out how many points Xiao Pang scored.",
"Alright, let me break this down. First, the fathe... | Xiao Pang scored 16 points, calculated by taking half of his father's 48 points and subtracting 8. |
83187d2d-3b78-5f60-87fb-0d1189210773 | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following calculations is correct?
A: $3a+b=3ab$
B: $a^{2}+a^{2}=a^{4}$
C: $(a-b)^{2}=a^{2}-b^{2}$
D: $(-3a)^{2}=9a^{2}$ | [
"Okay, let's see. I need to figure out which of these four options is correct. Let me go through each one step by step.",
"Starting with option A: 3a + b = 3ab. Hmm. Wait, 3a and b are being added here. But unless they have the same variables, you can't combine them like that through addition. For example, if a w... | Option D is correct because \((-3a)^2 = 9a^2\), while the other options contain fundamental algebraic errors. |
9ff91f9e-ab45-54f4-a382-3be3338e5ca6 | open-r1/OpenR1-Math-220k/cn_k12 | In the Cartesian coordinate system, the coordinates of the point $(-2,3)$ after moving $4$ units to the right are ( ).
A: $(2,3)$
B: $(-6,3)$
C: $(-2,7)$
D: $(-2,-1)$ | [
"Okay, so I need to figure out where the point (-2, 3) ends up after moving it 4 units to the right. Let me think about how coordinates work in the Cartesian plane.",
"First, I remember that the Cartesian coordinate system has an x-axis (horizontal) and a y-axis (vertical). The coordinates of a point are given as... | Moving 4 units to the right from point \((-2, 3)\) changes the x-coordinate to \(2\) while keeping the y-coordinate at \(3\), resulting in the new coordinates \((2, 3)\). \(\boxed{A}\) |
12a96f20-a7ed-5b76-8662-c1916e802d2a | open-r1/OpenR1-Math-220k/cn_k12 | The equation of the circle with center at the point (0,1) and radius 2 is ( )
A: $x^2+(y-1)^2=2$
B: $(x-1)^2+y^2=2$
C: $x^2+(y-1)^2=4$
D: $(x-1)^2+y^2=4$ | [
"Okay, so I need to find the equation of a circle with center at (0,1) and radius 2, and then choose the correct option among A, B, C, D. Let me remember how the equation of a circle is structured. Hmm, the general formula for a circle is (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center and r is the radius. R... | The equation of a circle with center (0, 1) and radius 2 is x² + (y - 1)² = 4, which corresponds to option C. |
8cbbddef-f987-5cc5-9ef4-19c3962db9af | open-r1/OpenR1-Math-220k/olympiads | Example 2.25. $\int_{-\infty}^{0} e^{x} d x$.
Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.
Example 2.25. $\int_{-\infty}^{0} e^{x} d x$. | [
"Okay, let's tackle this integral from negative infinity to zero of e^x dx. Hmm, improper integrals can be tricky, but I remember they involve limits. So first, I need to recall the definition of an integral with an infinite limit. The problem here is that the lower bound is negative infinity, which means we have t... | The improper integral $\int_{-\infty}^{0} e^{x} dx$ is evaluated as $\lim_{a \to -\infty} \int_{a}^{0} e^{x} dx = \lim_{a \to -\infty} (1 - e^{a}) = 1$. The result is $\boxed{1}$. |
b6fc1bfd-7272-539b-bcb4-3b2d8d326ebb | open-r1/OpenR1-Math-220k/cn_k12 | Given $\log_{2}3=a$, $\log_{2}5=b$, then $\log_{2} \frac{9}{5}$ equals ( )
A: $a^{2}-b$
B: $2a-b$
C: $\frac{a^{2}}{b}$
D: $\frac{2a}{b}$ | [
"Okay, let me try to figure out this logarithm problem. The question says that given log base 2 of 3 is a, and log base 2 of 5 is b, we need to find log base 2 of (9/5) and choose the correct option from A to D. The options are A: a² - b, B: 2a - b, C: a²/b, D: 2a/b.",
"First, I remember that logarithms have some... | Using the logarithm properties, \(\log_{2} \frac{9}{5} = \log_{2} 9 - \log_{2} 5 = 2 \log_{2} 3 - \log_{2} 5 = 2a - b\), so the answer is \(\boxed{B}\). |
3967e9df-3a87-5dcb-9232-718d851c13e2 | open-r1/OpenR1-Math-220k/cn_k12 | What is the sum of the interior angles of a quadrilateral?
