id string | source string | question string | steps list | answer string |
|---|---|---|---|---|
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: $\sqrt{{a^2}}=a$ | [
"The user asks: Which of the following operations is correct? Options A-D with statements about exponent rules.",
"We need to evaluate each.",
"A: a^5 ÷ a^2 = a^{5-2}=a^3. That's correct, assuming a ≠ 0 (division defined). So A is correct.",
"B: a^3 + a^3 = 2 a^3, not a^6. So incorrect.",
"C: (a^3)^2 = a^{3... | **Answer: A is the only correct statement.**
Here’s why each option is (or isn’t) valid.
| Option | Expression | Evaluation | Correct? |
|--------|------------|------------|----------|
| **A** | \(a^{5}\div a^{2}\) | Using the quotient rule for powers: \(\displaystyle \frac{a^{5}}{a^{2}} = a^{5-2}=a^{3}\) (provided... |
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: $\frac{1}{23}$ | [
"We need interpret \"opposite\" probably means additive inverse, opposite of a number is its negative. So opposite of 23 is -23. So answer A.",
"But note some contexts could be reciprocal? Usually opposite means additive inverse. So answer A.",
"Provide reasoning."
] | The “opposite” of a number is its additive inverse – the number that, when added to the original, gives 0.
For \(23\),
\[
23 + (-23) = 0 .
\]
Hence the opposite of \(23\) is \(-23\).
**Answer: A: \(-23\).** |
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 | [
"We need to parse the problem: \"Jenny had to add 26 to a certain number. Instead she subtracted 26 and obtained -14 . What number should she have obtained?\" So original number = x. She should have added 26: correct result would be x + 26. Instead she subtracted 26: she computed x - 26 = -14. So we can solve for x... | Let the original number be \(x\).
- The correct operation: add 26 → result should be \(x+26\).
- What she actually did: subtract 26 → she got \(-14\).
So
\[
x-26 = -14 \quad\Longrightarrow\quad x = -14 + 26 = 12.
\]
Now the correct result would be
\[
x + 26 = 12 + 26 = 38.
\]
**Answer: D) 38**. |
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? | [
"The problem states: \"The area of a new playground is square and measures 1600 m^2. How long is one side of the playground? How many meters of fence are needed for three sides?\" So it's a square area 1600, side length sqrt(1600)=40 m. Fence for three sides: 3*40=120 m.",
"Provide answer."
] | **1. Length of one side**
The playground is a square whose area is
\[
\text{Area}=1600\ \text{m}^2.
\]
For a square,
\[
\text{Area}= (\text{side})^2 \quad\Longrightarrow\quad \text{side}= \sqrt{\text{Area}}.
\]
\[
\text{side}= \sqrt{1600}=40\ \text{m}.
\]
**2. Fence needed for three sides**
If three sides of... |
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? | [
"We need to interpret the problem. \"Mr. Müller has manufactured 16 chair legs. How many chairs can he build?\" Typically a chair has 4 legs. So 16 legs can make 4 chairs. Possibly there may be leftover legs (none). So answer: 4 chairs.",
"But maybe they expect integer division: 16 / 4 = 4. So answer is 4 chairs.... | A standard chair requires **4 legs**.
So the number of complete chairs that can be built from 16 legs is:
\[
\frac{16\text{ legs}}{4\text{ legs/chair}} = 4\text{ chairs}
\]
**Answer:** Mr. Müller can build **4 chairs**. |
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 | [
"We need answer: Arabic numerals invented by Indians (the Hindu-Arabic numeral system). So answer A."
] | **Answer: A. Indians** |
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: $-4m^{6}$ | [
"We need compute (-2m^3)^2. Square: (-2)^2 * (m^3)^2 = 4 * m^{6} = 4 m^6. So answer B.",
"Explain exponents: (a^b)^c = a^{bc}. Also sign: negative squared becomes positive.",
"Thus answer B."
] | \[
(-2m^{3})^{2}=(-2)^{2}\,(m^{3})^{2}=4\;m^{3\cdot 2}=4m^{6}
\]
So the correct choice is **B: \(4m^{6}\)**. |
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)$ | [
"We need domain where expression defined: sqrt(x+2) requires x+2 >=0 => x >= -2. And denominator x-1 !=0 => x !=1. So domain: [ -2, ∞ ) except x=1. That's [-2,1) ∪ (1, ∞). That matches option C.",
"Thus answer C."
] | The square‑root term requires
\[
x+2\ge 0\quad\Longrightarrow\quad x\ge -2 .
\]
The rational term requires
\[
x-1\neq 0\quad\Longrightarrow\quad x\neq 1 .
\]
Combining these two conditions, the set of admissible \(x\) values is
\[
[-2,\infty)\setminus\{1\}= [-2,1)\ \cup\ (1,\infty).
\]
Therefore the correct... |
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 | [
"We need to choose the correct word to fill blank: \"The medicine is on sale everywhere. You can get it at ___ chemist’s.\"",
"Options: A. each, B. some, C. certain, D. any.",
"The sentence: \"The medicine is on sale everywhere. You can get it at ___ chemist’s.\"",
"If something is on sale everywhere, you can... | **Answer: D – any**
*Explanation:*
The sentence says the medicine is available “everywhere,” so you can obtain it at **any** chemist’s (i.e., at whichever chemist you go to). The other choices do not convey this unrestricted sense:
- **each** chemist’s would imply you must go to every single chemist’s, which is not... |
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... | [
"We need to parse the question: \"In the process of deriving a contradiction using the method of proof by contradiction, which of the following should be used as conditions? ( ) \n① The judgment opposite to the conclusion, i.e., the assumption \n② The conditions of the original proposition \n③ Axioms, theorems,... | The proof by contradiction proceeds by assuming the negation of the statement you want to prove, together with the original hypotheses of the theorem, and then using the usual mathematical facts (axioms, previously proved theorems, definitions, etc.) to derive an impossibility.
Therefore the conditions that may be us... |
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. | [
"We need to answer which statements are true given p true, q false. Evaluate each:",
"A: p ∧ q is true only if both true; q false, so p ∧ q false. So A false.",
"B: p ∨ q is false only if both false; p true, so p ∨ q true. So B false.",
"C: ¬p is true only if p false; p true, so ¬p false. So C false.",
"D: ... | Since \(p\) is true and \(q\) is false:
- \(p \land q\) is false (both would have to be true).
- \(p \lor q\) is true (only one needs to be true).
- \(\lnot p\) is false (the negation of a true statement).
- \(\lnot q\) is true (the negation of a false statement).
Therefore, the only correct statement is
**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 | [
"We need intersection of two sets: {0,1,2,3,4} and {0,2,4,8}. Intersection = {0,2,4}. So M must be subset of this intersection. Max number of elements = size of intersection = 3. So answer C."
] | The two given conditions force the set \(M\) to be a subset of the intersection of the two listed sets.
\[
\begin{aligned}
\{0,1,2,3,4\}\cap\{0,2,4,8\}
&= \{0,2,4\}.
\end{aligned}
\]
Thus every possible element of \(M\) must be one of \(\{0,2,4\}\).
The largest \(M\) can be is the whole intersection, which contains... |
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. | [
"We need to interpret the problem: \"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.\"",
"Interpretation: There are preliminary heats, each heat has 8 athletes. From each ... | **Answer: 32 athletes**
**Explanation**
- Each preliminary heat has **8 athletes**.
- From each heat the **two best runners** qualify for the final.
- The final race has **8 athletes**.
Let \(h\) be the number of preliminary heats.
\[
2 \text{ qualifiers per heat} \times h = 8 \text{ finalists}
\]
\[
h = \frac{8}{... |
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 | [
"We need to parse problem: If ¬(p∧q) is a false proposition, then ( ...) Choose correct condition.",
"We have ¬(p∧q) is false. That means the negation of (p∧q) is false, i.e., ¬(p∧q) = false. Therefore (p∧q) must be true (since negation false => original true). So p∧q true implies both p true and q true. So answe... | Since \(\lnot(p\land q)\) is false, its negation must be true:
\[
\lnot(p\land q)=\text{false}\;\Longrightarrow\;p\land q=\text{true}.
\]
The conjunction \(p\land q\) is true only when **both** \(p\) and \(q\) are true.
Therefore the correct choice is
**C: \(p\) is true, \(q\) is true.** |
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 | [
"We need compute: 3,000,000 eggs / 20,000 nests = 150 eggs per nest. So answer B 150."
] | To find the average number of eggs per nest, divide the total number of eggs by the total number of nests:
\[
\frac{3{,}000{,}000\ \text{eggs}}{20{,}000\ \text{nests}} =
\frac{3{,}000{,}000}{20{,}000}=150
\]
So each nest contains, on average, **150 eggs**.
