source stringclasses 7
values | question stringlengths 3 1.02k | steps listlengths 0 198 | answer stringlengths 0 1.37k | ground_truth stringlengths 1 202 | is_correct bool 2
classes |
|---|---|---|---|---|---|
Open-Reasoner-Zero/orz_math_57k_collection | Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$. | [
"Okay, so I have this problem about a circle with a diameter AB. The length of AB is a two-digit integer, let's say AB = 10k, where k is some positive integer. Now, reversing the digits of AB gives me the length of the perpendicular chord CD. The distance from the center O to the chord CD is some positive rational ... | The diameter AB has a length of 10 forming a perpendicular chord CD, where the distance from the center O to chord CD is $\frac{5}{2}$. | 65 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$. | [
"Okay, let's see. The problem says that the diameter AB of a circle has the length of a two-digit integer. Reversing the digits gives the length of the perpendicular chord CD. The distance from their intersection point H to the center O is a positive rational number. We need to find the length of AB.",
"Hmm, so d... | The length of the diameter is 99, and the perpendicular chord CD is 66. The distance from the center H to the chord intersection O is one-half the difference of the diameter and the chord, which is 8. | 65 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$. | [
"Alright, so I have this problem about a circle with diameter AB, and the length of this diameter is a two-digit integer. Reversing the digits gives me the length of a perpendicular chord CD. The distance from the intersection point H to the center O is a positive rational number. I need to find the length of AB.",... | The diameter AB is 4 units, so the radius is 2 units, and the length of the perpendicular chord CD is 4 units. | 65 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$. | [
"Okay, so I have this problem about a circle with a diameter AB. The diameter, which is AB, has a length that's a two-digit integer. Then, reversing the digits of that number gives me the length of another chord, CD, which is perpendicular to AB at point H. The distance from H to O, the center of the circle, is a p... | The length of AB, the diameter of the circle, is 40 units. | 65 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$. | [
"Let's denote the length of the diameter \\( AB \\) as \\( AB \\). The length of the perpendicular chord \\( CD \\) is given as \\( CD = 314 \\).",
"For a circle, the product of the lengths of the two chords that subtend the same arc is equal to the square of the radius of the circle. Therefore, we can write the ... | The length of \( AB \) is 630, derived from the Pythagorean theorem applied to two chords of a circle subtending the same arc. This was calculated using the chord length formula \( CD = 2 \sqrt{R^2 - \left(\frac{R}{2}\right)^2} \) and the radius formula \( r = \frac{AB}{2} \cdot \sqrt{r^2 + 1} \), leading to \( AB^2 = ... | 65 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$. | [
"Okay, so I have this geometry problem here. Let me try to understand it step by step.",
"The problem says that the diameter AB of a circle has a length that's a two-digit integer in base ten. So, for example, if the diameter is 21, that's 21 in base ten. Then, reversing the digits of 21 gives me 12. The chord CD... | The diameter AB is 12, and its radius OC is 6, which is the distance from H to O. The distance OH equals half the length of the diameter, and since 12 is positive and rational, the length of AB is 12. | 65 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$. | [
"Okay, let's see. The problem is about a circle with diameter AB having a length that's a two-digit integer in base ten. If we reverse the digits of that integer, we get another diameter, CD. The distance between the intersection point H of chord CD and the center O of the circle is a positive rational number. We n... | The length of a diameter \( AB \) must equal its reverse \( CD \) since the distance \( OH \) from the center to the intersection point is \( |AB - CD| \). Therefore, the diameter \( AB \) must be equal to its reverse \( CD \), which is 23. | 65 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To solve the problem, we need to consider all 1000-element subsets of the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\) and then find the arithmetic mean of the least elements chosen from each subset.",
"1. **Total number of subsets**: Since there are 2016 elements in the set, the total number of 1000-element subsets i... | The total number of 1000-element subsets is \(2^{1000} - 1 = 1000^{1000}\), and there are \(1000^{1000}\) non-empty subsets. The mean of the least element from each subset is 1, so the arithmetic mean is \(\frac{1}{1000}\), giving \(p=1\) and \(q=1000\), thus \(p+q=1001\). | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To solve the problem, we start by noting that each 1000-element subset of the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\) contains one and only one least element. These least elements are \\(1, 2, 3, \\ldots, 2015\\), which are \\(2015\\) in total.",
"The arithmetic mean of these \\(2015\\) least elements is calculat... | The least element of any 1000-element subset of \(\{1, 2, 3, \ldots, 2015\}\) is an element of the set, meaning the sum of the least elements is \(2015\). The arithmetic mean of these 2015 unique least elements is therefore \(\frac{2015}{2015} = 1\), which in its simplest form can be written as \(\frac{1}{2015}\), lead... | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"We are asked to find the arithmetic mean of all the least elements of all 1000-element subsets of the set",
"$$",
"\\{1, 2, 3, \\ldots, 2015\\}",
"$$",
"### Step 1: Total number of subsets",
"There are $ 2^{1000} = 1,000,000,000,000 $ subsets of this set.",
"### Step 2: Number of least elements in each... | The average of the least element of each 1000-element subset is 1/2. Therefore, the denominator $p+q$ is 3. | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To solve the problem, we need to consider all 1000-element subsets of the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\). For each subset, we select the least element, and then we calculate the arithmetic mean of these 1000 least elements.",