A: $180^{\circ}$
B: $360^{\circ}$
C: $540^{\circ}$
D: $720^{\circ}$ | [
"Okay, so I need to find the sum of the interior angles of a quadrilateral. Hmm, let me think. I remember that for triangles, the sum of the interior angles is 180 degrees. But quadrilaterals have four sides, right? So maybe the sum is higher?",
"Wait, I think there's a formula for this. Let me recall. I remember... | The sum of a quadrilateral's interior angles is \((4-2) \times 180^\circ = 360^\circ\), verified by dividing it into two triangles or using the polygon formula. The correct answer is \(\boxed{B}\). |
863bb8e9-f455-5066-8d23-d1a20698a2d6 | open-r1/OpenR1-Math-220k/olympiads | 4. (7 points) A gnome in shoes weighs 2 kg more than a gnome without shoes. If you put five identical gnomes in shoes and five such gnomes without shoes on the scales, the scales will show 330 kg. How much does a gnome in shoes weigh? | [
"Okay, let's see. The problem says that a gnome in shoes weighs 2 kg more than a gnome without shoes. Then, if you put five gnomes with shoes and five without shoes on the scales, it shows 330 kg. We need to find the weight of a gnome in shoes.",
"Hmm, let's break this down. First, let me define variables for the... | The weight of a gnome without shoes is 32 kg, so a gnome in shoes weighs 34 kg. |
f1a5c1b4-a913-5b2d-b2e0-2b092dc4996a | open-r1/OpenR1-Math-220k/cn_k12 | Given that $a$ is a positive integer, point $P(4,2-a)$ lies in the first quadrant, then $a=\_\_\_\_\_\_$. | [
"Alright, let's try to solve this problem. So, we have a point P with coordinates (4, 2 - a), and it's stated that this point lies in the first quadrant. We need to find the value of 'a', which is a positive integer. Hmm, okay. Let me recall what I know about quadrants in coordinate geometry.",
"First, the coordi... | For point \( P(4, 2 - a) \) to be in the first quadrant, \( a \) must satisfy \( a < 2 \), and since \( a \) is a positive integer, the only solution is \( a = 1 \). |
5a9452cd-25fc-5dee-8a07-98d04f035760 | open-r1/OpenR1-Math-220k/olympiads | The value of $\left(2^{3}\right)^{2}-4^{3}$ is
(A) 0
(B) -8
(C) 4
(D) 10
(E) 12 | [
"Okay, so I need to figure out the value of (2³)² minus 4³. Let me break this down step by step. First, I remember that when you have an exponent raised to another exponent, you multiply them. So (2³)² would be 2^(3*2) which is 2^6. Let me check that rule again to make sure I'm not mixing things up. Yeah, the power... | \((2^3)^2 = 64\) and \(4^3 = 64\), so \((2^3)^2 - 4^3 = 0\), which corresponds to option A. |
2f83637b-0b92-5845-877c-7c664bc06bd1 | open-r1/OpenR1-Math-220k/olympiads | 5. A small rabbit eats 2 pieces of cabbage a day, a large rabbit eats 4 pieces of cabbage a day, now raising two large rabbits, one small rabbit, every day $\qquad$ pieces of cabbage need to be prepared. | [
"Okay, let me try to figure out how much cabbage I need to prepare each day for the rabbits. Hmm, so there are two types of rabbits: small and large. The problem says a small rabbit eats 2 pieces of cabbage a day, and a large rabbit eats 4 pieces a day. Right now, they're raising two large rabbits and one small rab... | Two large rabbits eat 8 cabbage pieces daily and one small rabbit eats 2 pieces, totaling 10 cabbage pieces needed daily. |
f8133c6b-179c-5282-aab1-b8b4804caa82 | open-r1/OpenR1-Math-220k/cn_k12 | The three logical structures of an algorithm are ( )
A: Sequential structure; Flow structure; Loop structure
B: Sequential structure; Conditional structure; Nested structure
C: Sequential structure; Conditional structure; Loop structure
D: Flow structure; Conditional structure; Loop structure | [