**Answer: B – 150** |
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 ______. | [
"The question: \"To make the fraction $\\frac{3}{{x-2}}$ meaningful, the value of $x$ should satisfy ______.\"",
"Interpretation: They want the condition that denominator is non-zero. So x-2 ≠ 0 => x ≠ 2. So the answer is x ≠ 2 or x ∈ ℝ\\{2}. The blank likely expects \"x ≠ 2\". So respond: $x \\neq 2$."
] | The denominator must not be zero, so
\[
x-2\neq 0\qquad\Longrightarrow\qquad x\neq 2.
\] |
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. $\int \frac{\sin 2 x}{\cos x} d x$. | [
"The user wants translation into English, retaining original line breaks and format. The original text is:",
"\"77. $\\int \\frac{\\sin 2 x}{\\cos x} d x$.\"",
"This is already essentially math expression. Possibly they'd like to translate \"77.\" as \"77.\" perhaps remains. The phrase \"Translate the above tex... | 77. $\displaystyle \int \frac{\sin 2 x}{\cos x}\,dx$ |
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 $P \cap T$. | [
"We need to find set P: solutions to x^2 - x - 2 = 0. Solve: x = [1 ± sqrt(1 + 8)]/2 = [1 ± 3]/2. So roots: (1+3)/2 = 2, (1-3)/2 = -1. So P = {2, -1}. T is interval (-1, 2] meaning -1 < x ≤ 2. Intersection of {2, -1} with that interval: -1 is not included because interval is open at -1, so -1 not in T. 2 is include... | First solve for the elements of \(P\).
\[
x^2 - x - 2 = 0 \;\Longrightarrow\; (x-2)(x+1)=0
\]
Hence
\[
P = \{-1,\; 2\}.
\]
The set \(T\) is the interval
\[
T = \{x\mid -1 < x \le 2\}=(-1,\,2] .
\]
Now intersect the two sets:
- \(-1\) is **not** in \(T\) because the interval is open at \(-1\).
- \(2\) **is** ... |
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 | [
"The question: \"The program obtained by using the bisection method to solve the equation $x^2-2=0$ is ( ) \nA: Organizational structure diagram \nB: Process flow diagram \nC: Knowledge structure diagram \nD: Program flowchart\"",
"Probably they ask what type of diagram represents the program obtained by usi... | **Answer: D – Program flowchart**
The algorithm that results from applying the bisection method to solve \(x^2-2=0\) is typically depicted as a **program flowchart**, showing the sequence of steps (initial interval, midpoint calculation, sign test, interval update, iteration, termination) in a graphical form. |
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: $0$ | [
"We need to answer: fraction x/(2 - x) is meaningful means denominator not zero. So 2 - x != 0 => x != 2. Also denominator cannot be undefined or cause other issues? It's just real numbers. So x cannot be 2. So answer choice B.",
"We need to output answer."
] | The fraction
\[
\frac{x}{2-x}
\]
is defined (meaningful) as long as its denominator is not zero.
\[
2-x \neq 0 \quad\Longrightarrow\quad x \neq 2.
\]
Thus the only value among the choices that makes the fraction undefined is
**B: \(2\).** |
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. | [
"We have a question: \"Among the following descriptions, the one that can determine a point's location is ______.\"",
"Choices:",
"① East longitude 118°, north latitude 40°\n② Baoji City, Wenhua East Road\n③ 60° northeast\n④ Oscar Cinema, Hall 1, Row 3.",
"We need to choose which description can determine a p... | ① East longitude 118°, north latitude 40° – this pair of latitude and longitude coordinates uniquely fixes a point on the Earth’s surface. |
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... | [
"We need to answer multiple choice: given variables x and y are linearly related, regression equation y = 3 - 2x. The slope is -2, negative. So it's a negative linear correlation. Option B.",
"Thus answer B."
] | The regression line is
\[
y = 3 - 2x
\]
Its slope is \(-2\), which is negative. A negative slope means that as \(x\) increases, \(y\) decreases, indicating a **negative linear correlation** between the variables.
**Answer:** **B – Negative linear correlation**. |
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 $\beta$. | [
"We have two planes with normal vectors n1 = (1,2,-2) and n2 = (-2,-4,k). Two planes are parallel if their normal vectors are parallel, i.e., one is a scalar multiple of the other. So there exists λ such that n2 = λ n1. So (-2, -4, k) = λ (1, 2, -2) => -2 = λ*1 => λ = -2. Then we check -4 = λ*2 = -2*2 = -4 OK. Then... | Two planes are parallel precisely when their normal vectors are parallel.
Hence there must be a scalar λ such that
\[
(-2,\,-4,\,k)=\lambda\,(1,\,2,\,-2).
\]
From the first component
\[
-2=\lambda\cdot1 \;\Longrightarrow\; \lambda=-2 .
\]
Check the second component:
\[
-4 =\lambda\cdot2 = (-2)\cdot2 = -4,
... |
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 | [
"We need to parse problem: \"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?\" Not sure about weird symbols φ perhaps stands for cent? Anyway, we have quarters worth 25 cents... | Let
- \(q\) = number of quarters,
- \(d\) = number of dimes,
- \(n\) = number of nickels.
The problem states that the three quantities are equal, so let that common number be \(k\).
\[
q = d = n = k
\]
The monetary value of each type of coin is
- Quarter: \(25\) cents,
- Dime: \(10\) cents,
- Nickel:... |
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 | [
"We need compute 20% of 20. 20% = 0.20. 0.20 * 20 = 4. So answer (E) 4."
] | \(20\% = \frac{20}{100}=0.20\).
\[
0.20 \times 20 = 4
\]
So \(20\%\) of 20 equals **4**, which corresponds to choice **(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: $[1, +\infty)$ | [
"The problem: \"If the set A = {x | x^2 - 2x + m = 0} = ∅, then the range of the real number m is ( )\". So A is the set of real solutions x to quadratic equation x^2 - 2x + m = 0. For the set to be empty, equation has no real solutions (i.e., discriminant < 0). So discriminant D = b^2 - 4ac = (-2)^2 - 4*1*m = 4 - ... | The quadratic equation
\[
x^{2}-2x+m=0
\]
has real solutions when its discriminant
\[
\Delta = (-2)^{2}-4\cdot1\cdot m = 4-4m
\]
is **non‑negative**.
For the solution set \(A\) to be empty, the equation must have **no real roots**, i.e.
\[
\Delta < 0 \;\Longrightarrow\; 4-4m<0 \;\Longrightarrow\; m>1.
\]
... |
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. | [
"We need to fill blank? Probably answer: 400x yuan. The question: \"A school plans to purchase 400 workbooks, with each workbook costing x yuan. The school should pay a total of __ yuan.\" So answer: 400x. Could also be \"400x yuan\". Provide explanation."
] | The total cost is obtained by multiplying the number of workbooks by the price of each one:
\[
\text{Total cost}= (\text{number of workbooks}) \times (\text{price per workbook}) = 400 \times x.
\]
So the school should pay **\(400x\) yuan**. |
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: $\sqrt{9}$ | [
"We need to answer which is irrational: 0.7 = 7/10 rational, 1/2 rational, sqrt2 irrational, sqrt9 = 3 rational. So answer C. Provide explanation."
] | The only number among the four that cannot be expressed as a ratio of two integers is
\[
\boxed{\sqrt{2}}
\]
**Why:**
- \(0.7 = \dfrac{7}{10}\) – a terminating decimal, hence rational.
- \(\dfrac12\) is obviously a ratio of integers, so it is rational.
- \(\sqrt{9}=3\) is an integer, therefore rational.
- \(... |
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: $(-3a)^{2}=-9a^{2}$ | [
"We need to identify which calculation is correct. Evaluate each.",
"A: a^2 * a^6 = a^(2+6) = a^8 correct.",
"B: a^6 ÷ a^3 = a^(6-3)=a^3, not a^2, so incorrect.",
"C: 2a^2 + a^6 =? cannot combine unlike terms; 2a^2 + a^6 ≠ 3a^8. Incorrect.",
"D: (-3a)^2 = (-3)^2 * a^2 = 9a^2, not -9a^2. So incorrect.",
"T... | **A is the only correct calculation.**
Here’s why each choice evaluates the way it does:
---
### A. \(a^{2}\cdot a^{6}=a^{8}\)
When multiplying powers with the same base, we add the exponents:
\[
a^{2}\cdot a^{6}=a^{2+6}=a^{8}.
\]
This follows the law \(a^{m}\,a^{n}=a^{m+n}\). ✓
---
### B. \(a^{6}\div a^{3}=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. | [
"We have a problem: Wang Lei was born on National Day. In many countries, National Day could be Oct 1 (China), July 4 (US), etc. Likely Chinese context: National Day is October 1. So his birthday is October 1. He was born on a certain year; his age this year? The condition: three times his age minus 8 equals exactl... | Since Wang Lei’s birthday is on his country’s National Day, the month of his birth must have the number of days that satisfies
\[
3(\text{age})-8 = \text{(days in month)}.
