problem string | solution string | answer string | problem_type string | question_type string | source string | uuid string | is_reasoning_complete sequence | generations sequence | correctness_math_verify sequence | correctness_llama sequence | finish_reasons sequence | correctness_count int64 | messages list |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Given the proposition: "There exists $x \in [1,2]$ such that $x^{2}+2x+a \geqslant 0$" is true, the range of values for $a$ is _______ . | The proposition is: There exists $x \in [1,2]$ such that $x^{2}+2x+a \geqslant 0$
The negation of the proposition is: For all $x \in [1,2]$, $x^{2}+2x+a < 0 \quad \quad ①$
Regarding ①, it can be transformed into: $x^{2}+2x < -a$ holds true on $[1,2]$
However, $y=x^{2}+2x$ is monotonically increasing on $[1,2]$
$\th... | a \geqslant -8 | Algebra | math-word-problem | cn_k12 | b80b3b48-3a62-5375-94ac-ac353c1ce80c | [
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"<think>\nOkay, let's see. The problem says that there exists an x in the interval [1,2] such that x squared plus 2x plus a is greater than or equal to zero. We need to find the range of values for a that makes this proposition true. Hmm, so basically, we need to find all a such that at least one x between 1 and 2 ... | [
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"content": "Given the proposition: \"There exists $x \\in [1,2]$ such that $x^{2}+2x+a \\geqslant 0$\" is true, the range of values for $a$ is _______ .",
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"content": "<think>\nOkay, let's see. The problem says that there exists an x in the interval [1,2] such that x squared plu... |
$\mathbf{5 . 2} 020$ balls are placed into 2020 different boxes, with each ball having an equal probability of landing in any box, and the probabilities of different balls landing in the boxes are independent of each other. The expected number of empty boxes is $\qquad$ | 5. $\frac{2019^{2020}}{2020^{2019}}$.
Define the random variable $x_{i}$: if the $i$-th box has a ball, then $x_{i}=0$, otherwise, $x_{i}=1$.
Thus, $P\left(x_{i}=1\right)=\frac{2019^{2020}}{2020^{2020}}$,
$P\left(x_{i}=0\right)=1-\frac{2019^{2020}}{2020^{2020}}$.
Therefore, $\mathrm{E}\left(x_{i}\right)=1 \times \frac... | \frac{2019^{2020}}{2020^{2019}} | Combinatorics | math-word-problem | olympiads | 4148ae3d-3e6f-5dfd-993b-776445447634 | [
true,
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"<think>\nOkay, so I need to find the expected number of empty boxes when 2020 balls are placed into 2020 different boxes. Each ball has an equal chance of going into any box, and the placements are independent. Hmm, expectation problems can sometimes be tricky, but maybe I can use linearity of expectation here. Le... | [
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"content": "$\\mathbf{5 . 2} 020$ balls are placed into 2020 different boxes, with each ball having an equal probability of landing in any box, and the probabilities of different balls landing in the boxes are independent of each other. The expected number of empty boxes is $\\qquad$",
"role": "user"
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Let $a=\log_{3}2$, $b=\log_{2} \frac{1}{8}$, $c= \sqrt{2}$. Then ( )
A: $a > b > c$
B: $c > b > a$
C: $a > c > b$
D: $c > a > b$ | Since $a=\log_{3}2$ is in the interval $(0,1)$, $b=\log_{2} \frac{1}{8} 1$,
it follows that $c > a > b$.
Therefore, the correct choice is $\boxed{D}$.
This conclusion can be reached by utilizing the monotonicity of the logarithmic function.
This question tests the understanding of the monotonicity of logarithm... | D | Algebra | MCQ | cn_k12 | e0cd2ee1-5b72-5cab-9e28-9ae51895ef72 | [
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"<think>\nOkay, let's see. I need to compare the values of a, b, and c to determine their order. The problem states:\n\nLet a = log base 3 of 2, b = log base 2 of 1/8, and c = sqrt(2). Then, which option is correct: A, B, C, or D?\n\nFirst, I should recall how logarithms work and how to evaluate them. Let me start ... | [
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"content": "Let $a=\\log_{3}2$, $b=\\log_{2} \\frac{1}{8}$, $c= \\sqrt{2}$. Then ( )\nA: $a > b > c$\nB: $c > b > a$\nC: $a > c > b$\nD: $c > a > b$",
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"content": "<think>\nOkay, let's see. I need to compare the values of a, b, and c to determine their order. The problem states... |
Given that the eccentricity of the ellipse $C$ with foci on the $x$-axis is $\frac{\sqrt{3}}{2}$, and it passes through the point $A(-2,1)$.
(1) Find the standard equation of the ellipse $C$;
(2) Line $l:x-y+2=0$ intersects the ellipse $C$ at points $P$ and $Q$, with $O$ being the origin, find the area of $\Delta OPQ... | (1) According to the question, let the ellipse $C$ be represented by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$.
Since the ellipse $C$ passes through point $A(-2,1)$,
$\therefore \frac{4}{a^2} + \frac{1}{b^2} = 1 \quad \quad \quad \quad (1)$
Also, $e = \frac{c}{a} = \sqrt{\frac{a^2 - b^2}{a^2}}... | \frac{4\sqrt{6}}{5} | Geometry | math-word-problem | cn_k12 | 1506baf7-2598-5072-bdaf-04d439e97943 | [
true,
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"<think>\nOkay, so I need to solve this problem about an ellipse. Let me start with part (1). The problem says that the ellipse C has foci on the x-axis and an eccentricity of √3/2. It also passes through the point A(-2,1). I need to find the standard equation of this ellipse.\n\nFirst, recalling the standard form ... | [
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"content": "Given that the eccentricity of the ellipse $C$ with foci on the $x$-axis is $\\frac{\\sqrt{3}}{2}$, and it passes through the point $A(-2,1)$.\n\n(1) Find the standard equation of the ellipse $C$;\n\n(2) Line $l:x-y+2=0$ intersects the ellipse $C$ at points $P$ and $Q$, with $O$ being the origin, ... |
1. A natural number has a digit sum of 2013. The next number has a smaller digit sum and is not divisible by 4. What is the digit sum of the next natural number. | Answer: 2005.
