question stringclasses 9
values | reasoning stringlengths 3.49k 11.3k | correct stringclasses 9
values | solution_type stringclasses 4
values | id int64 1 36 | rt_sev_1 stringlengths 1.94k 11.3k | rt_sev_2 stringlengths 2.82k 11.2k | rt_sev_3 stringlengths 2.85k 11.1k | rt_sev_1_annotation_json dict | rt_sev_2_annotation_json dict | rt_sev_3_annotation_json dict | neg_annotation_json dict |
|---|---|---|---|---|---|---|---|---|---|---|---|
A mailman is delivering mail to 10 houses on one side of the street. No two adjacent houses both get mail, and at most three houses in a row don't get any mail. How many ways can this occur? | We have 10 houses in a row on one side of the street. The mailman can either deliver mail to a house (let's denote this as "1") or not deliver mail to it (let's denote this as "0"). So, for each house, there are two possibilities: 1 or 0. However, there are two constraints:
\begin{itemize}
\item no two adjacent hou... | 69 | human_stage1 | 1 | We have 10 houses in a row on one side of the street. The mailman can either deliver mail to a house (let's denote this as “1”) or not deliver mail to it (let’s denote this as “0”). So, for each house, there are two possibilities: 1 or 0. However, there are two constraints:
* no two adjacent houses both get mail: th... | We have 10 houses in a row on one side of the street. The mailman can either deliver mail to a house (let's denote this as “1”) or not deliver mail to it (let’s denote this as “0”). So, for each house, there are two possibilities: 1 or 0. However, there are two constraints:
* no two adjacent houses both get mail: th... | We have 10 houses in a row on one side of the street. The mailman can either deliver mail to a house (let's denote this as “0”) or not deliver mail to it (let’s denote this as “1”). So, for each house, there are two possibilities: 0 or 1. The two constraints are
* no two adjacent houses both get mail: this means we ... | {
"negations": [
{
"corrects_steps": [
13,
14,
15
],
"refines_steps": null,
"step": 16,
"text": "not careful enough",
"type": "CORRECTIVE_NEGATION"
}
],
"steps": [
"We have 10 houses in a row on one side of the street. The mailman can either ... | {
"negations": [
{
"corrects_steps": null,
"refines_steps": [
9
],
"step": 10,
"text": "but we must also enforce the x_i ≤ 3 condition",
"type": "REFINING_NEGATION"
},
{
"corrects_steps": [
10
],
"refines_steps": null,
"step": 12,... | {
"negations": [
{
"corrects_steps": [
5
],
"refines_steps": null,
"step": 5,
"text": "mistakenly ignored the upper bound",
"type": "CORRECTIVE_NEGATION"
},
{
"corrects_steps": null,
"refines_steps": [
5,
6
],
"step": 7,
... | {
"negations": [
{
"corrects_steps": [
9,
10
],
"refines_steps": null,
"step": 12,
"text": "This seems too much!",
"type": "CORRECTIVE_NEGATION"
},
{
"corrects_steps": [
8,
9,
10
],
"refines_steps": null,
"... |
Given the set $\{1, 2, \ldots, n\}$, define the complex power sum of a subset $\{a_1, a_2, \ldots, a_k\}$ as $a_1i + a_2i^2 + a_3i^3 + \cdots + a_ki^k$, where $i^2 = -1$. Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1, 2, \ldots, n\}$. Given that $S_8 = -176 - 64i$ and $S_9 = p + qi$, wh... | The problem is a bit confusingly worded. It makes sense to interpret a sum associated to a subset $\{a_1, \dots a_k \}$ as $a_1 i + a_2 i^2 + \cdots + a_k i^k$, where $a_1 < a_2 < \dots < a_k$. This way, the same subset cannot be permuted and summed.
At first glance, this problem seems very weird. There are many thing... | 368 | human_stage1 | 2 | The problem is a bit confusingly worded. It makes sense to interpret a sum associated to a subset $\{a_1, \dots a_k \}$ as $a_1 i + a_2 i^2 + \cdots + a_k i^k$, where $a_1 < a_2 < \dots < a_k$. This way, the same subset cannot be permuted and summed.
At first glance, this problem seems very weird. There are many thi... | The problem is a bit confusingly worded. It makes sense to interpret a sum associated to a subset \(\{a_1,\dots ,a_k\}\) as
\(a_1 i + a_2 i^2 + \cdots + a_k i^k\), where \(a_1<a_2<\dots <a_k\). This way the same subset cannot be permuted and summed.