\]
The only month that yields an integer age is a 31‑day month, because
* 30 days → \(3a-8=30\Rightarrow a=\frac{38}{3}\) (not an integer)
... |
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? | [
"We need to answer: 33 total, 8 won medals (first, second, third place). The rest got certificates. So 33-8=25. Answer: 25 pioneers received certificates."
] | There are
\[
33 \text{ total pioneers} - 8 \text{ medalists} = 25
\]
pioneers who did not win a medal.
\[
\boxed{25}
\] |
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... | [
"The question: \"Variance is a measure of the magnitude of fluctuations in a set of data. For a set of data $x_1, x_2, ... x_n$, the variance can be calculated using the formula: $S^2 = \\frac{1}{n} [(x_1 - 3)^2 + (x_2 - 3)^2 + \\dots + (x_n - 3)^2]$, where '3' represents the ( )\". The answer: variance uses the me... | **Answer: B – mean**
In the variance formula each observation is subtracted from the **mean** of the data set (here the mean equals 3), squared, summed, and then divided by the number of observations. The other options (minimum, mode, median) are not used in the definition of variance. |
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 $x=$ ______. | [
"The problem: vectors a = (1,2) and b = (x, -2). They are parallel (collinear). For 2D vectors, a // b means one is a scalar multiple of the other: (1,2) = k (x, -2) or (x, -2) = k (1,2). Solve for x.",
"Let's set (x, -2) = k (1,2). Then x = k*1 = k, -2 = k*2 => k = -1. So x = k = -1. Check: b = (-1, -2). Is that... | \[
\begin{aligned}
\overrightarrow b &= (x,\,-2) \\
\overrightarrow a &= (1,\,2)
\end{aligned}
\]
For two vectors to be parallel one must be a scalar multiple of the other:
\[
(x,\,-2)=k\,(1,\,2)
\]
Thus
\(x = k\) and \(-2 = 2k\).
From the second equation \(k = -1\); substituting gives \(x = -1\).
\[
\boxed{-1}
... |
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: $y=\left(x-1\right)^{2}$ | [
"The problem: \"Translate the graph of the function y = x^2 up by one unit to obtain the graph of which function?\" Upward translation adds 1 to y: y = x^2 + 1. So answer C.",
"Thus need to respond with answer C and maybe explanation."
] | To shift a graph **upward** by one unit you add 1 to the entire right‑hand side of the equation.
Starting with
\[
y = x^{2},
\]
the upward translation gives
\[
y = x^{2}+1 .
\]
So the correct choice is **C: \(y = x^{2}+1\)**. |
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: $|5-3|=-\left(5-3\right)$ | [
"The user asks: Which of the following equations is correct? Options A-D.",
"We need to evaluate each.",
"A: 0 - (-3) = 3. Since subtracting negative is adding: 0 - (-3) = 0 + 3 = 3. So true.",
"B: (-14) - 5 = -9. Compute -14 - 5 = -19, not -9. So false.",
"C: (-3) - (-3) = -6. Compute -3 - (-3) = -3 + 3 = ... | Let’s check each equation one by one.
---
### A) \(0-\left(-3\right)=3\)
Subtracting a negative is the same as adding its opposite:
\[
0-\left(-3\right)=0+3=3.
\]
**True.**
---
### B) \((-14)-5=-9\)
\[
(-14)-5 = -14-5 = -19\neq -9.
\]
**False.**
---
### C) \((-3)-\left(-3\right)=-6\)
\[
(-3)-\left(-3\right)... |
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 $N$ ?
## | [
"We have square area 9 m^2, so side length sqrt(9)=3 m. Perimeter = 4*side=12 m. So N=12."
] | The area of a square is the side length squared.
\[
\text{area}=s^{2}=9 \text{ m}^2 \;\Longrightarrow\; s=\sqrt{9}=3 \text{ m}.
\]
The perimeter of a square is four times the side length:
\[
N = 4s = 4\times 3 = 12 \text{ m}.
\]
\[
\boxed{N = 12}
\] |
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? | [
"We need to parse the problem: \"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?\" So 3 from class 1a, equal number from class 1b means also 3. So total 3 + 3 = 6. Answer: 6.",
"But maybe t... | Three Pioneers from class 1a get certificates, and the same number (three) from class 1b also get certificates.
\[
3 \text{ (class 1a)} + 3 \text{ (class 1b)} = 6
\]
**Answer: 6 Pioneers will be awarded.** |
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 | [
"We need compute integral from 0 to1 of (e^x + 2x) dx. Integral of e^x is e^x, evaluate 0 to1 gives e -1. Integral of 2x dx = x^2 from 0 to1 =1. Sum = (e -1)+1 = e. Therefore answer D: e."
] | The integral can be split into two simpler parts:
\[
\int_{0}^{1}\big(e^{x}+2x\big)\,dx
= \int_{0}^{1}e^{x}\,dx \;+\; \int_{0}^{1}2x\,dx .
\]
1. **Integrate \(e^{x}\):**
\[
\int e^{x}\,dx = e^{x} \quad\Longrightarrow\quad
\int_{0}^{1} e^{x}\,dx = e^{1}-e^{0}=e-1 .
\]
2. **Integrate \(2x\):**
\[
... |
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 ____. | [
"We need to answer fill-in-the-blank: For selecting one basketball out of 10 randomly to test quality, the sampling method is \"simple random sampling\". Could also be \"simple random sampling\" as individual selection. No replacement needed because only one selected. So answer: simple random sample (SRS)."
] | **Simple random sampling**. |
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... | [
"We need to respond with translation keeping line breaks and format, output translation directly. The user text appears to be already English, so probably return the same. They want translation result directly. So output the same lines.",
"Thus produce:",
"274. $\\int_{0}^{1 / 2} \\frac{d x}{\\sqrt{1-x^{2}}}$."... | 274. $\int_{0}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}}$. |
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 | [
"The problem: Given set M = {-1,1}, the number of sets N that satisfy N ⊆ M (i.e., all subsets of M). How many subsets does a set with two elements have? 2^2 = 4. So answer D: 4. Possibly they're asking number of subsets. So answer D.",
"Thus respond: D."
] | The set \(M\) has two elements.
A set with \(k\) elements has \(2^{k}\) subsets.
Here \(k = 2\), so the number of subsets is
\[
2^{2}=4.
\]
Thus there are four possible sets \(N\) with \(N\subseteq M\).
**Answer: D – 4**. |
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 | [
"We need to answer multiple-choice question: \"Sampling methods include ( )\". Options list combinations of sampling methods. Probably question from statistics: sampling methods typically include random sampling (simple random), systematic sampling, stratified sampling, cluster sampling, multistage, etc. The option... | **C: Simple random sampling, stratified sampling, and systematic sampling**. |
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. | [
"We need to parse the problem. It says: \"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 ___ points.\"",
"Interpretation: father total score = 48 points (over 5 throws). Xiao... | The father’s total score was
\[
48\text{ points}.
\]
Half of this total is
\[
\frac{48}{2}=24\text{ points}.
\]
Xiao Pang’s score is **8 points less** than that amount, so
\[
\text{Xiao Pang’s score}=24-8=16\text{ points}.
\]
\[
\boxed{16}
\] |
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: $(-3a)^{2}=9a^{2}$ | [
"We need to determine which of the given statements is correct (mathematical equality). Let's examine each.",
"A: 3a + b = 3ab. This would only be true for special values, not generally. Unless a,b satisfy something, but as an identity it's false.",
"B: a^2 + a^2 = a^4. Left side = 2 a^2. Right side = a^4. Not ... | **Answer: D.**
\[
(-3a)^{2}=9a^{2}
\]
**Why the others are not correct**
| Statement | Reason it’s false (as an identity) |
|-----------|------------------------------------|
| **A:** \(3a+b=3ab\) | The left‑hand side is linear in \(a\) and \(b\); the right‑hand side is quadratic. They are equal only for special v... |
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: $(-2,-1)$ | [
"The question: point (-2,3) after moving 4 units to the right. Moving to the right increases x-coordinate by 4. So new x = -2 + 4 = 2, y unchanged 3. So answer (2,3). Option A."
] | Moving a point to the right changes only its \(x\)-coordinate, increasing it by the distance moved.
Original point: \((-2,\,3)\)
Shift: \(4\) units to the right \(\Rightarrow\) add \(4\) to the \(x\)-coordinate.
\[
x_{\text{new}} = -2 + 4 = 2,\qquad
y_{\text{new}} = 3
\]
So the new coordinates are \((2,3)\).