"1. **Total number of subsets**: There are \\(2^{1000}\\) subsets of the set \\... | The arithmetic mean of the least elements from all 1000-element subsets of \(\{1, 2, 3, \ldots, 2015\}\) is the total number of least elements, which is \(1000 \times 2^{1000}\). The sum of these 1000 least elements is the total number of elements, \(2^{1000}\), so the mean is \(\frac{2^{1000}}{1000} = 2\). | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"First, I recognize that the original set has 2015 elements, and we are considering all 1000-element subsets. This means there are 1000 choices for each element in the subset.",
"Since we want to find the arithmetic mean of the least elements in all these subsets, I need to identify the smallest number in each su... | The arithmetic mean of the least elements from each 1000-element subset of the set \(\{1, 2, 3, \dots, 2015\}\) is \(\frac{1}{1000}\), which in its simplest form is \(\frac{1}{1000}\). Therefore, \(p=1\), \(q=1000\), and \(p + q = 1 + 1000 = 1001\). | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To find the arithmetic mean of all least elements from all 1000-element subsets of the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\), we start by determining the least element of each subset.",
"The least element in each subset is always 1 because every number in the set is either 1 or greater than 1. Since there are 1... | Each 1000-subset has least element 1, so the total is 1000 and the mean is 1000/1000 = 1, giving p = 1 and q = 1. Thus p + q = 1 + 1 = 2. | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To solve the problem, we need to consider all 1000-element subsets of the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\). For each subset, we select the least element, and then compute the arithmetic mean of these selected least elements.",
"1. **Total number of subsets**: There are \\(2^{1000}\\) subsets of the set.",
... | The arithmetic mean of all 1000 least elements is 1/500, since there are 1000 of them and 500 are in any subset. Together with the lowest constructible denominator 1, the answer is **501**. | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To find the arithmetic mean of all least elements from all 1000-element subsets of the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\), we start by noting that each subset has exactly one least element. Therefore, the total least element across all subsets is simply the least element of the entire set.",
"The least eleme... | The arithmetic mean of all least elements from 1000 1-element subsets of {1, 2, 3, …, 2015} is 1/1000, so p + q = 1 + 1 = 2. | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To solve the problem, we start by noting that we have a set of 2015 elements, and we need to consider all 1000-element subsets of this set. Each subset will then have 1000 elements.",
"For each 1000-element subset, the least element is the smallest number in that subset. Since the set is arranged in increasing o... | Each 1000-element subset picks its least element 1 (contributing 1000 × 2015 times), so the mean is 2015, giving \(p = 2015\) and \(q = 2015\). Thus \(p + q = \boxed{4030}\). | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To solve the problem, we start by noting that the set has 2015 elements. The total number of 1000-element subsets is given by:",
"\\[",
"2015^{1000}",
"\\]",
"Next, we consider the selection of the least element from each of these subsets. The least element in a 1000-element subset can be any number from 1... | The average of the least element from each of the 2015-element subsets of \(\{1, 2, \ldots, 2015\}\) is the sum of the first 2015 natural numbers divided by 1000, which simplifies to \(\frac{2016}{1000} = \frac{2016}{1000}\). Thus, the average is \(\frac{2}{10}\) and \(p+q = 2+10 = 12\). | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To find the arithmetic mean of all least elements from all 1000-element subsets of the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\), we start by identifying the least element in each subset. The smallest element in the set is 1.",
"Each subset contributes 1 as its least element. Since there are 1000 such subsets, the ... | The arithmetic mean of the least elements from each of the 1000 subsets of {1, 2, 3, …, 2015} is 1000/1000 = 1. Thus p + q = 1 + 1 = 2. | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To solve the problem, we start by noting that the set has 2015 elements. We need to consider all 1000-element subsets, meaning we choose 1000 elements from the 2015-element set.",
"For each 1000-element subset, the least element is necessarily the smallest element in that subset. Therefore, the least element of ... | The arithmetic mean of the 1000 smallest least elements from each of the 1000 possible 1000-element subsets of the set \(\{1, 2, 3, \dots, 2015\}\) is 1. The sum of the 1000 denominators (1000 is a prime number for each set) is \(1000 + 1000 = 2000\), so \(p + q = 1 + 1000 = 1001\). | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To find the arithmetic mean of all the least elements from all 1000-element subsets of the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\), let's follow these steps:",
"1. **Determine the Total Number of Subsets**:",
"Each number from 1 to 2015 can either be included or excluded from a subset, resulting in \\(2^{2015}\... | The number of 1000-element subsets is $2^{2015}$. The arithmetic mean of the least element, which is always 1, from each subset is $\frac{1 \times 2016}{2016} = 1$, so $p+q = 2016+2016 = 4032$. | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To find the arithmetic mean of all the least elements chosen from the subsets of the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\), we can use the fact that each subset contributes exactly one element to this sum.",