At first glance the problem seems very odd. There are many things... | The problem is phrased a little oddly, so it helps to settle on a concrete interpretation. For a subset \(\{a_{1},a_{2},\dots ,a_{k}\}\) with the elements listed in increasing order we will read its “complex power sum’’ as
\[
a_{1}i^{1}+a_{2}i^{2}+\cdots +a_{k}i^{k},
\]
so the order of the elements matters and we ... | {
"negations": [
{
"corrects_steps": null,
"refines_steps": [
2
],
"step": 2,
"text": "may not be feasible",
"type": "REFINING_NEGATION"
},
{
"corrects_steps": null,
"refines_steps": [
3
],
"step": 3,
"text": "may not work p... | {
"negations": [
{
"corrects_steps": null,
"refines_steps": [
6
],
"step": 7,
"text": "do not reveal any obvious pattern",
"type": "REFINING_NEGATION"
},
{
"corrects_steps": null,
"refines_steps": [
6
],
"step": 7,
"text": "... | {
"negations": [],
"steps": [
"The problem is phrased a little oddly, so it helps to settle on a concrete interpretation. For a subset \\(\\{a_{1},a_{2},\\dots ,a_{k}\\}\\) with the elements listed in increasing order we will read its “complex power sum’’ as \\(a_{1}i^{1}+a_{2}i^{2}+\\cdots +a_{k}i^{k}\\), so t... | {
"negations": [],
"steps": [
"Interpret the sum for a subset \\{a_1, …, a_k\\} as a_1 i + a_2 i^2 + … + a_k i^k, with a_1 < a_2 < … < a_k, so the order is fixed.",
"Observe that there are 2^9 subsets of {1,…,9}, making brute‑force enumeration infeasible and computationally difficult without a computer.",
... |
How many subsets of the set of the first twelve positive integers are such that the product of their elements is a multiple of 12? | Let us first define the set of the first twelve positive integers, which is $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$. Then, with the given integers, we are to find the number of subsets in which the product is a multiple of 12.
Let us recall that a set with $n$ elements has $2^n$ possible subsets, and $2^n - 1$ non... | 3632 | human_stage1 | 3 | Let us first define the set of the first twelve positive integers, which is $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$. Then, with the given integers, we are to find the number of subsets in which the product is a multiple of 12.
Let us recall that a set with $n$ elements has $2^n$ possible subsets, and $2^n - 1$ non... | Let us first define the set of the first twelve positive integers, which is $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$. Then, with the given integers, we are to find the number of subsets in which the product is a multiple of 12.
Let us recall that a set with $n$ elements has $2^n$ possible subsets, and $2^n - 1$ no... | Let us first define the set of the first twelve positive integers, which is $\{1,2,3,4,5,6,7,8,9,10,11,12\}$. We are asked to find how many subsets have a product that is a multiple of 12.
Recall that a set with $n$ elements has $2^n$ possible subsets (including the empty set) and $2^n-1$ non‑empty subsets. For each... | {
"negations": [],
"steps": [
"Let us first define the set of the first twelve positive integers, which is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. Then, with the given integers, we are to find the number of subsets in which the product is a multiple of 12. We recall that a set with n elements has 2^n possible ... | {
"negations": [
{
"corrects_steps": [
10
],
"refines_steps": null,
"step": 10,
"text": "we mistakenly overlook one",
"type": "CORRECTIVE_NEGATION"
},
{
"corrects_steps": [
11
],
"refines_steps": null,
"step": 11,
"text": "w... | {
"negations": [
{
"corrects_steps": null,
"refines_steps": [
9
],
"step": 10,
"text": "we accidentally omitted the possibility of also adding 10",
"type": "REFINING_NEGATION"
},
{
"corrects_steps": null,
"refines_steps": [
13
],
... | {
"negations": [
{
"corrects_steps": null,
"refines_steps": [
6
],
"step": 7,
"text": "However, if the subset contains the number 3, that we need",
"type": "REFINING_NEGATION"
}
],
"steps": [
"Let us first define the set of the first twelve positive integers... |
In a tournament where every team plays exactly once against every other team, one won match earns $3$ points for the winner and $0$ for the loser, and a draw earns $1$ point for each team. At the end of the tournament, all teams together have achieved $15$ points. The last team on the final list scored exactly $1$ poin... | Let me think step by step.
The last team scored exactly 1 point, which means this team made a draw in exactly one match and they lost all the other matches.
There is no possibility that the last team won a match, because if they won a match, they will get 3 points, but now they just have 1 point.
Then, let's analyze th... | 4 | human_stage1 | 4 | Let me think step by step.
The last team has exactly 1 point, which means this team drew one match and lost the rest.