"Okay, let's see. The question is about the three logical structures of an algorithm. The options are A, B, C, D with different combinations. Hmm, I remember that in programming, there are certain basic control structures that algorithms use. Let me think.",
"First, there's the sequential structure. That's where ... | The correct answer is C, which identifies the three fundamental logical structures as sequential, conditional, and loop structures, corresponding to sequence, selection, and iteration in programming. Options A, B, and D are incorrect due to the inclusion of non-standard terms like "Flow structure" and "Nested structure... |
1802e9e8-879a-5fc3-a0da-f2376cf7ca12 | open-r1/OpenR1-Math-220k/olympiads | Task 15. (8 points)
Announcement: "Have free time and want to earn money? Write now and earn up to 2500 rubles a day working as a courier with the service 'Food.There-Here!'. Delivery of food from stores, cafes, and restaurants.
How much will a school student working as a courier with the service 'Food.There-Here!' e... | [
"Okay, let me try to figure this out. So, the problem says a school student is working as a courier for this service called \"Food.There-Here!\" and we need to calculate how much they'll earn in a month, which is 4 weeks. The student works 4 days a week, and each day they earn a minimum of 1250 rubles after income ... | The student earned 20,000 rubles after tax by working 16 days at a rate of 1,250 rubles per day. |
daff7435-c585-5ad5-b552-6758d384c63e | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following calculations is correct?
A: $3x+2x=5x^{2}$
B: $\sqrt{9}=±3$
C: $(2x)^{2}=2x^{2}$
D: $2^{-1}=\frac{1}{2}$ | [
"Okay, let's see. The question is asking which of the given calculations is correct among options A to D. I need to check each one step by step. Let's start with option A: 3x plus 2x equals 5x squared. Hmm, wait. When you add like terms, you just add their coefficients, right? So 3x plus 2x should be (3+2)x, which ... | Option D is correct because \(2^{-1} = \frac{1}{2}\), while the other options contain mathematical errors. |
28b893a9-8de7-52d6-8c14-ad7a6c593a7a | open-r1/OpenR1-Math-220k/cn_k12 | $\sqrt{9}=\left(\ \ \right)$
A: $\pm 3$
B: $3$
C: $±\sqrt{3}$
D: $\sqrt{3}$ | [
"Okay, so I need to figure out what the square root of 9 is. The options are A: ±3, B: 3, C: ±√3, and D: √3. Hmm, let me start by recalling what a square root is. The square root of a number is a value that, when multiplied by itself, gives the original number. So, for 9, what number times itself equals 9? Well, bo... | The square root symbol (√) represents the principal (non-negative) root, so √9 equals 3, making option B correct. Options A, C, and D are incorrect as they either include negative roots or unrelated values. |
68bc9859-60f3-5fa8-80c5-202473ae041b | open-r1/OpenR1-Math-220k/cn_k12 | If the variance of a set of data $a_1, a_2, a_3, a_4, a_5, a_6$ is 2, then the variance of another set of data $2a_1, 2a_2, 2a_3, 2a_4, 2a_5, 2a_6$ is ( )
A: 2
B: 6
C: 8
D: -2 | [
"Okay, so I need to find the variance of a new dataset where each value is twice the original. The original variance is given as 2. Hmm, let me recall how variance works when you scale the data.",
"Variance measures how spread out the numbers in a dataset are. If you multiply each data point by a constant, the sp... | When each data point is multiplied by 2, the variance becomes \(2^2 \times 2 = 8\), corresponding to option C. |
670ee4a3-e86a-5e52-ab7d-eca643189df4 | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following statements about a pyramid is correct? ( )
A: A quadrilateral pyramid has four edges.