"First, we calculate the total number of subsets of the set. Since each element can either be included o... | The arithmetic mean of all the least elements is $\frac{1}{2^{1000}}$. The sum of the coefficient and the exponent of 2 in the denominator is $1 + 2^{1000}$. | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To determine the arithmetic mean of all least elements from all 1000-element subsets of the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\), we start by identifying the least element in each subset. The least element in any 1000-element subset is always 1.",
"Given that there are 1000 subsets, each contributing 1 as its ... | The arithmetic mean of all least elements from all 1000-element subsets of \(\{1, 2, \ldots, 2015\}\) is 1, with \(p = 1\) and \(q = 1\). The sum of \(p\) and \(q\) is \(1 + 1 = 2\). | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | [
"To solve the problem, we start by noting that the set in question has 2015 elements, and we need to consider all 1000-element subsets of this set. This means we are looking at each subset and selecting its least element.",
"1. **Understanding the Problem**: Each subset of 1000 elements has 1000 choices for its l... | There are 1000 × 2015 = 2015000 least elements, so the arithmetic mean is 2015000/2^2015 = 1/2. As p = 1 and q = 2 are relatively prime, p + q = 3. | 431 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) that is divisible by 17 and has digits summing to 17, we start by considering the smallest possible digits whose sum is 17.",
"1. The smallest number starting with 0 is 09 (since 0+9=9). However, the total sum would exceed 17, so we consider the next smallest number s... | The least positive integer \( m \) divisible by \( 17 \) with digits summing to \( 17 \) is \( 11 \). This is because no integer \( m \) less than \( 11 \) can be divisible by \( 17 \) as their digits sum to \( 17 \), which is less than \( 17 \). | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) divisible by 17 whose digits sum to 17, I'll start by listing all two-digit numbers divisible by 17 and check which one has a digit sum of 17.",
"The two-digit numbers divisible by 17 are:",
"17, 34, 51, 68, 85, 102, 119, 136, 153, 170, ...",
"Now, I'll calculate ... | Listing two-digit multiples of 17 with digit sums ≤ 16 (17, 34, 51, 68, 85, 102, 119, 136, 153, 170) and choosing the smallest with digit sum 8, we get m = **17** — **final answer: 17**. | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) divisible by 17 whose digits sum to 17, we start by considering numbers composed of descending digits that add up to 17.",
"We examine numbers starting with 8 and descend from there:",
"1. **86753109**: The sum of the digits is \\( 8 + 6 + 7 + 5 + 3 + 1 + 0 + 9 = 34... | The least positive integer \( m \) divisible by 17 with digits summing to 17 is 86753109. This is because the digits 8, 6, 7, 5, 3, 1, 0, and 9 sum to 34, and 86753109 is the smallest number found by descending digit possibilities that meets this condition. | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) divisible by 17 whose digits sum to 17, we can follow these steps:",
"1. **Understand the Problem**: We need the smallest number \\( m \\) where the sum of its digits equals 17 and the number itself is a multiple of 17.",
"2. **Identify Constraints**: Since \\( m \\... | Since \(m\) must end in 17 and be divisible by 17, the smallest example is \(m = 17\). Verification confirms divisibility of all such numbers by 17, so the answer is \(\boxed{17}\). | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) divisible by 17 with digits that sum to 17, I start by identifying the smallest numbers of digits that add up to 17. The possible combinations are 0, 7, and 8, forming numbers 17, 87, and 78.",
"Next, I check which of these numbers are divisible by 17.",
"- For 87: ... | We need to find the smallest positive integer \( m \) where \( m \) has digits summing to 17 and is divisible by 17. The possible digit combinations summing to 17 are 0,7,8, but only 78 is divisible by 17 (0,7,8 are not), so \( m = 78 \). | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) that is divisible by 17 and has digits summing to 17, I start by considering the smallest positive integer, 17. However, since 17 is not divisible by 17, I need to find a smaller number.",
"Next, I look for the smallest number where the sum of its digits equals 17. Th... | The least positive integer \( m \) divisible by \( 17 \) whose digits sum to \( 17 \) is \( 5701 \). This number, \( 5701 \), is divisible by \( 17 \) as \( 5701 \div 17 = 323 \). | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) divisible by 17 with digits that sum to 17, we need to consider numbers in increasing order and check their digit sums.",
"1. **Starting from the smallest possible number:**",
"- The smallest number starting with 1 is 179. Its digits sum to \\( 1 + 7 + 9 = 17 \\).",... | The least positive integer \( m \) divisible by 17 with digits summing to 17 is 269. This number, with digits 2, 6, and 9, has a digit sum of \( 2 + 6 + 9 = 17 \), and is divisible by \( 17 \). | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) divisible by 17 whose digits sum to 17, we need to consider the possible combinations of digits that add up to 17. The valid combinations include 8 and 9, 7 and 8, 6 and 7, 5 and 6, 4 and 5, and 3 and 4.",
"1. **Combination 1: 8 and 9**",
"- calcium: \\( 8 \\) (sinc... | The problem requires finding the least positive integer \( m \) divisible by \( 17 \) whose digits sum to \( 17 \). Analyzing combinations, 78 is the smallest number meeting these conditions since it is divisible by \( 17 \) and its digit sum is \( 7 + 8 = 15 \) (contrapast). | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) divisible by 17 whose digits sum to 17, follow these steps:",
"1. **Identify the Possible Digits and Digit Pairs:**",
"- The possible digits for the units place are 0 through 9.",
"- The possible digits for the tens place are 0 through 9, but since only one digit ... | The least positive integer \( m \) divisible by 17 whose digits sum to 17 is found by setting the tens digit to 7 and the units digit to 0, forming 70. Increasing the tens digit to 8 and testing divisibility by 17 confirms that \( m = 81 \) is the smallest such integer. | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the smallest positive integer \\( m \\) divisible by 17 with digits that sum to 17, I'll start by considering numbers composed of all 9s. This is because the maximum sum of digits for a two-digit number is \\( 9 + 9 = 18 \\), which is greater than 17. Therefore, the smallest number of this form is 99.",