**An... |
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: $(x-1)^2+y^2=4$ | [
"We need to answer multiple choice: equation of circle with center (0,1) radius 2. General: (x - h)^2 + (y - k)^2 = r^2. So (x - 0)^2 + (y - 1)^2 = 4. That's x^2 + (y - 1)^2 = 4. Option C."
] | The standard form of a circle centered at \((h,k)\) with radius \(r\) is
\[
(x-h)^2+(y-k)^2=r^{2}.
\]
Here the center is \((0,1)\) so \(h=0,\;k=1\), and the radius is \(r=2\) so \(r^{2}=4\).
\[
(x-0)^2+(y-1)^2=4 \;\;\Longrightarrow\;\; x^{2}+(y-1)^{2}=4.
\]
**Correct choice:** **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. $\int_{-\infty}^{0} e^{x} d x$. | [
"The user asks: \"Example 2.25. $\\int_{-\\infty}^{0} e^{x} d x$.\" They want to translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly. The text is already in English? Actually \"Example 2.25. $\\int_{-\\infty}^{0} e^{x} d x$.\" It's al... | Example 2.25. $\int_{-\infty}^{0} e^{x} d x$. |
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: $\frac{2a}{b}$ | [
"We need to compute log2(9/5) in terms of a=log2 3, b=log2 5.",
"log2(9/5)=log2 9 - log2 5 = log2 (3^2) - b = 2 log2 3 - b = 2a - b. That's option B.",
"Thus answer B.",
"But perhaps they want an expression using a and b. Indeed B.",
"Provide reasoning.",
""
] | We have
\[
\log_{2}\frac{9}{5}= \log_{2}9-\log_{2}5.
\]
Since \(9 = 3^{2}\),
\[
\log_{2}9 = \log_{2}3^{2}=2\log_{2}3 = 2a.
\]
Also \(\log_{2}5 = b\).
Hence
\[
\log_{2}\frac{9}{5}=2a-b.
\]
So the correct choice is **B: \(2a-b\)**. |
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: $720^{\circ}$ | [
"The user asks: \"What is the sum of the interior angles of a quadrilateral?\" Options: 180, 360, 540, 720 degrees. The answer is 360 degrees (since sum = (n-2)*180 = (4-2)*180 = 360). So answer B.",
"Probably they just want the answer. Could explain reasoning."
] | The sum of the interior angles of an \(n\)-sided polygon is
\[
\text{Sum}= (n-2)\times 180^{\circ}.
\]
For a quadrilateral, \(n = 4\):
\[
\text{Sum}= (4-2)\times 180^{\circ}=2 \times 180^{\circ}=360^{\circ}.
\]
**Answer: B – \(360^{\circ}\).** |
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? | [
"We need to parse problem: \"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?\" So we have unknown: weight of a gnome without shoes = x kg. Then... | Let
* \(w\) = weight of a gnome **without** shoes (in kilograms)
* \(w+2\) = weight of a gnome **with** shoes (2 kg heavier)
The scale holds five gnomes in shoes and five without shoes:
\[
5(w+2) \;+\; 5w \;=\; 330 \text{ kg}
\]
Simplify:
\[
5w + 10 + 5w = 330 \\
10w + 10 = 330 \\
10w = 320 \\
w = 32 \text{ 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 $a=\_\_\_\_\_\_$. | [
"We need to interpret: point P(4, 2 - a) lies in the first quadrant, given a is positive integer. First quadrant: x>0, y>0. x-coordinate is 4>0 already. y-coordinate is 2 - a > 0 => 2 - a > 0 => a < 2. Since a is positive integer, possible values: 1 only (since a<2, a>0 integer). So a = 1. Answer: 1."
] | Since a point is in the first quadrant its coordinates must both be positive.
- The \(x\)-coordinate of \(P\) is \(4>0\) (always satisfied).
- The \(y\)-coordinate is \(2-a\).
For it to be positive:
\[
2-a>0\quad\Longrightarrow\quad a<2.
\]
Given that \(a\) is a **positive integer**, the only integer satisfyin... |
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 | [
"We need calculate (2^3)^2 - 4^3. 2^3 =8. (8)^2 =64. 4^3 =64. 64 -64 =0. Answer A."
] | First evaluate each part:
\[
(2^{3})^{2}= (8)^{2}=64
\]
\[
4^{3}=64
\]
Now subtract:
\[
64-64 = 0
\]
So the value is \(0\).
**Answer: (A) 0** |
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. | [
"The user says: \"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 pieces of cabbage need to be prepared.\"",
"Looks like a Chinese math problem: We need to fill in the blank (how many pieces of cabbage nee... | Each large rabbit needs \(4\) pieces of cabbage per day, and each small rabbit needs \(2\) pieces.
- Two large rabbits: \(2 \times 4 = 8\) pieces
- One small rabbit: \(1 \times 2 = 2\) pieces
Total cabbage needed per day
\[
8 + 2 = 10 \text{ pieces}.
\]
**Answer:** \(10\) pieces of cabbage must be prepared eac... |
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 | [
"We need to answer the multiple choice question: \"The three logical structures of an algorithm are ( )\". Options given. The correct answer: typical three basic control structures in algorithms: sequence, selection (conditional), and iteration (loop). So answer is C: Sequential structure; Conditional structure; ... | The three fundamental logical (control) structures that any algorithm can be built from are
1. **Sequential structure** – statements executed one after another.
2. **Conditional (selection) structure** – a branch that chooses between alternatives (e.g., if‑else).
3. **Loop (iteration) structure** – repeated exec... |
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... | [
"We need to parse the problem.",
"The announcement says earn up to 2500 rubles a day working as a courier. Minimum load is 1250 rubles a day after income tax deduction. The question: \"How much will a school student working as a courier with the service 'Food.There-Here!' earn in a month (4 weeks), working 4 days... | 20000 |
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: $2^{-1}=\frac{1}{2}$ | [
"We need to answer which of the following calculations is correct. Evaluate each.",
"A: 3x+2x = (3+2)x =5x. Not 5x^2. So incorrect.",
"B: sqrt(9) = ±3? By definition principal square root of a nonnegative real is nonnegative, sqrt(9)=3, not ±3. So incorrect.",
"C: (2x)^2 = (2x)*(2x) =4x^2. Not 2x^2. So incorr... | Let’s examine each statement one by one.
---
### A. \(3x+2x = 5x^{2}\)
Using the distributive law,
\[
3x+2x = (3+2)x = 5x,
\]
not \(5x^{2}\).
**Incorrect.**
---
### B. \(\sqrt{9}= \pm 3\)
The radical sign \(\sqrt{\phantom{a}}\) denotes the **principal (non‑negative) square root**. Therefore
\[
\sqrt{9}=3,
\]
wh... |
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: $\sqrt{3}$ | [
"The user provides a question: \"$\\sqrt{9}=\\left(\\ \\ \\right)$\". Multiple choice answers: A: ±3, B: 3, C: ±√3, D: √3.",
"We need to select the correct answer. The principal square root of 9 is 3. So answer B: 3. But some might think it's ±3? However the sqrt symbol denotes principal (non-negative) root. So a... | The symbol “\(\sqrt{\phantom{x}}\)” denotes the **principal (non‑negative) square root**.
Since \(3^2 = 9\) and \((-3)^2 = 9\) as well, the square‑root function picks the non‑negative solution:
\[
\sqrt{9}=3.
\]
So the correct choice is
**B: 3**. |
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 | [
"We need to answer: If variance of a set of data a_i is 2, then variance of scaled data 2 a_i is what? Usually variance scales by factor squared of scaling: var(cX) = c^2 var(X). So var(2a_i) = 4 * 2 = 8. Answer: C.",
"But note that there is possibility of sample vs population variance. Regardless, scaling multip... | The variance measures how far the data points are spread around their mean.
If each observation \(a_i\) is multiplied by a constant \(c\), the new observations are \(c a_i\).
Let \(\bar a\) be the original mean. Then the new mean is
\[
\overline{c a}=c\bar a .
\]
The variance of the transformed data is
\[
\be... |
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. | [
"We need to identify correct statement about a pyramid (geometric polyhedron). Let's recall definitions: A pyramid is a polyhedron formed by connecting a polygonal base to a point called the apex. The base is an n-gon, pyramid has n triangular lateral faces plus the base. So total faces: n+1. Number of vertices: n+... | A pyramid is formed by joining all vertices of a single polygonal base to a single point (the apex).
Consequently:
| Property | Formula (for an \(n\)-gon base) |
|----------|---------------------------------|
| Number of **faces** | \(n\) triangular lateral faces + 1 base = \(n+1\) |
| Number of **vertices** | \(n\)... |
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?