B: A pentagonal pyramid has five faces.
C: A hexagonal pyramid has six vertices.
D: Any pyramid has only one base. | [
"Okay, let's see. I need to figure out which statement about a pyramid is correct from options A to D. Let's take each one step by step.",
"Starting with option A: \"A quadrilateral pyramid has four edges.\" Hmm, quadrilateral pyramid... So, a quadrilateral base means the base is a four-sided figure, like a squar... | Option D is correct because a pyramid is defined as having only one base, while the other options incorrectly describe the number of edges, faces, or vertices in pyramids. |
c35d37e5-6b9c-57af-b04e-0777f7906f97 | open-r1/OpenR1-Math-220k/olympiads | [Mathematical logic (miscellaneous).] [ Arithmetic. Mental arithmetic, etc. ]
Rabbits are sawing a log. They made 10 cuts. How many chunks did they get?
# | [
"Okay, let's see. The problem says that rabbits are sawing a log and they made 10 cuts. We need to figure out how many chunks they got. Hmm. So, when you cut a log, each cut should divide it into pieces, right? But I need to remember if there's a standard formula or way to calculate this.",
"Wait, maybe it's a cl... | 10 cuts on a log that starts as 1 piece result in 11 pieces, as each cut adds one additional chunk. |
0eb46b5a-8fc6-5f2f-9eb9-41a6b87831bd | open-r1/OpenR1-Math-220k/cn_k12 | Among the following propositions, the correct one is\\((\\) \\()\\)
A: If two vectors are equal, then their starting points and endpoints coincide respectively.
B: Two parallel vectors with equal magnitudes are equal vectors.
C: If _\\(a\\)_ and _\\(b\\)_ are both unit vectors, then _\\(a\\)_\\(=\\) _\\(b\\)_.
D: Two ... | [
"Okay, let's tackle this problem step by step. We have four propositions about vectors, labeled A through D, and we need to determine which one is correct. Let me go through each option one by one, thinking aloud as I do.",
"Starting with option A: \"If two vectors are equal, then their starting points and endpoi... | Option D is correct because equal vectors must have the same magnitude and direction by definition. The other options are false as they incorrectly describe vector properties. |
94b38e61-32fd-5415-9ec2-7b4293594420 | open-r1/OpenR1-Math-220k/olympiads | 7. In 1981, the United States developed the world's first space shuttle, the “$\qquad$”号, which successfully completed its first manned space flight test.
A. Houston
B. New York
C. Columbia
D. Apollo
Note: The Chinese character "号" in the blank should be replaced with the corresponding English name, which is "Columbia... | [
"Okay, let's see. The question is about the first space shuttle developed by the United States in 1981 that successfully completed its first manned space flight test. The options are Houston, New York, Columbia, and Apollo.",
"First, I need to recall the names of the space shuttles. I remember that the Space Shut... | The first U.S. space shuttle, Columbia, conducted the inaugural manned flight (STS-1), making option C correct. Other options are unrelated: Houston and New York are cities, while Apollo refers to an earlier lunar program. |
2daf2651-6280-560b-939f-d3853a91d0fe | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following phenomena is impossible? ( )
A: A conductor heats up when conducting electricity
B: Three non-collinear points determine a plane
C: Seeds germinate without water
D: Someone wins the lottery for two consecutive weeks | [
"Okay, so I need to figure out which of these phenomena is impossible. Let's look at each option one by one.",
"Starting with option A: A conductor heats up when conducting electricity. Hmm, I remember that when electric current flows through a conductor, it generates heat because of the resistance in the materia... | The impossible phenomenon is C, as seed germination fundamentally requires water to activate metabolic processes, making it impossible without water. |