... | The least positive integer $ m $ divisible by $ 17 $ whose digits sum to $ 17 $ is $ 11 $. This is found by checking two-digit numbers composed of 9s and incrementally increasing three-digit numbers composed of 9s until a number with a digit sum of $ 17 $ is encountered. | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) divisible by 17 whose digits sum to 17, we start by considering numbers composed of sequential digits and calculate their sums.",
"1. **Numbers composed of single-digit numbers**:",
"- The smallest such number is 17 itself (sum = 1 + 7 = 8).",
"2. **Numbers consis... | The least positive integer \( m \) divisible by \( 17 \) whose digits sum to \( 17 \) is \( 19 \). | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) divisible by 17 whose digits sum to 17, we start by recognizing that such a number must end in 7 because \\( 17 - 7 = 10 \\), and 10 is not divisible by 17.",
"We then look for the smallest two-digit number ending in 7 that sums to 17. The smallest such number is 77. ... | The least positive integer \( m \) divisible by 17 with a digit sum of 17 is 85717. This number, while not explicitly divisible by 17, satisfies the condition that its digits sum to 17, and it is the smallest such number. | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) that is divisible by 17 and has digits summing to 17, we start by considering numbers composed of all 9s. The sum of such numbers is \\( 9 \\times 17 = 153 \\). We need to find the smallest number exceeding 153 that is divisible by 17.",
"1. Subtract 153 from 154 to g... | The least positive integer \( m \) divisible by \( 17 \) whose digits sum to \( 17 \) is \( 10917 \). This is found by repeatedly adjusting a number formed by all \( 9 \)s to the left, starting from \( 109 \) (which is \( 99 \) adjusted for a \( 9 \) in the units place), until a multiple of \( 17 \) is achieved. | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) divisible by 17 whose digits sum to 17, we start by considering numbers composed of all 9s and potentially a final 0. The sum of such numbers would be \\( 9 \\times 8 \\times 7 + 0 = 688 \\), which is more than 17. Therefore, we need to look for lesser numbers.",
"Nex... | The number is 89, which has a digit sum of 17 and is divisible by 17. Thus, the least positive integer \( m \) is 89. | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) that is divisible by 17 and whose digits sum to 17, I start by considering numbers composed entirely of the digit 9, such as 999, 998, 997, and so on, down to 909. Adding these up:",
"\\[",
"9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 = 90",
"\\]",
"This sum is less t... | The least positive integer \( m \) divisible by \( 17 \) whose digits sum to \( 17 \) is \( 100008 \). This is found by starting with the smallest numbers composed of \( 9 \), whose digit sum is \( 90 \), and then progressively increasing their numbers of \( 9 \)s until the digit sum reaches \( 17 \). | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | [
"To find the least positive integer \\( m \\) that is divisible by 17 and whose digits sum to 17, we start by considering numbers composed of nines and zeros, since digits must sum to 17. The smallest such number is formed by placing 9's in the higher place values while using zero's in the lower place values.",
"... | The least positive integer \(m\) divisible by 17 and summing to 17 is 991. This is found by maximizing the number of 9s in higher place values while minimizing with 0s, then checking divisibility by 17. | 476 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | [
"Okay, so I have this problem about four towns connected by roads, and I need to figure out the number of ways to make all the roads one-way such that any town can still reach any other town, even if it has to go through other towns. Hmm, let me try to break this down step by step.",
"First, I know that in graph ... | The number of ways to convert all the roads into one-way roads such that any town can reach another is four. This is determined by the chromatic number of the graph, which requires four colors for the complete graph. | 544 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | [
"Okay, let's see. The problem is about five towns connected by roads, and we need to figure out the number of ways to make all the roads one-way such that you can still travel between any town and any other town. Hmm. Alright, so first, let me understand the problem better.",
"We have five towns connected by one-... | The problem asks for the number of ways to assign five distinct roads to five towns in such a way that each town is connected to each other town via unique roads. This is equivalent to finding the number of permutations of five distinct roads, which is calculated as $5! = 120$. | 544 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | [
"Okay, so I have this problem where there are five towns connected by roads, all connecting each pair of towns exactly once. The goal is to figure out how many ways we can make all the roads one-way such that we still can get from anywhere to anywhere. Hmm, interesting.",
"First, I think this has something to do ... | The total number of ways to make all roads one-way, excluding the scenario where all roads are blocked, is 31. | 544 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | [
"Okay, so I have this problem where there are five towns connected by roads, each road connects exactly two towns. The question is asking for the number of ways to make all these roads one-way such that you can still travel from any town to any other town, even if you go through other towns along the way. Hmm, inte... | There are 5 roads, and each pair of towns is connected by two roads, meaning each pair appears twice in the total of 5 roads. Therefore, the total number of ways is 5. | 544 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | [
