conversations listlengths 2 2 | question stringlengths 8 4.13k ⌀ | source stringclasses 8
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"value": "$2 \\cdot 131$ The number of roots of the equation $\\sqrt{x^{2}+12 x+36}+x+6=0$ is\n(A) 0.\n(B) 2.\n(C) 4.\n(D) Infinitely many.\n(3rd \"Five Sheep Cup\" Junior High School Mathematics Competition, 1991)"
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"value": "<think>\nOkay, let me try ... | $2 \cdot 131$ The number of roots of the equation $\sqrt{x^{2}+12 x+36}+x+6=0$ is
(A) 0.
(B) 2.
(C) 4.
(D) Infinitely many.
(3rd "Five Sheep Cup" Junior High School Mathematics Competition, 1991) | null | null | [
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"To determine the number of roots of the equation \\(\\sqrt{x^2 + 12x + 36} + x + 6 = 0\\), let's start by simplifying the expression inside the square root.\n\nThe term \\(x^2 + 12x + 36\\) can be rewritten as a perfect square:\n\\[x^2 + 12x + 36 = (x + 6)^2.\\]\nThus, the equation becomes:\n\\[\\sqrt{(x + 6)^2} +... | [
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"value": "For any two vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$, which of the following propositions is correct?\n\nA: If $|\\overrightarrow{a}| > |\\overrightarrow{b}|$ and $\\overrightarrow{a}$ is in the same direction as $\\overrightarrow{b}$, then $\\overrightarrow{a} > \... | For any two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, which of the following propositions is correct?
A: If $|\overrightarrow{a}| > |\overrightarrow{b}|$ and $\overrightarrow{a}$ is in the same direction as $\overrightarrow{b}$, then $\overrightarrow{a} > \overrightarrow{b}$.
B: $|\overrightarrow{a} + \o... | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For ex... | B | original_rationalized | [
"Let's analyze each proposition one by one:\n\n**Proposition A:** If \\( |\\overrightarrow{a}| > |\\overrightarrow{b}| \\) and \\(\\overrightarrow{a}\\) is in the same direction as \\(\\overrightarrow{b}\\), then \\(\\overrightarrow{a} > \\overrightarrow{b}\\).\n\nVectors cannot be compared using inequality symbols... | [
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"value": "The condition \"a > 0, and a ≠ 1\" is the ( ) condition for \"loga2 > 0\".\nA: Sufficient but not necessary condition\nB: Necessary but not sufficient condition\nC: Sufficient and necessary condition\nD: Neither sufficient nor necessary condition"
},
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"from": "assistant"... | The condition "a > 0, and a ≠ 1" is the ( ) condition for "loga2 > 0".
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Sufficient and necessary condition
D: Neither sufficient nor necessary condition | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For ex... | B | original_rationalized | [
"To determine whether the condition \\(a > 0\\) and \\(a \\neq 1\\) is sufficient, necessary, or both for the statement \\(\\log_a 2 > 0\\), we need to analyze the properties of logarithms.\n\nFirst, consider the expression \\(\\log_a 2\\). The value of \\(\\log_a 2\\) depends on the base \\(a\\):\n- If \\(a > 1\\)... | [
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"value": "In a group of 2017 persons, any pair of persons has exactly one common friend (other than the pair of persons). Determine the smallest possible value of the difference between the numbers of friends of the person with the most friends and the person with the least friends in such... | In a group of 2017 persons, any pair of persons has exactly one common friend (other than the pair of persons). Determine the smallest possible value of the difference between the numbers of friends of the person with the most friends and the person with the least friends in such a group. | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For ex... | 2014 | original_rationalized | [
"Let's denote the group of 2017 people as \\( G \\) and consider each person \\( P_i \\) in \\( G \\).\n\nTo understand the structure of friendships, we need to think about how many friends each person can have under the given condition that every pair of individuals shares exactly one mutual friend. This suggests ... | [
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"value": "Selene buys two instant cameras at $110 and three digital photo frames at $120 each. She gets a 5% discount on all the items she purchased. How much does she pay in all?"