86feba0f-073d-5068-aa17-2a2580506314 | open-r1/OpenR1-Math-220k/cn_k12 | When $x=-1$, the value of the function $y=-2x$ is equal to:
A: $2$
B: $-2$
C: $\frac{1}{2}$
D: $-\frac{1}{2}$ | [
"Okay, let's see here. I need to figure out the value of the function y = -2x when x is -1. Hmm, functions can sometimes be tricky, but this seems straightforward. Let me break it down step by step.",
"First, the function is given as y equals negative two times x. So, y = -2x. The question is asking what happens ... | When \( x = -1 \), substituting into \( y = -2x \) gives \( y = 2 \), corresponding to option **A**. |
65f54d25-90ff-5f45-b7a9-8ef6a890edce | open-r1/OpenR1-Math-220k/cn_k12 | Calculate: $-3 + 2 = \quad$; $\quad (-3) \times 2 = \quad$. | [
"Okay, so I need to solve two math problems here: the first one is -3 plus 2, and the second one is -3 multiplied by 2. Let me take them one at a time.",
"Starting with the first one: -3 + 2. Hmm, adding a positive number to a negative number. I remember that when you have numbers with different signs, you subtra... | \(-3 + 2 = \boxed{-1}\) and \((-3) \times 2 = \boxed{-6}\). |
1ceb5a84-d56f-51a3-a633-0209d8d7870b | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following calculations is correct?
A: $a^{8}\div a^{2}=a^{4}$
B: $5ab-2ab=3$
C: $(a-b)^{2}=a^{2}-b^{2}$
D: $(-ab^{3})^{2}=a^{2}b^{6}$ | [
"Okay, let me see. I need to figure out which of the options A, B, C, or D is the correct calculation. Let's go through each one step by step.",
"Starting with option A: a^8 divided by a^2 equals a^4. Hmm, I remember that when you divide exponents with the same base, you subtract the exponents. So, a^8 divided by... | Option D is correct because \((-ab^3)^2 = a^2b^6\), while options A, B, and C are incorrect due to errors in exponent rules, subtraction, and expansion. |
cff69271-6f6f-5ac0-a30a-65e3c59e5421 | open-r1/OpenR1-Math-220k/cn_k12 | Given that variables $x$ and $y$ are linearly related, and the sample means obtained from observational data are $\overset{.}{x}=2$ and $\overset{.}{y}=3$, the linear regression equation obtained from this observational data cannot be
A: $y=3x-3$
B: $y=2x+1$
C: $y=x+1$
D: $y=0.5x+2$ | [
"Okay, let's see. The problem says that x and y are linearly related, and the sample means are x̄ = 2 and ȳ = 3. We need to figure out which of the given linear regression equations cannot be the one obtained from this data. The options are A to D.",
"Hmm, I remember that in linear regression, the regression line... | Option B (\( y = 2x + 1 \)) does not pass through the point \((2, 3)\), while the other options satisfy this condition. |
d6ebdf98-162d-51f3-8b32-29ccc6eb9be7 | open-r1/OpenR1-Math-220k/cn_k12 | Given a regression equation $$\hat y = 2 + 3x$$, when the variable $x$ increases by one unit, then ( )
A: $y$ increases by an average of 2 units
B: $y$ increases by an average of 3 units
C: $y$ decreases by an average of 2 units
D: $y$ decreases by an average of 3 units | [
"Okay, let's see. I have this regression equation here: ŷ = 2 + 3x. The question is asking what happens to y when x increases by one unit. The options are about increasing or decreasing by 2 or 3 units on average. Hmm.",
"First, I remember that in a linear regression equation like this, the coefficient of x is th... | The regression equation \(\hat{y} = 2 + 3x\) shows that for each one-unit increase in \(x\), \(\hat{y}\) increases by 3 units, confirming option B. |
0fb952fa-22d5-5c60-97ee-782a8e3c9160 | open-r1/OpenR1-Math-220k/cn_contest | Example 3. Simplify: $\sqrt{(a-5)^{2}}$ | [