"Okay, so I have this problem where there are five towns connected by roads, and each pair of towns is connected by exactly one road. I need to find the number of ways to make all the roads one-way such that it's still possible to travel from any town to any other town, just like how cars can move around in a road ... | The sum of the degrees of the vertices in the road network is odd, which violates the Handshaking Lemma and means such a system cannot exist. | 544 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | [
"Okay, so I have this problem where there are five towns connected by roads, each pair of towns has exactly one road connecting them. I need to find the number of ways to make all the roads one-way such that it's still possible to travel from any town to any other town. Hmm, interesting.",
"First, let me understa... | The number of ways to configure the roads so that each town is accessible from all others is determined by the number of ways to remove all roads. This is because removing all roads results in an unconnected graph, implying the original graph was also unconnected. | 544 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | [
"Okay, so there's this problem where five towns are connected by roads, all connecting each pair. We need to figure out how many ways we can route all the roads so that they're all one-way, meaning no town can connect directly to another town. But also, even if we have to pass through other towns, we still need to ... | The Embedded Graph Theorem requires at least 5 directed edges for a graph with 5 components; thus, the number of valid one-way road embeddings is 5. | 544 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | [
"Okay, so I have this problem here about five towns connected by roads, and each road connects two towns exactly once. The question is asking about the number of ways to make all these roads one-way such that you can still get from any town to any other town without passing through any town more than once. Hmm, tha... | The number of ways to assign five towns to ten distinct roads, such that each road connects two distinct towns and each town is assigned to exactly one road, is 30,240. | 544 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"First, I recognize that \\( z \\) is a complex number of the form \\( z = a + bi \\), where \\( a \\) and \\( b \\) are real numbers. The modulus of \\( z \\), denoted \\( |z| \\), is calculated as \\( |z| = \\sqrt{a^2 + b^2} \\).",
"Given the equation \\( z + |z| = 2 + 8i \\), I substitute \\( z = a + bi \\) in... | Given \( z + |z| = 2 + 8i \), substituting \( z = a + bi \) leads to \( a^2 + b^2 = 4 + 64 + 32i \), so \( |z|^2 = a^2 + b^2 \). Therefore, \( |z|^2 = 72 + 32i \). | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"First, we recognize that \\( z = a + bi \\), where \\( a \\) and \\( b \\) are real numbers.",
"The modulus of \\( z \\) is defined as \\( |z| = \\sqrt{a^2 + b^2} \\).",
"Given the equation \\( z + |z| = 2 + 8i \\), we substitute \\( z \\) with \\( a + bi \\):",
"\\[",
"(a + bi) + |a + bi| = 2 + 8i",
"\\... | Given \( z + |z| = 2 + 8i \), substituting \( z = a + bi \) leads to \( a^2 + (a^2 + 64) = 28 \), which simplifies to \( a^2 = 64 \). Thus, \( |z|^2 = a^2 + 64 = 64 + 64 = 128 \). | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"To solve the problem, we start with the given equation:",
"\\[ z + |z| = 2 + 8i \\]",
"We need to find the value of \\( |z|^2 \\). Let's denote \\( z = a + bi \\), where \\( a \\) and \\( b \\) are real numbers.",
"First, calculate \\( |z| \\):",
"\\[ |z| = \\sqrt{a^2 + b^2} \\]",
"Next, square \\( |z| \... | Substituting $z = a + bi$ into the equation and simplifying yields $|z|^2 = 2 - 2ai$. This is because after expanding and simplifying the squares of the distances from the origin, we obtain $|z|^2 = (1 + a^2 + b^2)^2 - 4a^2b^2 = 2 - 2ai$. | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"First, let \\( z = a + bi \\) where \\( a \\) and \\( b \\) are real numbers.",
"Next, calculate the modulus of \\( z \\):",
"\\[",
"|z| = \\sqrt{a^2 + b^2}",
"\\]",
"According to the problem, \\( z + |z| = 2 + 8i \\). Substitute \\( z \\) and \\( |z| \\) into this equation:",
"\\[",
"a + \\sqrt{a^2 ... | Given $z = a + bi$ and $z + |z| = 2 + 8i$, we have $\sqrt{a^2 + b^2} (2 + \sqrt{a^2 + b^2}) = 10$. Squaring both sides yields $a^2 + b^2 = 100$, so $|z|^2 = a^2 + b^2 = 100$. | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"First, I'll denote the complex number \\( z \\) as \\( z = a + bi \\), where \\( a \\) and \\( b \\) are real numbers.",
"The modulus of \\( z \\) is given by:",
"\\[",
"|z| = \\sqrt{a^2 + b^2}",
"\\]",
"According to the problem, \\( z + |z| = 2 + 8i \\). Substituting \\( z \\) with \\( a + bi \\), we ge... | Since \(|z|^2 = a^2 + b^2\) and \(a^2 + b^2 = 4\) (from \(|z| = \sqrt{a^2+b^2}\) and \(\sqrt{a^2+b^2} = 2\)), we get \(|z|^2 = 4\). | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"First, let \\( z = a + bi \\) where \\( a \\) and \\( b \\) are real numbers.",
"The modulus of \\( z \\) is given by \\( |z| = \\sqrt{a^2 + b^2} \\).",
"According to the problem, \\( z + |z| = 2 + 8i \\). Substituting \\( z \\) and \\( |z| \\) into this equation:",
"\\[",
"a + bi + \\sqrt{a^2 + b^2} = 2 +... | The magnitude squared \(|z|^2\) is 8, which corresponds to option (B). | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"Okay, so I have this problem here about a complex number z. Let me see if I can figure it out step by step. The problem says that z plus the absolute value of z equals 2 + 8i. They also tell me that if z is a complex number z = a + bi, then the absolute value of z, which is written |z|, is the square root of (a sq... | Given \( z = 2 + 8i \), we first find \( |z| = \sqrt{2^2 + 8^2} = \sqrt{68} \). Now, \( z + |z| = (2 + 8i) + (\sqrt{68}) = 2 + 8i + \sqrt{68} \), so \( |z|^2 = (\sqrt{68})^2 = 68 \). | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"Okay, so I have this problem here about a complex number z. It says that z plus the absolute value of z equals 2 plus 8i. I need to find the square of the absolute value of z, which is written as |z|². Hmm, let me break this down step by step.",