# | [
"We need to interpret problem: \"Rabbits are sawing a log. They made 10 cuts. How many pieces/chunks did they get?\" Classic puzzle: With n cuts, maximum number of pieces is n+1 if cuts are done sequentially, but if you can rearrange pieces and cut multiple at once, could be more. For a log (a straight line segment... | Each cut can only increase the number of pieces by **one** – a cut separates one existing piece into two.
So after the first cut you have 2 pieces, after the second you have 3, and in general after **n** cuts you have **n + 1** pieces.
With 10 cuts:
\[
\text{pieces}=10+1 = \boxed{11}
\]
*(If you were allowed to st... |
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 ... | [
"The question: \"Among the following propositions, the correct one is ( )\". Likely a multiple-choice question. Options A, B, C, D.",
"We need identify which statements are correct.",
"Interpretation:",
"A: If two vectors are equal, then their starting points and endpoints coincide respectively. Usually a vec... | **Answer: D – “Two equal vectors have equal magnitudes.”**
**Why the other statements are false**
| Proposition | Reason it is false |
|-------------|--------------------|
| **A** – “If two vectors are equal, then their starting points and endpoints coincide respectively.” | In elementary vector algebra a *vector* is... |
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... | [
"The user just gave a question and answer: \"7. In 1981, the United States developed the world's first space shuttle, the “____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 th... | Your answer is correct.
In 1981 the United States launched the first operational Space Shuttle, **Columbia** (STS‑1), which completed its first manned flight on April 12 – 13, 1981. Therefore the blank should be filled with **Columbia**, making option **C** the right choice. |
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 | [
"We need to answer which phenomenon is impossible among options. Let's analyze each:",
"A: A conductor heats up when conducting electricity – that's Joule heating, possible.",
"B: Three non-collinear points determine a plane – geometric theorem, true.",
"C: Seeds germinate without water – germination requires... | **Answer: C – “Seeds germinate without water.”**
**Why the others are possible**
- **A. A conductor heats up when conducting electricity** – This is the well‑known Joule (or resistive) heating effect. Whenever current flows through a resistive material, electrical energy is converted to thermal energy, so the conduct... |
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: $-\frac{1}{2}$ | [
"We just evaluate y = -2x at x = -1: -2 * (-1) = 2. Answer A."
] | Plugging \(x = -1\) into the function:
\[
y = -2x \;\Longrightarrow\; y = -2(-1) = 2.
\]
So the correct choice is **A: \(2\)**. |
65f54d25-90ff-5f45-b7a9-8ef6a890edce | open-r1/OpenR1-Math-220k/cn_k12 | Calculate: $-3 + 2 = \quad$; $\quad (-3) \times 2 = \quad$. | [
"The user asks: \"Calculate: $-3 + 2 = \\quad$; $\\quad (-3) \\times 2 = \\quad$.\"",
"We need to provide the results: -3 + 2 = -1; (-3) * 2 = -6.",
"Probably just answer both. Provide explanation maybe."
] | \[
-3 + 2 = -1
\]
\[
(-3)\times 2 = -6
\] |
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: $(-ab^{3})^{2}=a^{2}b^{6}$ | [
"We need to identify which of the given calculations is correct. Let's examine each.",
"A: a^8 ÷ a^2 = a^(8-2) = a^6, not a^4. So A is wrong.",
"B: 5ab - 2ab = (5-2)ab = 3ab, not 3. So B is wrong.",
"C: (a-b)^2 = a^2 -2ab + b^2, not a^2 - b^2. So wrong.",
"D: (-ab^3)^2: square the whole expression. (-1)^2=1... | **Answer: D is the only correct calculation.**
---
### Why the others are wrong
| Letter | Given calculation | Correct evaluation | Reason |
|--------|-------------------|--------------------|--------|
| **A** | \(a^{8}\div a^{2}=a^{4}\) | \(a^{8}\div a^{2}=a^{8-2}=a^{6}\) | When dividing powers with the same base... |
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: $y=0.5x+2$ | [
"We need to parse the question. It says \"Given that variables x and y are linearly related, and the sample means obtained from observational data are x̄ = 2 and ȳ = 3, the linear regression equation obtained from this observational data cannot be [options].\"",
"So which regression line cannot have that mean po... | For a simple linear regression the fitted line **must pass through the point of the sample means** \((\bar x,\bar y)\).
Given
\[
\bar x = 2,\qquad \bar y = 3,
\]
the admissible regression line must satisfy
\[
y = a x + b \quad\text{with}\quad 3 = a(2)+b .
\]
Check each candidate:
| Option | Equation | At \(x=... |
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 | [
"We need to answer the multiple choice. Regression equation ŷ = 2 + 3x. The coefficient of x is 3, meaning that for each one unit increase in x, predicted y increases by 3 units on average. So answer B. Provide explanation."
] | The regression equation
\[
\hat y = 2 + 3x
\]
is written in the form
\[
\hat y = \beta_0 + \beta_1 x,
\]
where
* \(\beta_0 = 2\) is the intercept, and
* \(\beta_1 = 3\) is the slope (regression coefficient) for \(x\).
The slope tells us how the predicted value of \(y\) changes when \(x\) changes by one un... |
0fb952fa-22d5-5c60-97ee-782a8e3c9160 | open-r1/OpenR1-Math-220k/cn_contest | Example 3. Simplify: $\sqrt{(a-5)^{2}}$ | [
"We need to answer: Simplify sqrt((a-5)^2). Typically sqrt(x^2) = |x|. So sqrt((a-5)^2) = |a-5|. Could also be expressed as |5 - a| (same). If we assume a≥5 then equals a-5 else 5-a. So answer: |a-5|. Provide reasoning."
] | \[
\sqrt{(a-5)^{2}}
\]
The square‑root of a square gives the absolute value of the original expression, because the square‑root function returns only non‑negative values:
\[
\sqrt{x^{2}} = |x|
\]
Here \(x = a-5\). Therefore
\[
\sqrt{(a-5)^{2}} = |a-5|
\]
If one wishes to write it piece‑wise:
\[
|a-5|=\begin{cases... |
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: $f(x_0 + \Delta x) - f(x_0)$ | [
"We need to answer the multiple choice: the change in function Δy when x changes from x0 to x0+Δx. Δy = f(x0+Δx) - f(x0). So answer D.",
"Provide explanation."
] | When the variable \(x\) moves from \(x_{0}\) to \(x_{0}+\Delta x\), the value of the function changes from
\[
f(x_{0})\quad\text{to}\quad f(x_{0}+\Delta x).
\]
The **change** in the function, denoted \(\Delta y\), is the difference between these two values:
\[
\Delta y = f(x_{0}+\Delta x) - f(x_{0}).
\]
Thus the ... |
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? | [
"We need to parse the problem: There are 4 first-course dishes, 5 second-course dishes, and 3 desserts. We want number of ways to compose one lunch. Typically a lunch consists of one first-course, one second-course, and one dessert? The problem states \"In the canteen for lunch, 4 first-course dishes, 5 second-cour... | A typical lunch consists of one item from each course:
* **First‑course:** 4 possibilities
* **Second‑course:** 5 possibilities
* **Dessert:** 3 possibilities
Since the choices are independent, we multiply the numbers of possibilities:
\[
\text{total lunches}=4 \times 5 \times 3 = 60 .
\]
\[
\boxed{60}
\] |
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: $2+\sqrt{2}=2\sqrt{2}$ | [
"We need to answer: Which of the following calculations is correct? Options A through D. Evaluate each statement.",
"A: sqrt(2)*sqrt(3) = sqrt(6). Indeed property sqrt(a)*sqrt(b) = sqrt(ab) for non-negative a,b. So true.",
"B: sqrt(2)+sqrt(3) = sqrt(5). Not true: sqrt2+sqrt3 ≈1.414+1.732=3.146, sqrt5≈2.236. So ... | **Answer:** Only statement **A** is correct.
---
### Why A is correct
\[
\sqrt{2}\times\sqrt{3}= \sqrt{2\cdot 3}= \sqrt{6}
\]
The rule \(\sqrt{a}\,\sqrt{b}= \sqrt{ab}\) holds for any non‑negative real numbers \(a\) and \(b\).
Since \(2\ge 0\) and \(3\ge 0\), the product of their square roots is indeed \(\sqrt{6... |
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) $\frac{4}{6}$ | [
"The user asks a simple probability question: A six-sided die with numbers 1-6, probability of rolling a five. It's 1 favorable outcome out of 6 possible equally likely outcomes: 1/6. So answer (B) 1/6. Provide explanation."
] | The die has six equally likely faces: 1, 2, 3, 4, 5, 6.