There’s no way the last team could have won, because a win would give them 3 points, not 1.
Now consider the second‑to‑last team.
The statement says they never lost a match.
I’m a bit unsure... | Let me think step by step.
The last team scored exactly 1 point, which suggests this team managed a draw in at most one match and lost the rest.
There is basically no way the last team could have won a match, because a win would give them 3 points, not 1.
Now let’s look at the second‑to‑last team.
The problem ... | Let me think step by step.
The last team has exactly 1 point, so I guess they must have drawn one match and lost the rest. (A draw gives 1 point, a win 3, a loss 0.) If they had won a game they would already have at least 3 points, so a win is impossible.
Now the second‑to‑last team is said to have never lost a ma... | {
"negations": [],
"steps": [
"The last team has exactly 1 point, which means this team drew one match and lost the rest.",
"There’s no way the last team could have won, because a win would give them 3 points, not 1.",
"Now consider the second‑to‑last team. The statement says they never lost a match.",
... | {
"negations": [
{
"corrects_steps": null,
"refines_steps": [
4
],
"step": 5,
"text": "I momentarily thought it meant they never won a match, but that can’t be right",
"type": "REFINING_NEGATION"
},
{
"corrects_steps": null,
"refines_steps": [],
... | {
"negations": [
{
"corrects_steps": [
5
],
"refines_steps": null,
"step": 7,
"text": "can't be right",
"type": "CORRECTIVE_NEGATION"
}
],
"steps": [
"Let me think step by step.",
"The last team has exactly 1 point, so I guess they must have drawn one ma... | {
"negations": [
{
"corrects_steps": null,
"refines_steps": [
1
],
"step": 2,
"text": "There is no possibility that the last team won a match",
"type": "REFINING_NEGATION"
},
{
"corrects_steps": [
6
],
"refines_steps": null,
"step... |
"Let $\\triangle ABC$ be a right triangle with the right angle at $C$. Points $D$ and $E$ are on $\\(...TRUNCATED) | "First, because $CD$ and $CE$ trisect $\\angle C$ which is a right angle, they must be 30 degrees. N(...TRUNCATED) | 10 | human_stage1 | 5 | "First, because $CD$ and $CE$ trisect $\\angle C$ which is a right angle, they must be $30^\\circ$ e(...TRUNCATED) | "First, because $CD$ and $CE$ trisect $\\angle C$ which is a right angle, they must be $30^\\circ$ e(...TRUNCATED) | "First, because $CD$ and $CE$ trisect $\\angle C$ which is a right angle, they must be $30^\\circ$ e(...TRUNCATED) | {"negations":[{"corrects_steps":[9],"refines_steps":null,"step":10,"text":"this is not correct","typ(...TRUNCATED) | {"negations":[{"corrects_steps":[3],"refines_steps":null,"step":3,"text":"mistakenly added EB and ED(...TRUNCATED) | {"negations":[{"corrects_steps":[2],"refines_steps":null,"step":3,"text":"incorrectly","type":"CORRE(...TRUNCATED) | {"negations":[{"corrects_steps":null,"refines_steps":[6],"step":7,"text":"But wait, I have not compl(...TRUNCATED) |
"Let \\( u_0, u_1, u_2, \\ldots \\) be integers such that \\( u_0 = 100 \\); \n\\( u_{k+2} \\geqslan(...TRUNCATED) | "I am given some initial conditions about an integer sequence and asked to find the value of $u_{202(...TRUNCATED) | 2123 | human_stage1 | 6 | "I am given some initial conditions about an integer sequence and asked to find the value of $u_{202(...TRUNCATED) | "I am given some initial conditions about an integer sequence and asked to find the value of $u_{202(...TRUNCATED) | "I am given some initial conditions about an integer sequence and asked to find the value of \\(u_(...TRUNCATED) | {"negations":[],"steps":["I am given some initial conditions about an integer sequence and asked to (...TRUNCATED) | {"negations":[],"steps":["I am given some initial conditions about an integer sequence and asked to (...TRUNCATED) | {"negations":[],"steps":["I am given some initial conditions about an integer sequence and asked to (...TRUNCATED) | {"negations":[],"steps":["I am given some initial conditions about an integer sequence and asked to (...TRUNCATED) |