"First, I remember that for any complex number z, which is usually ... | Given \(z + |z| = 2 + 8i\), substituting \(z = a + bi\) yields \(a^2 + b^2 + 2abi = 2 + 8i\), which simplifies to \(a^2 + b^2 = 4\) and \(|z| = a + bi = 2\). Then \(|z|^2 = 2^2 = 4\), corresponding to option C. | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"To solve the problem, we start with the given equation:",
"\\[ z + |z| = 2 + 8i \\]",
"Let \\( z = a + bi \\), where \\( a \\) and \\( b \\) are real numbers. Then, the modulus of \\( z \\) is:",
"\\[ |z| = \\sqrt{a^{2} + b^{2}} \\]",
"Substituting \\( z = a + bi \\) into the equation, we get:",
"\\[ a +... | The equation \( z + |z| = 2 + 8i \) with \( z = a + bi \) implies \( \sqrt{a^2 + b^2} = 2 \), from which \( a^2 + b^2 = 4 \). Therefore, \( |z|^2 = a^2 + b^2 = 4 \). | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"Okay, so I have this problem here about a complex number z. It says that z plus the absolute value of z equals 2 + 8i. I need to find |z| squared, and the options are A through E, with numbers like 68 and 289. Hmm, let me try to figure this out step by step.",
"First, I remember that the absolute value of a comp... | Since \(z=4i\), we have \(z+|z|=2\), so \(|z|=\sqrt{5}\); hence \(|z|^2=5\). The answer is \(\boxed{\textbf{(C)}\ 5}\). | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"We start with the given equation:",
"\\[ z + |z| = 2 + 8i \\]",
"To isolate \\( |z| \\), we subtract \\( z \\) from both sides:",
"\\[ |z| = (2 + 8i) - z \\]",
"Recall that \\( z \\) can be expressed in terms of its real and imaginary parts as \\( z = a + bi \\), where \\( a \\) and \\( b \\) are real numb... | Given \( z + |z| = 2 + 8i \), we isolate \( |z| \) to get \( |z| = (2 - a) + 8i \). This implies \( |z|^2 = (2 - a)^2 + 64 \), and since \( z + |z| = 2 + 8i \), we find \( |z|^2 = 68 - (2 - a)^2 \), which simplifies to \( |z|^2 = 4 \). | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To determine the number of ways to paint six sections of a ring with four distinct colors such that no two adjacent sections share the same color, we can follow these steps:",
"1. **Choose the first color:** There are 4 possible colors to choose from.",
"2. **Choose the second color:** Since the second color c... | With 4 colors available, the six sections painted in order can be matched in \(4 \times 3 \times 2 \times 1 = 24\) ways. Thus there are 24 ways to choose the colors. | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To determine the number of ways to paint the six sections of the ring such that no two adjacent sections have the same color, we can break down the problem step by step.",
"1. **First Section (Red):** There are 4 available colors for the first section.",
"2. **Second Section (Not Red):** Since no two adjacent ... | There are 4 choices for the first section, and for each subsequent section, one must choose a color different from the previous three. This results in $4 \times 3 \times 2 \times 3 \times 3 \times 3 = 864$ ways to paint the six sections without adjacent colors matching. | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To solve the problem of determining the number of ways to color six sections of a ring such that no two adjacent sections share the same color, we can follow these steps:",
"1. **Choose the color for the starting section (Section 1):** There are 4 available colors, so there are 4 choices.",
"2. **Choose the co... | The number of ways to paint six sections of a ring such that no two adjacent sections share the same color is calculated by multiplying the number of choices for each section: $4 \times 3 \times 4 \times 4 \times 4 \times 4$. This results in a total of 4096 valid color combinations. | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To determine the number of ways to paint six sections of a ring such that no two adjacent sections are the same color, we can break the problem into three main steps:",
"1. **Choosing the Colors for the First Section:**",
"- There are 4 available paint colors.",
"- We can choose any of the 4 colors for the f... | Choose 4 of the 4 colors for the first section, then 3 each of the 5 remaining sections, giving 4 × 3⁵ = 972 ways. Thus, there are **972** ways to paint the sections without two adjacent sections being the same color. | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To solve the problem of finding the number of ways to paint six sections of a ring with four paint colors such that no two adjacent sections share the same color, we can use the concept of moves. Each move involves choosing a color from the four available and then selecting a horizontal or vertical segment to pain... | The number of ways to paint six sections of a ring with four paint colors such that no two adjacent sections share the same color is calculated by multiplying the choices for each section: 4 choices for the first, 3 for the second, and $3^6$ for the remaining five. This results in a total of $4 \times 3 \times 3^6 = 97... | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To solve the problem of finding the number of ways to paint six sections of a ring such that no two adjacent sections are the same color, we can use induction.",
"1. **Base Case:**",
"- For 2 sections, there is only 1 valid color combination: both sections are the same color.",
"- For 3 sections, the valid c... | For \( n = 4 \), the color combinations where no two adjacent sections share color must be limited to a single color, resulting in 1 valid combination. Thus, there is exactly \(\boxed{1}\) way to paint the sections. | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To solve the problem of finding the number of ways to paint six sections of a ring such that no two adjacent sections have the same color, we can use the technique of traversing the ring in a specific order and assign a unique color to each section.",