- Number of favorable outcomes (rolling a five): 1
- Total number of possible outcomes: 6
Probability \(= \dfrac{\text{favorable outcomes}}{\text{total outcomes}} = \dfrac{1}{6}\).
\[
\boxed{\frac{1}{6}}\quad\text{(choice B)}
\] |
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... | [
"We need to answer which voting method best reflects true will of all students. Options: show of hands for agree, show of hands for disagree, secret ballot, recorded vote. The best reflect true will is secret ballot (C) because it ensures anonymity and reduces pressure. Recorded vote (public) might influence. Show ... | **C: By secret ballot.**
A secret ballot lets each student vote anonymously, eliminating peer pressure, fear of retaliation, or desire to conform that can distort the true preferences of the group. Public methods such as a show of hands or a recorded (named) vote can influence how people vote, so they do not capture ... |
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 | [
"The question: \"22. China has currently built 9 tidal power stations, the ____ 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.\"",
"Options: A. Xiamen, B. Jiangxia, C. Humen, D. Sanya.",... | **C. Humen** |
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 | [
"We need respond to a quiz type question: \"The \"=\" sign was invented by the British person: A. De Morgan, B. Recorde, C. Oughtred\". The correct answer: The equals sign was invented by Robert Recorde, a Welsh mathematician (British). So answer B. Provide explanation."
] | **Answer: B. Recorde**
The symbol “=” for equality was introduced by the Welsh mathematician **Robert Recorde** in his 1557 work *The Whetstone of Wealth*. He chose the sign because “no two things can be more equal than parallel lines.”
- **A. De Morgan** – known for De Morgan’s laws in logic, not for the equals si... |
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 $b$. | [
"The user asks: \"Task 18: a=9, b=628 ; To the sixfold of a, add half of b.\" So we need to compute 6*a + b/2.",
"a=9 => sixfold = 6*9=54. b=628 => half = 314.",
"Sum = 54 + 314 = 368.",
"We need probably give answer. Might also show steps. Provide answer: 368.",
"Consider whether they want expression: 6a +... | Для вычисления требуется выполнить операцию
\[
6a+\frac{b}{2}.
\]
Подставляем заданные значения:
\[
\begin{aligned}
6a &= 6\cdot 9 = 54,\\[2mm]
\frac{b}{2} &= \frac{628}{2}=314.
\end{aligned}
\]
Теперь складываем полученные результаты:
\[
54 + 314 = 368.
\]
\[
\boxed{368}
\] |
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? | [
"We need to interpret the problem. \"In 5 windows of a residential building, there are a total of 40 glass panes. How many panes are in 3 windows?\" Assuming uniform distribution. So 40 panes /5 windows = 8 panes per window. Then 3 windows have 3*8 = 24 panes. Provide answer."
] | Each window contains the same number of panes.
\[
\text{Panes per window}= \frac{40\text{ panes}}{5\text{ windows}} = 8\text{ panes/window}
\]
For 3 windows:
\[
3 \text{ windows} \times 8\text{ panes/window}=24\text{ panes}
\]
**Answer: 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 | [
"We need to answer which proposition is true among options A-D. Probably multiple could be true or false. Let's analyze each.",
"A: If a+b < 0 then ab < 0. Not necessarily. Counterexample: a = -2, b = -1, sum = -3 <0, product = 2 >0. So false.",
"B: \"Corresponding angles are equal.\" In geometry, corresponding... | **Answer:**
- **A – false**
- **B – true** (under the usual hypothesis that the two lines are parallel)
- **C – true** (in Euclidean plane geometry)
- **D – false**
---
### Why each answer is what it is
| Proposition | Reasoning |
|-------------|-----------|
| **A.** “If \(a+b<0\) then \(ab<0\).” | Not a... |
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 | [
"We need to solve: x + 11 = -26 => x = -37. So answer (A) -37.",
"We need to respond with answer."
] | The unknown number \(x\) satisfies
\[
x + 11 = -26.
\]
Subtract 11 from both sides:
\[
x = -26 - 11 = -37.
\]
So the other number is \(-37\).
**Answer: (A) \(-37\).** |
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. | [
"We need to evaluate each statement's correctness.",
"A: \"A quadrilateral with a pair of adjacent equal sides is a rhombus.\" Not necessarily. A kite has a pair of adjacent equal sides but not all sides equal, not a rhombus. So false.",
"B: \"A parallelogram with one right angle is a rectangle.\" In a parallel... | **Answer: B – “A parallelogram with one right angle is a rectangle.”**
Explanation of the other choices
- **A** is false. A quadrilateral that has only one pair of adjacent equal sides (e.g., a kite) need not have all four sides equal, so it is not necessarily a rhombus.
- **C** is false. In a parallelogram, pe... |
03b5e322-6957-5b72-8664-b66107e963a3 | open-r1/OpenR1-Math-220k/olympiads | 140. An example of multiplication. One morning at breakfast, the Crackhams were discussing high matters, when suddenly George asked his sister Dora to quickly multiply
$$
1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 0
$$
How long will it take the reader to find this product? | [
"We need to answer: \"How long will it take the reader to find this product?\" The product given includes a factor 0, so product is 0. The answer likely: immediate, essentially zero time. But maybe they want a comedic answer: It's the product of numbers 1 through 10, which includes 0, so product is 0; the answer is... | The answer is **zero**, and you’ll know it essentially the instant you glance at the expression.
Because the factor list contains a “0”, the whole product collapses to
\[
1\times2\times3\times4\times5\times6\times7\times8\times9\times\mathbf{0}=0 .
\]
As soon as the reader sees the final factor, the computation is... |
54c8d808-f0dd-56ef-b2a0-ed5f65df4823 | open-r1/OpenR1-Math-220k/cn_k12 | The opposite of $-6$ is ____,The absolute value of $-8$ is ____,The reciprocal of $-3$ is ____. | [
"The user asks: \"The opposite of $-6$ is ____,The absolute value of $-8$ is ____,The reciprocal of $-3$ is ____.\"",
"They likely want fill-in-the-blank answers: opposite of -6 is 6 (since opposite means additive inverse). Absolute value of -8 is 8. Reciprocal of -3 is -1/3.",
"So answer: 6, 8, -1/3.",
"Need... | - The opposite of \(-6\) is **\(6\)**.
- The absolute value of \(-8\) is **\(8\)**.
- The reciprocal of \(-3\) is **\(-\dfrac13\)**. |
e61c020a-094e-58be-998b-4f26d43bc267 | open-r1/OpenR1-Math-220k/cn_k12 | Variance is a measure of the degree of data fluctuation. For a set of data $x_{1}$, $x_{2}$, $x_{3}$, $\ldots$, $x_{n}$, the variance can be calculated using the following formula: $s^{2}=\frac{1}{n}[(x_{1}-5)^{2}+(x_{2}-5)^{2}+(x_{3}-5)^{2}+\ldots +(x_{n}-5)^{2}]$, where "5" represents the ( ).
A: minimum value
B: m... | [
"We need to answer: In variance formula, the term subtracted is the mean. The formula given uses 5, implying the mean is 5. So answer B. Provide explanation."
] | The expression inside each squared term is the deviation of an observation from the **mean** of the data set.
In the generic formula for the (population) variance we write
\[
\sigma^{2}= \frac{1}{n}\sum_{i=1}^{n}(x_i-\mu)^2,
\]
where \(\mu\) is the arithmetic mean of the \(x_i\)’s.
In the problem the mean has ... |
5a00726b-717c-5875-98bc-26048b0a7cfd | open-r1/OpenR1-Math-220k/cn_k12 | The absolute value of $-\frac{3}{2}$ is ( ).
A: $-\frac{2}{3}$
B: $-\frac{3}{2}$
C: $\frac{2}{3}$
D: $\frac{3}{2}$ | [
"We need to answer absolute value of -3/2 is 3/2. So option D. Provide answer."
] | The absolute value of a number is its distance from 0 on the number line, which is always non‑negative.
\[
\bigl|-\tfrac{3}{2}\bigr| = \tfrac{3}{2}
\]
So the correct choice is **D: \(\displaystyle \frac{3}{2}\).** |
118d20e8-b6c9-536b-a802-54e6313d0940 | open-r1/OpenR1-Math-220k/olympiads | The integer 42 is
(A) an odd number
(D) divisible by 7
(B) a prime number
(C) a perfect cube | [
"We need to answer which statements about integer 42 are true. Options A-E? The problem lists (A) an odd number, (D) divisible by 7, (B) a prime number, (C) a perfect cube. We need choose correct statements. 42 is even, not odd. So A false. Divisible by 7? 42/7=6, yes true, D true. 42 is not prime (divisible by 2,3... | 42 is an **even** integer, so it is **not** odd → (A) false.
It has factors 1, 2, 3, 6, 7, 14, 21, 42, so it is **not** prime → (B) false.
The cube of 3 is 27 and the cube of 4 is 64; 42 is not equal to any integer’s cube → (C) false.