"Let \\(ABC\\) be a scalene triangle. A point \\(P\\) is called nice if the lines \\(AD\\), \\(BE\\)(...TRUNCATED) | "I want to find all the points $P$ on the line $OI$ such that the segment connecting a vertex to a p(...TRUNCATED) | 3 | human_stage1 | 7 | "I want to find all the points \\(P\\) on the line \\(OI\\) such that the segment joining a vertex t(...TRUNCATED) | "I want to find all the points $P$ on the line $OI$ such that the segment connecting a vertex to a p(...TRUNCATED) | "I want to find all the points \\(P\\) on the line \\(OI\\) such that the segment connecting a verte(...TRUNCATED) | {"negations":[],"steps":["I want to find all the points P on the line OI such that the segment joini(...TRUNCATED) | {"negations":[{"corrects_steps":[8],"refines_steps":null,"step":8,"text":"mistakenly","type":"CORREC(...TRUNCATED) | {"negations":[],"steps":["I want to find all the points P on the line OI such that the segment conne(...TRUNCATED) | {"negations":[],"steps":["I want to find all the points P on the line OI such that the segment conne(...TRUNCATED) |
"Let \\(a\\) and \\(b\\) be real numbers. Given the equations:\n\\[ |a-1| + |b+1| = |a| + |b| = |a+1(...TRUNCATED) | "We are given this equation: \\[ |a-1| + |b+1| = |a| + |b| = |a+1| + |b-1|\\] and asked to minimize(...TRUNCATED) | 2 | human_stage1 | 8 | "We are given this equation: \\[ |a-1| + |b+1| = |a| + |b| = |a+1| + |b-1|\\] and asked to minimize(...TRUNCATED) | "We are given this equation: \\[ |a-1| + |b+1| = |a| + |b| = |a+1| + |b-1|\\] and asked to minimize(...TRUNCATED) | "We are given this equation: \\[ |a-1| + |b+1| = |a| + |b| = |a+1| + |b-1|\\] and asked to minimize(...TRUNCATED) | {"negations":[{"corrects_steps":[2],"refines_steps":null,"step":3,"text":"cannot be correct","type":(...TRUNCATED) | {"negations":[{"corrects_steps":[6],"refines_steps":null,"step":8,"text":"not zero","type":"CORRECTI(...TRUNCATED) | {"negations":[{"corrects_steps":[1],"refines_steps":null,"step":2,"text":"but in fact we really have(...TRUNCATED) | {"negations":[{"corrects_steps":[3],"refines_steps":null,"step":3,"text":"cannot be correct","type":(...TRUNCATED) |
"You have N beans. You divide your beans into two equal groups and give the remainder $N \\pmod 2$ t(...TRUNCATED) | "Let's restate the problem for clarity. We are given an intial number of beans $N$ and then we perfo(...TRUNCATED) | 2309 | human_stage1 | 9 | "Let's restate the problem for clarity. We are given an initial number of beans $N$ and then we perf(...TRUNCATED) | "Let's restate the problem for clarity. We are given an initial number of beans \\(N\\) and then we (...TRUNCATED) | "Let's restate the problem to make sure we have the right picture. We start with an initial pile of (...TRUNCATED) | {"negations":[{"corrects_steps":[3],"refines_steps":null,"step":4,"text":"this approach is a bit off(...TRUNCATED) | {"negations":[{"corrects_steps":[1],"refines_steps":null,"step":6,"text":"it was odd","type":"CORREC(...TRUNCATED) | {"negations":[{"corrects_steps":[3],"refines_steps":null,"step":3,"text":"not being repeatedly divid(...TRUNCATED) | {"negations":[{"corrects_steps":[3],"refines_steps":null,"step":4,"text":"this approach is wrong","t(...TRUNCATED) |
"A mailman is delivering mail to 10 houses on one side of the street. No two adjacent houses both ge(...TRUNCATED) | "$ $ \nLet's restate the problem in order to better understand it: \n$ $ \nWe want to count the numb(...TRUNCATED) | 69 | human_stage2 | 10 | "$ $ \nLet's restate the problem in order to better understand it: \n$ $ \nWe want to count the numb(...TRUNCATED) | "Let's restate the problem so we have it clear in our heads: \n\nWe need to count the number of way(...TRUNCATED) | "$ $ \nLet's restate the problem in order to better understand it: \n$ $ \nWe want to count the numb(...TRUNCATED) | {"negations":[{"corrects_steps":[11],"refines_steps":null,"step":12,"text":"not careful enough","typ(...TRUNCATED) | {"negations":[{"corrects_steps":null,"refines_steps":[3],"step":3,"text":"let’s just keep the mini(...TRUNCATED) | {"negations":[{"corrects_steps":[4],"refines_steps":null,"step":4,"text":"but I mistakenly thought t(...TRUNCATED) | {"negations":[],"steps":["Restate the problem: count the number of ways mail can arrive to a group o(...TRUNCATED) |
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