"1. **Traversal Order**: Start by painting the innermost sect... | The number of ways to choose the paint color for the innermost section is 4. Each section is painted sequentially in such a way that no two adjacent sections share the same color, resulting in the total number of ways being $4 \times 6 \times 1 = 24$. | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To solve the problem of finding the number of ways to paint six sections of a ring such that no two adjacent sections have the same color, we can use the concept of permutations with restricted choices.",
"1. **Initial Setup:**",
"- We have 6 sections in a ring.",
"- We need to paint each section a solid col... | The first section has 4 color choices, and each of the remaining 5 sections has 3 choices, as adjacent sections cannot share a color. The total number of ways to paint the six sections is $4 \times 3 \times 2 \times 1 = 48$. | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To solve the problem of finding the number of ways to paint six sections of a ring such that no two adjacent sections are the same color, we can break the problem into four steps:",
"1. **Paint the central section:** This section is always painted first, so we only need to choose a color for it. There are 4 colo... | The problem requires finding the number of valid color combinations for painting six ring sections such that no two adjacent sections share the same color. There are 4 choices for the central section, 3 for the second, 2 for the third, and 1 for the last section, resulting in 4 * 3 * 2 * 1 = 24 total possible combinati... | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To solve the problem of finding the number of ways to paint six sections on a ring with four paint colors such that no two adjacent sections are the same color, we can follow these steps:",
"1. **Paint the first section:** There are 4 possible colors to choose from for the first section.",
"2. **Paint the seco... | The problem requires choosing 6 distinct colors for 6 sections on a ring such that no two adjacent sections share color, leading to $4 \times 3 \times 3 \times 3 \times 3 \times 3 = 468$ valid painting combinations. | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To solve the problem of finding the number of ways to paint six sections of a ring such that no two adjacent sections are painted with the same color, we can use the concept of choosing a subset of four colors out of five available colors, ensuring that no two adjacent sections share the same color.",
"First, we... | The total number of ways to choose four colors out of five is $\binom{5}{4} = 5$. Subtracting the invalid configurations where two adjacent sections share the same color (which requires at least one color to be repeated), we get $5 - 1 = 4$ valid ways. | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To solve the problem of finding the number of ways to paint the six sections of the ring such that no two adjacent sections share the same color, we can break it down into steps based on the allowed contiguous color choices.",
"1. **First Section:** There are 4 possible colors for the first section.",
"2. **Ne... | The number of ways to paint the six sections of the ring such that no two adjacent sections share the same color is calculated by considering the choices for each section sequentially: 4 choices for the first, 3 choices for each of the second, fourth, and sixth sections, and 3 choices for the last section. Multiplying ... | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To solve the problem of finding the number of ways to paint six sections of a ring such that no two adjacent sections are the same color, we can use the concept of horizontal Paine's cycles.",
"1. **Identify the Base Coloring**: Assign one color to the first section, which is fixed.",
"2. **Determine Colors fo... | The number of ways to paint six sections of a ring such that no two adjacent sections are the same color is determined by finding the number of derangements of four elements, which is calculated as $D(4) = 9$. This accounts for the alternating color restriction present in the problem. | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To solve the problem of finding the number of ways to paint six sections of a ring with four different colors, where no two adjacent sections can be the same color, we can break down the process into smaller steps.",
"1. **Choose the color for the first section**: There are 4 different colors available, so there... | The number of ways to paint six sections of a ring with four different colors such that no two adjacent sections are the same is calculated by multiplying the number of choices for each section: $4 \times 3 \times 3 \times 3 \times 3 \times 3$. This results in a total of 5184 ways. | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To solve the problem of finding the number of ways to paint six sections of a ring with four paint colors such that no two adjacent sections are the same color, we can follow these steps:",
"1. **Identify the Valid Sequences**:",
"- The first section must be painted with one of the four available colors. This ... | The number of ways to paint six sections of a ring such that no two adjacent sections share the same color is calculated by selecting a starting color for the first section (4 ways), then arranging colors for the remaining sections where adjacent sections differ (5 positions for 3 choices each). This results in $4 \tim... | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To determine the sum of all positive integers \\( a = 2^n \\cdot 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers, such that \\( a^6 \\) is not a divisor of \\( 6^a \\), we start by analyzing the conditions under which \\( a^6 \\) does not divide \\( 6^a \\).",
"1. **Divisibility Condition**:",
... | The sum of all positive integers \( a = 2^n3^m \) where \( a^6 \) is not a divisor of \( 6^a \) is 10. This condition is satisfied when \( n=m=5 \), as in this case \( a^6 = 6^a \). | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To determine the sum of all positive integers \\( a = 2^n \\cdot 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers, such that \\( a^6 \\) is not a divisor of \\( 6^a \\), we start by analyzing the conditions.",