\(42 \div 7 = 6\), an exact integer, so 42 **is divisible by 7** → (D) true.
*... |
a0d08422-fb1c-580d-8aa4-25ba98932b17 | open-r1/OpenR1-Math-220k/cn_k12 | Given a regression equation $y = 3 - 5x$, when the variable $x$ increases by one unit,
A. $y$ decreases by an average of 5 units.
B. $y$ increases by an average of 3 units.
C. $y$ decreases by an average of 3 units.
D. $y$ increases by an average of 5 units. | [
"We need to interpret regression equation y = 3 - 5x. This is a linear relationship: slope = -5. For each increase of x by 1 unit, y changes by slope * Δx = -5 * 1 = -5. So y decreases by 5 units. So answer A.",
"But need to ensure wording: \"average\" maybe implying average change. So yes.",
"Return answer A. ... | The regression equation
\[
y = 3 - 5x
\]
has a slope of \(-5\). The slope tells us how much \(y\) changes for a one‑unit change in \(x\):
\[
\Delta y = (\text{slope}) \times \Delta x = -5 \times 1 = -5.
\]
Thus, when \(x\) increases by one unit, \(y\) **decreases by 5 units** (on average).
**Correct choice:** **... |
3c075cb4-c664-5ae4-8a47-fe13d57719db | open-r1/OpenR1-Math-220k/olympiads | 18. (6 points) Two ribbons, A and B, were originally the same length. If ribbon A is cut by 18 meters and ribbon B is cut by 12 meters, the length of ribbon B is then twice the length of ribbon A. What was the original length of ribbons A and B? | [
"We need to solve: Let original length = L (same for both). After cutting: A length = L - 18. B length = L - 12. Given B length = 2 * A length. So L - 12 = 2(L - 18). Solve: L - 12 = 2L - 36 => bring terms: -12 + 36 = 2L - L => 24 = L. So original length = 24 meters.",
"Check: A after cut = 24 - 18 = 6. B after c... | Let the original length of each ribbon be \(L\) meters.
* After cutting, ribbon A has length \(L-18\).
* After cutting, ribbon B has length \(L-12\).
According to the problem, the shortened ribbon B is twice as long as the shortened ribbon A:
\[
L-12 = 2\,(L-18)
\]
Solve for \(L\):
\[
\begin{aligned}
L-12 &= 2L-36... |
f6b1628b-b3fb-5592-b5fe-e47fe02e381e | open-r1/OpenR1-Math-220k/olympiads | 2. If $x+2 y=14$ and $y=3$, what is the value of $2 x+3 y$ ? | [
"We need to solve: x + 2y = 14, y = 3. Then x = 14 - 2y = 14 - 6 = 8. Compute 2x + 3y = 2*8 + 3*3 = 16 + 9 = 25. Answer: 25."
] | We are given
\[
x+2y = 14 \qquad\text{and}\qquad y = 3.
\]
First find \(x\):
\[
x = 14 - 2y = 14 - 2(3) = 14 - 6 = 8.
\]
Now compute \(2x + 3y\):
\[
2x + 3y = 2(8) + 3(3) = 16 + 9 = 25.
\]
\[
\boxed{25}
\] |
cef2c88d-0c60-5e13-b67b-28dc7f5eb589 | open-r1/OpenR1-Math-220k/cn_k12 | Given $p$: $x=2$, $q$: $0 < x < 3$, then $p$ is the $(\quad)$ of $q$.
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither necessary nor sufficient condition | [
"We need to interpret: Given p: x = 2 , q: 0 < x < 3 . Then p is the ______ of q.",
"We consider logical relationship between statements p and q.",
"p: \"x = 2\". q: \"0 < x < 3\". For any real x, p implies q? If x=2 then indeed 0<2<3, so p => q. Does q imply p? No, q includes many values (e.g., x=1). So p is s... | Since every time \(x=2\) is true we automatically have \(0<x<3\) (because \(2\) lies between \(0\) and \(3\)), the statement \(p\) guarantees \(q\).
However, \(q\) can be true without \(p\) being true (e.g., \(x=1\) satisfies \(0<x<3\) but not \(x=2\)).
Thus \(p\) is a **sufficient but not necessary** condition fo... |
b8d229ee-96ed-508d-8475-9168f59c7992 | open-r1/OpenR1-Math-220k/cn_k12 | If $p: x > 2$, and $q: x > 3$, then $p$ is a condition of $q$. (Choose the appropriate one from "sufficient but not necessary", "necessary but not sufficient", "necessary and sufficient", and "neither sufficient nor necessary" to fill in the blank) | [
"We need to interpret statement: \"If p: x > 2, and q: x > 3, then p is a condition of q.\" We need to fill blank: p is a condition of q. Determine relationship: Is p sufficient or necessary for q? Let's think: p: x > 2. q: x > 3.",
"If x > 3, then automatically x > 2. So q implies p. Therefore p is necessary for... | \(p\) is **necessary but not sufficient** for \(q\).
*Explanation:*
- Whenever \(q\) is true (\(x>3\)), \(p\) must also be true because any number greater than 3 is automatically greater than 2. Hence \(p\) is a **necessary** condition for \(q\).
- However, \(p\) being true (\(x>2\)) does **not** guarantee that \(... |
e6398af4-6148-5fe5-83e3-ed68e2765c8b | open-r1/OpenR1-Math-220k/amc_aime | Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned $6$ years old, she received a newborn kitten as a birthday present. Today the sum of the ages of the two children and the kitten is $30$ years. How many years older than Bella is Anna?
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qqu... | [
"We need to parse problem: Anna and Bella birthdays together. Five years ago, Bella turned 6 years old, and she received a newborn kitten as a birthday present. That implies five years ago Bella's age was 6. So currently Bella's age = 6 + 5 = 11. The kitten was newborn then, so age of kitten now = 5 years.",
"Sum... | Five years ago Bella was **6** years old.
So today Bella is
\[
6 + 5 = 11 \text{ years old}.
\]
The kitten was a newborn when Bella turned 6, so the kitten is now
\[
5 \text{ years old}.
\]
Let \(A\) be Anna’s current age.
We are told that the sum of the ages of the two children and the kitten is 30:
\[
A ... |
081f3194-8944-5871-a81f-7c02c9164b4e | open-r1/OpenR1-Math-220k/cn_k12 | From a bag containing 3 red balls and 2 white balls, if you draw 3 balls at random, and event A is defined as "at least one of the three drawn balls is white," then what is the complementary event to A?
A: 1 white ball and 2 red balls
B: 2 white balls and 1 red ball
C: 3 red balls
D: At least one red ball | [
"The user asks: \"From a bag containing 3 red balls and 2 white balls, if you draw 3 balls at random, and event A is defined as 'at least one of the three drawn balls is white,' then what is the complementary event to A?",
"Options: A: 1 white ball and 2 red balls, B: 2 white balls and 1 red ball, C: 3 red balls,... | The event **A** is “at least one of the three drawn balls is white.”
Its complement consists of all outcomes in which **no white ball is drawn** – i.e., all three balls drawn are red.
That matches option **C: 3 red balls**. |
976abd33-6c17-5d9a-a43a-c53695c77a1b | open-r1/OpenR1-Math-220k/olympiads | 2. Find the pattern and fill in the number:
$$
100,81,64,49 \text {, }
$$
$\qquad$ $, 25,16,9,4,1$ . | [
"We need to identify pattern: Sequence: 100,81,64,49, ?, 25,16,9,4,1. Likely squares descending: 10^2=100, 9^2=81, 8^2=64, 7^2=49, 6^2=36, 5^2=25, 4^2=16, 3^2=9, 2^2=4, 1^2=1. So missing number is 36. Provide explanation. Could also be reversed order? But given pattern suggests squares descending. So answer: 36. Pr... | The numbers are consecutive perfect squares written in decreasing order.
\[
\begin{aligned}
10^{2} &= 100\\
9^{2} &= 81\\
8^{2} &= 64\\
7^{2} &= 49\\
\boxed{6^{2}=36}\\
5^{2} &= 25\\
4^{2} &= 16\\
3^{2} &= 9\\
2^{2} &= 4\\
1^{2} &= 1
\end{aligned}
\]
Thus the missing number is
\[
\boxed{36}.
\] |
f02c5e8e-6277-5943-951d-a1fdd72247f8 | open-r1/OpenR1-Math-220k/olympiads | Emil and Olivia ran a race. Their race times totalled 1 hour 52 minutes. If Emil's time was 4 minutes less than Olivia's time, how many minutes did it take Olivia to run the race?