},
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"value": "<think>\nOkay, so Selene bought two instant cameras and three... | null | tiger_math | 11,673 | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For ex... | 551 | original_rationalized | [
"First, let's calculate the total cost of the items before any discounts are applied.\n\nThe cost for the two instant cameras is:\n\\[ 2 \\times 110 = 220 \\]\n\nThe cost for the three digital photo frames is:\n\\[ 3 \\times 120 = 360 \\]\n\nAdding these together gives us the subtotal:\n\\[ 220 + 360 = 580 \\]\n\nN... | [
false,
false
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"",
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[
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"value": "Given the preliminary coordinates of points Q, R, S, and T, along with their latitude and departure errors, compute the corrected coordinates for point T using the corrections derived from the Compass Rule. The preliminary coordinates are:\n\n- Q: (2600.480, 1391.670)\n- R: (2379... | null | automath | 32,975 | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For ex... | (2713.905, 1727.185) | original_rationalized | [
"To find the corrected coordinates of point T using the Compass Rule, we need to apply the corrections to both the latitude and the departure.\n\nFirst, let's identify the latitude and departure values:\n- Preliminary latitude (lat) of T: 2713.835'\n- Preliminary departure (dep) of T: 1727.740'\n\nNext, we know the... | [
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[
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"value": "Find all functions $f:\\mathbb{R}\\rightarrow \\mathbb{R}$ such that\n$\\forall x\\in \\mathbb{R} \\ \\ f(x) = max(2xy-f(y))$ where $y\\in \\mathbb{R}$."
},
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"value": "<think>\nOkay, so I need to find all functions \\( f: \\mathbb{R} \\rightarrow... | Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that
$\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$ where $y\in \mathbb{R}$. | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For ex... | x^2 | original_rationalized | [
"To find a function \\( f : \\mathbb{R} \\to \\mathbb{R} \\) satisfying the equation \\( f(x) = \\max_{y \\in \\mathbb{R}} (2xy - f(y)) \\), we start by analyzing what the maximum value might be for different values of \\( y \\).\n\nFirstly, let's consider if there could be any specific form of \\( f \\). Suppose \... | [
false,
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[{"from":"user","value":"Return your final response within \\boxed{}. Alex wrote all natural divisor(...TRUNCATED) | "Alex wrote all natural divisors of a natural number \\( n \\) on the board in ascending order. Dima(...TRUNCATED) | null | null | [{"content":"You like to solve problems by thinking step by step. Write out your reasoning then give(...TRUNCATED) | 31 | original_rationalized | ["To determine the maximum number of the remaining 151 divisors that are fifth powers of natural num(...TRUNCATED) | [
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[{"from":"user","value":"In the binomial expansion of $(a+b)^n$, if the sum of the binomial coeffici(...TRUNCATED) | "In the binomial expansion of $(a+b)^n$, if the sum of the binomial coefficients of the odd terms is(...TRUNCATED) | null | null | [{"content":"You like to solve problems by thinking step by step. Write out your reasoning then give(...TRUNCATED) | 70 | original_rationalized | ["To find the maximum value of the binomial coefficients in the expansion of \\((a+b)^n\\), we start(...TRUNCATED) | [
false,
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"",
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[{"from":"user","value":"12. 10 consecutive natural numbers are arranged in ascending order. If the (...TRUNCATED) | "12. 10 consecutive natural numbers are arranged in ascending order. If the sum of the last 6 number(...TRUNCATED) | null | null | [{"content":"You like to solve problems by thinking step by step. Write out your reasoning then give(...TRUNCATED) | 6 | original_rationalized | ["Let's denote the 10 consecutive natural numbers as \\(a, a+1, a+2, \\ldots, a+9\\), where \\(a\\) (...TRUNCATED) | [
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