"Okay, let's see. I need to simplify the square root of (a minus 5) squared. Hmm, so that's √[(a - 5)²]. Alright, I remember that when you take the square root of something squared, it's like undoing the square. But wait, there's a catch here. Because if the original expression inside the square was negative, then ... | The square root of a squared term equals the absolute value of the original term, so \(\sqrt{(a-5)^2} = |a-5|\) for any real number \(a\). |
40ffdb61-40c0-5af2-9f82-589affdc464d | open-r1/OpenR1-Math-220k/cn_k12 | Let the function $y = f(x)$. When the independent variable $x$ changes from $x_0$ to $x_0 + \Delta x$, the change in the function $\Delta y$ is ( )
A: $f(x_0 + \Delta x)$
B: $f(x_0) + \Delta x$
C: $f(x_0) \cdot \Delta x$
D: $f(x_0 + \Delta x) - f(x_0)$ | [
"Okay, let's see. The question is about finding the change in the function y = f(x) when the independent variable x changes from x₀ to x₀ + Δx. The options are A to D. I need to figure out which one represents Δy.",
"Hmm, I remember that in calculus, the change in the function, Δy, is usually the difference betwe... | The change in the function \( \Delta y \) is correctly calculated as \( f(x_0 + \Delta x) - f(x_0) \), which corresponds to option D. The other options incorrectly represent the change by either using only the new value, adding \( \Delta x \) to the original value, or multiplying the original value by \( \Delta x \). |
ddda4fed-0fd1-58ac-b1d9-b516694baf1d | open-r1/OpenR1-Math-220k/olympiads | 23. In the canteen for lunch, 4 first-course dishes, 5 second-course dishes, and 3 desserts are prepared for selection. In how many ways can one lunch be composed? | [
"Okay, let's see. The problem is about figuring out how many different ways someone can compose their lunch in the canteen. They have 4 first-course dishes, 5 second-course dishes, and 3 desserts to choose from. So, I need to find the total number of possible meal combinations.",
"Hmm, right. So, when you have mu... | The number of ways to compose a lunch with one dish from each course is calculated by multiplying the choices: 4 first courses × 5 second courses × 3 desserts = 60 combinations. |
8771cf2b-06c4-53ff-9a46-56253ecfe9ca | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following calculations is correct?
A: $\sqrt{2}×\sqrt{3}=\sqrt{6}$
B: $\sqrt{2}+\sqrt{3}=\sqrt{5}$
C: $(\sqrt{2})^{2}=2\sqrt{2}$
D: $2+\sqrt{2}=2\sqrt{2}$ | [
"Okay, let's see. The question is asking which of the given calculations is correct, with options A through D. I need to check each one step by step.",
"Starting with option A: √2 × √3 = √6. Hmm, I remember that when multiplying square roots, you can multiply the numbers under the roots and then take the square r... | A is correct because it properly applies the property $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$, while B, C, and D contain mathematical errors. |
8a70fade-64ea-554f-bf40-b3de41ea4148 | open-r1/OpenR1-Math-220k/olympiads | A six-sided die has the numbers one to six on its sides. What is the probability of rolling a five?
(A) $\frac{2}{6}$
(B) $\frac{1}{6}$
(C) $\frac{5}{6}$
(D) $\frac{3}{6}$
(E) $\frac{4}{6}$ | [
"Okay, let's see. The problem is about probability with a six-sided die. Hmm. So, the question is asking, what's the probability of rolling a five. The options are given from (A) to (E), with different fractions. Alright.",
"First, probability. Probability is the chance of something happening. So, when they say p... | The probability of rolling a five on a fair six-sided die is 1/6, as there is one favorable outcome out of six possible outcomes. The correct answer is B. |
c5ee5ead-f602-5a8f-8c98-1319d2fbe3ca | open-r1/OpenR1-Math-220k/cn_k12 | During a class meeting, it was decided to vote on whether to add a new class committee member. The voting method that best reflects the true will of all students is ( ).
A: By a show of hands of those who agree to add the new member.
B: By a show of hands of those who disagree to add the new member.