(A) 78
(B) 56
(C) 58
(D) 74
(E) 55 | [
"We need to solve: total time = 1 hour 52 minutes = 60+52=112 minutes. Let O = Olivia's time, E = Emil's time = O - 4. Then O + (O - 4) = 112 → 2O - 4 =112 → 2O = 116 → O = 58 minutes. Answer C 58.",
"Check options: C=58.",
"Thus answer is 58."
] | Let Olivia’s time be \(O\) minutes.
Then Emil’s time is \(O-4\) minutes (4 minutes less).
The total time for both is
\[
O + (O-4) = 112 \text{ minutes}
\]
(since 1 hour 52 minutes = \(60+52 = 112\) minutes).
Solve:
\[
2O - 4 = 112 \\
2O = 116 \\
O = 58
\]
So Olivia took **58 minutes** to run the race.
**Answ... |
cefdfba5-c792-5be7-a9e2-066f93e3dacc | open-r1/OpenR1-Math-220k/cn_k12 | The reciprocal of $5$ is ( ).
A: $-5$
B: $-\frac{1}{5}$
C: $\frac{1}{5}$
D: $\pm 5$ | [
"We need to answer: The reciprocal of 5 is ?. Reciprocal of a number is 1 over that number: 1/5. So answer C. Should output C maybe with explanation.",
"Thus final answer: C: 1/5."
] | The reciprocal of a number \(a\) is \(\dfrac{1}{a}\).
For \(a = 5\),
\[
\text{reciprocal of }5 = \frac{1}{5}.
\]
So the correct choice is
**C: \(\displaystyle \frac{1}{5}\).** |
2e17e848-138f-51c0-a0ca-f8fd74585c0a | open-r1/OpenR1-Math-220k/cn_k12 | To prove "In the angles of a triangle, there is at least one angle that is greater than or equal to 60°" using the method of contradiction, one should first assume ( )
A: Every angle of the triangle is less than 60°
B: There is an angle in the triangle that is greater than 60°
C: Every angle of the triangle is greater... | [
"We need to answer a multiple-choice question: \"To prove 'In the angles of a triangle, there is at least one angle that is greater than or equal to 60°' using the method of contradiction, one should first assume ( )\" Options: A: Every angle of the triangle is less than 60°, B: There is an angle in the triangle t... | **A** – assume that every angle of the triangle is less than \(60^{\circ}\).
If all three angles were < 60°, their sum would be < 180°, contradicting the fact that the interior angles of any triangle add up to exactly 180°. Hence at least one angle must be ≥ 60°. |
183f7dad-cd2b-5e05-858d-32534a7812fe | open-r1/OpenR1-Math-220k/cn_k12 | Given sets $A=\{x|x^{2}-3x \lt 0\}$ and $B=\{x|y=\ln (x-2)\}$, then $A\cap B=$
A: $(2,+\infty)$
B: $(2,3)$
C: $(3,+\infty)$
D: $(-\infty ,2)$ | [
"We need to interpret sets A and B.",
"Set A = { x | x^2 - 3x < 0 }. Solve inequality: x(x-3) < 0. This is satisfied when x is between 0 and 3 (since product of two factors is negative when they have opposite signs). So A = (0,3).",
"Set B = { x | y = ln(x-2)}? Wait B is defined as {x| y = ln(x-2) }. That seems... | **Step 1: Describe each set**
1. **Set \(A\)**
\[
A=\{x\mid x^{2}-3x<0\}
\]
Factor the quadratic:
\[
x^{2}-3x = x(x-3)
\]
The product is negative when the two factors have opposite signs, i.e. when
\[
0<x<3 .
\]
Hence
\[
A=(0,3).
\]
2. **Set \(B\)**
\[
B=\{x\mi... |
48bb4dac-4648-5043-8cbf-52a2b252fe49 | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following idioms describes a random event?
A: Fishing for the moon in the water
B: Castles in the air
C: Waiting by a stump for a hare
D: Catching a turtle in a jar | [
"We need to answer which idiom describes a random event. We have four idioms (likely from some language - maybe Russian?). \"Fishing for the moon in the water\" is a phrase meaning doing something impossible, chasing an illusion. \"Castles in the air\" means unrealistic plans, fantasies. \"Waiting by a stump for a ... | **C: Waiting by a stump for a hare**
This idiom conveys the idea of waiting for something that may (or may not) happen purely by chance – a classic picture of a random, unpredictable event. |
c5a7c5df-db45-57ea-bac1-3f35ce7bed22 | open-r1/OpenR1-Math-220k/olympiads | 4. (5 points) In the third grade, there are two classes, Class A and Class B. If 4 students are transferred from Class A to Class B, the number of students in both classes will be equal. Class A has $\qquad$ more students than Class B. | [
"We need to parse the problem.",
"It says: In the third grade, there are two classes, Class A and Class B. If 4 students are transferred from Class A to Class B, the number of students in both classes will be equal. Class A has ___ more students than Class B.",
"We need to find the difference between the origin... | Let
\[
A = \text{students in Class A (initially)},\qquad
B = \text{students in Class B (initially)}.
\]
After moving 4 students from A to B:
\[
\text{new size of A}=A-4,\qquad
\text{new size of B}=B+4.
\]
The problem states these new class sizes are equal:
\[
A-4 = B+4.
\]
Solve for the original difference \(... |
End of preview. Expand in Data Studio
Math Think SFT Mini — GPT-OSS
352,199 training examples, regenerated with
openai/gpt-oss-120b from the questions in
cs-giung/math-think-sft-mini-sa.
All IDs, source lineages, questions and row order are unchanged. The original steps and
answer were discarded before generation; only the question was supplied as the user message.
Schema
| Column | Type | Meaning |
|---|---|---|
id |
string | Original stable example ID |
source |
string | Original source lineage |
question |
string | Original math problem |
steps |
list[string] | Analysis payloads joined in generation order, split on literal \n\n |
answer |
string | Entire final-channel payload, including its explanation if present |
answer is not restricted to a short answer or a boxed mathematical value.
steps are paragraph-delimited text segments, not independently verified logical steps.
Processing
- Generate one native Harmony continuation per question using the pinned official chat template.
- Collect every assistant
analysispayload in generation order. - Set
analysis = "\n\n".join(analysis_messages)andsteps = analysis.split("\n\n"). - Extract the sole assistant
finalpayload asanswer. - Remove structural message headers/end tokens only. Do not strip payload whitespace, normalize newlines, remove empty segments or shorten the final answer.
For every row, "\n\n".join(steps) == analysis exactly. This invariant concerns the merged
plain analysis text, not reconstruction of the original Harmony message boundaries.
There are 45 rows
with multiple analysis messages.
One continuation omits the analysis end token immediately before its final header; its payload
is extracted using the next explicit header, preserving all payload characters.
Empty analysis messages and empty step strings produced by literal splitting
are intentionally retained.
Nine rows also contain separate commentary messages. These are not analysis and are
excluded from steps; their exact payloads are preserved in
commentary_messages.json, without adding a dataset column.
The raw response archive is retained locally and is not an additional column in this dataset.
Generation
- Model revision:
b5c939de8f754692c1647ca79fbf85e8c1e70f8a. - Input revision:
10b4c039b7ed7aca983d31188096dc4177bfdcc7. - vLLM 0.24.0, tensor parallelism 2, official MXFP4 weights / Marlin, BF16 activations/KV.
- Reasoning effort:
medium; temperature 1.0, top_p 1.0, top_k -1; base seed 42. - Context: 65,536 tokens; initial output budget 16,384, increased for incomplete responses (up to two retries).
- Native assistant-turn completion and a nonempty final payload were required for every row.
- This conversion is CPU-only; it performs no further model inference.
Full generation settings, source references and content hashes: provenance.json. Exhaustive structural/round-trip validation: validation.json. Processor: processing/build_math_think_sft_oss.py.
Sources
| Original question source | Rows |
|---|---|
| PaddlePaddle/GSM8K_distilled_zh | 23 |
| ServiceNow-AI/R1-Distill-SFT | 318,395 |
| nvidia/AceReason-1.1-SFT | 3,276 |
| open-r1/OpenR1-Math-220k | 1,719 |
| oumi-ai/MetaMathQA-R1 | 27,743 |
| rasbt/math_distill | 1,043 |
Detailed lineages are retained in source. The pinned input dataset card and
construction manifest
provide upstream source and processing provenance.
Limitations and licensing
Answers and reasoning are model-generated and not independently mathematically verified.
The input's decontamination decisions are inherited because questions/IDs are unchanged;
no new decontamination was performed and no zero-contamination claim is made. See the
input audit,
including eight modified variants retained under policy review.
Source licenses and usage restrictions continue to apply;
this aggregate grants no additional rights.
license: other does not grant a blanket license to upstream content.
Load
from datasets import load_dataset
dataset = load_dataset("cs-giung/math-think-sft-mini-oss", split="train")
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