Solution. The sum of the digits of the next number decreases only if it ends in nine.
It cannot end in two or more nines, as the next number would end in two zeros and be divisible by 4.
Therefore, the 9 is replaced by 0, but the tens digit increases by 1. The sum of the digits of the next number is 20... | 2005 | Number Theory | math-word-problem | olympiads | 9316ca24-486c-5704-92c0-01c25857bb81 | [
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"<think>\nOkay, let's try to figure out this problem step by step. The question is about a natural number whose digit sum is 2013. The next number (i.e., the number that comes right after it) has a smaller digit sum and is not divisible by 4. We need to find the digit sum of this next natural number.\n\nFirst, let ... | [
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"content": "1. A natural number has a digit sum of 2013. The next number has a smaller digit sum and is not divisible by 4. What is the digit sum of the next natural number.",
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"content": "<think>\nOkay, let's try to figure out this problem step by step. The question is about a ... |
8,9
The area of triangle $ABC$ is $S, \angle BAC = \alpha, AC = b$. Find $BC$. | "Apply the Law of Cosines.\n\n## Solution\n\nSince $S=\\frac{1}{2} A B \\cdot A C \\sin \\alpha$, th(...TRUNCATED) | \sqrt{\frac{4S^{2}}{b^{2}\sin^{2}\alpha}+b^{2}-4S\cot\alpha} | Geometry | math-word-problem | olympiads | 83f5244f-e29d-590d-bf45-c7e2ff816add | [
true,
true
] | ["<think>\nOkay, so I need to find the length of BC in triangle ABC, given that the area is S, angle(...TRUNCATED) | [
false,
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] | null | [
"stop",
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] | 1 | [{"content":"8,9\n\nThe area of triangle $ABC$ is $S, \\angle BAC = \\alpha, AC = b$. Find $BC$.","r(...TRUNCATED) |
"Ten birds land on a $10$-meter-long wire, each at a random point chosen uniformly along the wire. ((...TRUNCATED) | "1. **Labeling and Segmenting the Wire:**\n - We label the birds from \\(1\\) to \\(10\\).\n - W(...TRUNCATED) | \frac{1}{10^{10}} | Combinatorics | math-word-problem | aops_forum | ce2a09dc-ab3a-5466-8989-9eec795b935f | [
true,
true
] | ["<think>\nOkay, so I need to figure out the probability that ten birds landing randomly on a 10-met(...TRUNCATED) | [
true,
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] | null | null | 2 | [{"content":"Ten birds land on a $10$-meter-long wire, each at a random point chosen uniformly along(...TRUNCATED) |
"PROBLEM 3. Consider the regular quadrilateral pyramid $V A B C D$, with $A B=12$. Let $A C \\cap B (...TRUNCATED) | "PROBLEM 3. Consider the regular quadrilateral pyramid $VABCD$, with $AB=12$. Let $AC \\cap BD=\\{O\(...TRUNCATED) | )\frac{3\sqrt{7}}{2},b)\frac{6\sqrt{161}}{23},)\frac{5\sqrt{7}}{7} | Geometry | math-word-problem | olympiads | f7042ecc-440c-5d9a-9af7-ef449156a4e7 | [
true,
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] | ["<think>\nOkay, let's tackle this problem step by step. It's about a regular quadrilateral pyramid (...TRUNCATED) | [
false,
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true,
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] | null | 1 | [{"content":"PROBLEM 3. Consider the regular quadrilateral pyramid $V A B C D$, with $A B=12$. Let $(...TRUNCATED) |
"Reduce $$\\frac{2}{\\sqrt{4-3\\sqrt[4]{5} + 2\\sqrt[4]{25}-\\sqrt[4]{125}}}$$ to its lowest form.\n(...TRUNCATED) | "Given the expression:\n\\[ T = \\frac{2}{\\sqrt{4 - 3\\sqrt[4]{5} + 2\\sqrt[4]{25} - \\sqrt[4]{125}(...TRUNCATED) | 1 + \sqrt[4]{5} | Algebra | proof | aops_forum | 7239874d-de61-5354-9519-fc61978cc2b3 | [
true,
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] | ["<think>\nOkay, so I need to reduce this fraction: 2 divided by the square root of (4 - 3 times the(...TRUNCATED) | [
true,
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] | null | null | 2 | [{"content":"Reduce $$\\frac{2}{\\sqrt{4-3\\sqrt[4]{5} + 2\\sqrt[4]{25}-\\sqrt[4]{125}}}$$ to its lo(...TRUNCATED) |
"Seven consecutive integers stood in a row, arranged from the smallest to the largest. After a while(...TRUNCATED) | "We will denote the numbers in ascending order as 1. to 7. Their arrangement changed as follows:\n\n(...TRUNCATED) | -4,-3,-2,-1,0,1,2or-5,-4,-3,-2,-1,0,1 | Logic and Puzzles | math-word-problem | olympiads | cc9f9bab-037c-57b2-9c72-893b6b850f39 | [
true,
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] | ["<think>\nOkay, so there's this problem about seven consecutive integers arranged in a row from sma(...TRUNCATED) | [
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] | null | 1 | [{"content":"Seven consecutive integers stood in a row, arranged from the smallest to the largest. A(...TRUNCATED) |
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