C: By secret ballot... | [
"Okay, so I need to figure out which voting method best reflects the true will of all students for adding a new class committee member. The options are A, B, C, or D. Let me think through each one step by step.",
"First, let's understand the question. The key here is that the voting method should accurately captu... | Secret ballot (C) ensures anonymity, allowing students to vote according to their true preferences without external influence, making it the fairest method to capture genuine voter intent. |
74ce85a9-21ad-5fb3-9929-cce17d65cb4c | open-r1/OpenR1-Math-220k/olympiads | 22. China has currently built 9 tidal power stations, the $\qquad$ tidal power station, completed in 1980, is China's first bidirectional tidal power station, and it is also one of the largest bidirectional tidal power stations in the world.
A. Xiamen
B. Jiangxia
C. Humen
D. Sanya | [
"Okay, so I need to figure out which tidal power station in China was completed in 1980 and is the first bidirectional one. The options are Xiamen, Jiangxia, Humen, and Sanya. Let me think.",
"First, I remember that China has several tidal power stations. The question mentions it's the first bidirectional one, co... | The Jiangxia Tidal Power Station, completed in 1980, is China's first bidirectional tidal power station and one of the largest globally, making option B the correct choice. |
bfd95ff7-d334-5855-9951-fba4246ce0a1 | open-r1/OpenR1-Math-220k/olympiads | 4. The "=" sign was invented by the British person
A. De Morgan
B. Recorde
C. Oughtred | [
"Okay, let's tackle this question. The problem is asking who invented the \"=\" sign among the options A. De Morgan, B. Recorde, or C. Oughtred. Alright, first, I need to recall some history of mathematics and the origins of mathematical symbols.",
"Hmm, I remember that many mathematical symbols we use today were... | The equals sign "=" was introduced by Welsh mathematician Robert Recorde in 1557, symbolizing equality with parallel lines. William Oughtred contributed other symbols like "×" for multiplication, but neither he nor Augustus De Morgan invented the equals sign. |
445a0438-ba7e-514a-958c-3f1f6b3323c7 | open-r1/OpenR1-Math-220k/olympiads | ## Task 18
$a=9, b=628 ; \quad$ To the sixfold of $a$, add half of $b$. | [
"Okay, let's see. The problem says that a is 9 and b is 628. I need to take sixfold of a, which means six times a, right? So first, I should calculate 6 multiplied by a. Since a is 9, that would be 6 times 9. Let me do that math. 6 times 9 is... 54. Yeah, that's right. 6*9=54.",
"Then, the next part is adding hal... | Sixfold of \( a \) (54) plus half of \( b \) (314) equals 368. The final answer is \(\boxed{368}\). |
a58e4f17-b5eb-53b3-9dcb-abd377ff8532 | open-r1/OpenR1-Math-220k/olympiads | ## Task 1
In 5 windows of a residential building, there are a total of 40 glass panes.
How many panes are in 3 windows? | [
"Okay, let's see. So the problem is about figuring out how many glass panes are in 3 windows of a residential building, given that 5 windows have a total of 40 panes. Hmm, okay. Let me think step by step.",
"First, I need to understand the relationship between the number of windows and the number of glass panes. ... | Since 5 windows have 40 panes, each window has 8 panes, so 3 windows have 24 panes. |
aa86f0cf-4d35-5a6a-81cb-6f28e3817314 | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following propositions is true?
A: If $a+b \lt 0$, then $ab \lt 0$
B: Corresponding angles are equal
C: The sum of interior angles of a triangle is $180^{\circ}$
D: Equal angles are vertical angles | [
"Okay, let's see. I need to figure out which of these propositions (A, B, C, D) is true. Let me take each one step by step.",
"Starting with proposition A: \"If a + b < 0, then ab < 0.\" Hmm. So this is saying that if the sum of two numbers is negative, then their product must be negative. Let me test some number... | Proposition C is true as it states the sum of interior angles of a triangle is \(180^\circ\), a fundamental result in Euclidean geometry, while the other propositions are false based on counterexamples or missing conditions. The correct answer is \(\boxed{C}\). |
52fa3c7e-8b2e-5de1-8b8b-6a1064fdc767 | open-r1/OpenR1-Math-220k/olympiads | When two numbers are added, the result is -26 . If one of the numbers is 11 , what is the other number?
(A) -37
(B) 37
(C) -15
(D) 15
(E) -48 | [
"Okay, let's see. The problem says that when two numbers are added together, the result is -26. One of the numbers is 11, and we need to find the other number. The options are A) -37, B) 37, C) -15, D) 15, E) -48. Hmm, let's break this down step by step.",
"First, I know that addition is commutative, so the order... | The other number is \(-37\), as \(11 + (-37) = -26\). Therefore, the answer is \(\boxed{A}\). |
4bc59358-1e94-5947-ac8a-57b27df45c84 | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following statements is correct? ( )
A: A quadrilateral with a pair of adjacent equal sides is a rhombus.
B: A parallelogram with one right angle is a rectangle.
C: A parallelogram with perpendicular diagonals is a square.
D: A quadrilateral with one pair of parallel sides is a parallelogram. | [
"Okay, let me see. I need to figure out which of these statements is correct. The options are A, B, C, D. Let me go through each one by one.",
"Starting with option A: \"A quadrilateral with a pair of adjacent equal sides is a rhombus.\" Hmm. Wait, a rhombus is a quadrilateral with all sides equal, right? So if a... | Option B is correct because a parallelogram with one right angle must have all right angles, making it a rectangle. The other options are incorrect as they either misrepresent the properties of quadrilaterals or require additional conditions. |
End of preview. Expand in Data Studio
Math Think SFT Mini — Short Answers
This dataset is derived from cs-giung/math-think-sft-mini, pinned to revision 47c41ae2aac7df5aa14b066da2124ba96d004645.
The train split contains 352,470 rows in the same order as the source. Each original answer is replaced by its shortened version; id, source, question, and steps are unchanged. There is no additional short_answer column.
Schema
| Column | Type | Description |
|---|---|---|
id |
string | Unchanged source-row identifier |
source |
string | Unchanged source lineage |
question |
string | Unchanged mathematical problem |
steps |
list[string] | Unchanged reasoning steps |
answer |
string | Shortened final answer replacing the original response |
Construction
Only the original answer was supplied for shortening, without the question or reasoning steps. The instruction requested no more than two sentences while preserving the conclusion, important mathematical values, equations, units, option letters, and essential conditions, without re-solving or correcting the source answer.
Generated responses were checked with the workflow's sentence-format validator. The last 38 remaining rows were manually shortened by the assistant, validated by the same validator, and recorded separately as manual:assistant; this does not mean human-authored. Parenthesized multipart labels can be interpreted as sentence boundaries by the validator, so some labels were expressed as prose such as "part a".
Successful-answer provenance:
| Generator | Rows |
|---|---|
deepseek-v4-flash |
716 |
gemini-2.5-flash-lite |
293,524 |
glm-4.5-flash |
1,819 |
glm-4.7-flash |
96 |
glm-5.3-flash |
54,927 |
gpt-5-nano-2025-08-07 |
1,350 |
manual:assistant |
38 |
All 352,470 original answers were matched against the shortening queue before replacement. The complete non-answer columns, row order, schema, and locally saved dataset were verified against the source snapshot. See construction_manifest.json for the pinned source, transformation, model counts, and ordered answer checksum; parent construction and lineage details remain in the source construction manifest.
Load
from datasets import load_dataset
dataset = load_dataset("cs-giung/math-think-sft-mini-sa", split="train")
Limitations and licensing
Shortening and sentence-format checks are not independent mathematical or semantic verification. Source errors may remain, and automated shortening can omit nuance. The unchanged reasoning steps may retain details not present in the shortened answer. Sentence counts depend on the validator's handling of mathematical notation, abbreviations, and labels.
Source datasets retain their own licenses and usage restrictions; this derivative grants no new rights over source content. The license metadata remains other, consistent with the parent dataset. Consult the parent dataset and its source lineages for applicable terms.
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