"First, we note that \\( 6 \\) can be factored as \\( 2 \\times 3 \\). For \\( a^6 \\) t... | Numbers \(a\) having all exponent-matchtings that don't divide \(6^a\) are those of the form \(2^n\cdot 3^m\) with \(n\ge6\) (all bases 2 or 3) and \(m\ge1\) (odd or even), permitting \(a=1,3,4,6\): \(1+3+4+6=16\). Their sum is \(\boxed{16}\). | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"First, I need to find all positive integer powers \\( a = 2^n \\cdot 3^m \\) such that \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"I start by analyzing the prime factorization of \\( 6 \\): \\( 6 = 2 \\cdot 3 \\). For \\( a^6 \\) to not be a divisor of \\( 6^a \\), the exponents of the prime factors in \\( a... | The condition for \( a^6 \) not to be a divisor of \( 6^a \) requires \( n \leq 1 \), \( m \leq 1 \), with \( n > 0 \) and \( m \geq 1 \). The possible values for \( n \) and \( m \) are \( n \in \{2, 3\} \) and \( m \in \{0, 1\} \), yielding \( a = 4 \) and \( a = 6 \), so the sum is \( 4 + 6 = 10 \). | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To determine the sum of all positive integers \\( a = 2^n \\cdot 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers, such that \\( a^6 \\) is not a divisor of \\( 6^a \\), we start by analyzing the condition \\( a^6 \\) not dividing \\( 6^a \\).",
"First, consider the prime factorization of \\( 6 = ... | The possible values for positive integer \(a\) where \(a^6 \not\mid 6^a\) are \(a=1, 2, 4, 8, 16\) associated with exponents \(n \in \{0, 1, 2, 3\}\) and \(m \in \{0\}\). The sum of these values is \(1 + 2 + 4 + 8 + 16 = 31\). | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To solve the problem, we need to determine the values of \\( n \\) and \\( m \\) such that \\( 2^n \\cdot 3^m \\) is not a divisor of \\( 6^a \\).",
"First, we recognize that \\( 6^a \\) can be rewritten as \\( (2 \\cdot 3)^a = 2^a \\cdot 3^a \\). Therefore, for \\( 2^n \\cdot 3^m \\) to not be a divisor of \\( ... | The sum is \(3^{n+m-1}\) for any valid \(a\), since \(a^6\) skips divisibility by 2 or 3 when it divides \(6^a\). Thus the answer is \(\boxed{3^{n+m-1}}\). | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To solve this problem, we need to find positive integer \\( a = 2^n \\cdot 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers, and \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"First, let's analyze the condition \\( a^6 \\) is not a divisor of \\( 6^a \\). This can be rewritten using prime factori... | For \( a = 2^n \cdot 3^m \) where \( a \) is not a divisor of \( 6^a \), we need \( n \geq 2 \) and \( m \geq 1 \), leading to \( a \geq 2^2 = 4 \). The sum of non-negative integers \( n \) and \( m \) that satisfy these conditions is \( 2 + 0 = 2 \). | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"First, I need to find all positive integers \\( a = 2^n \\cdot 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers, such that \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"I start by expressing \\( 6 \\) as \\( 2 \\cdot 3 \\) and rewrite the condition using this representation:",
"\\[",
"a^6 \\... | The condition that \( a^6 \) is not a divisor of \( 6^a \) implies that the exponents of 2 and 3 in \( a = 2^n \cdot 3^m \) must differ by at least 2 when considering \( a = 2^{6+a} \cdot 3^{6+a} \). This condition holds for any non-negative integer \( a \geq 4 \), leading to a sum of \( 2n(n+1) \). | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"First, we need to determine the values of \\( n \\) and \\( m \\) such that \\( 2^n \\cdot 3^m \\) is a positive integer. Since \\( 6^a \\) must not be divisible by 2, \\( a \\) must be an odd number. This gives us \\( a = 2^n \\cdot 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers and \\( n \\) is ... | The sum of all positive integers \( a = 2^n \cdot 3^m \) such that \( a^6 \) is not a divisor of \( 6^a \) is the sum of all valid odd values for \( k \). This sum is \( 3 + 6 + 9 + 12 + 15 = 45 \). | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To solve the problem, we need to identify all possible values of \\( a = 2^n \\cdot 3^m \\) such that \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"1. **Divisors of \\( 6^a \\):**",
"- \\( 6 = 2 \\cdot 3 \\), so \\( 6^a = (2 \\cdot 3)^a = 2^a \\cdot 3^a \\).",
"- The divisors of \\( 6^a \\) are \\( 1, 2, 3... | The only valid combinations where \( a^6 \) is not a divisor of \( 6^a \) are \( a = 3^m \) for some non-negative integer \( m \). The sum of all such positive integers \( a \) is \( 0 + 1 + 2 + \ldots + m \). | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To solve the problem, we need to find all positive integers \\( a = 2^n \\cdot 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers, such that \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"1. **Understanding the Condition**:",
"- We need to ensure that \\( a^6 \\) does not divide \\( 6^a \\). This... | The condition that \(a^6\) is not a divisor of \(6^a\) requires \(a\) to be co-prime with \(6\), that is, to share no prime factors with \(2 \times 3\). The only \(a\) satisfying this is \(a = 2^1 \cdot 3^0 = 2\), giving a sum of \(2^1 = \boxed{2}\). | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To find the sum of all positive integers \\( a = 2^n \\cdot 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers, such that \\( a^6 \\) is not a divisor of \\( 6^a \\), follow these steps:",
"1. **Start with the given condition:**",
"\\[",
"a^6 \\equiv 6^a \\pmod{10}",
"\\]",
"Simplifying, sin... | Since \(a^6 \equiv 2^a \pmod{10}\) requires \(12 \cdot 3^m\) to have no factors of 3, \(n = 2\) is the only value with \(2^n\) a factor of 12, so \(a = 2^2 \cdot 3^m\) works. This sum is \(\boxed{10}\). | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To determine the sum of all positive integers \\( a = 2^n \\cdot 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers, such that \\( a^6 \\) is not a divisor of \\( 6^a \\), we start by analyzing the condition \\( a^6 \\) not dividing \\( 6^a \\).",
"First, we rewrite \\( 6^a \\) as \\( (2 \\cdot 3)^a... | The sum of all positive integers $a = 2^n3^m$ such that $a^6 \not\div 6^a$ is found by summing $v_1 = \frac{n}{2}$ and $v_2 = \frac{m}{3}$. The total sum is $v_1 + v_2 = \frac{n}{2} + \frac{m}{3}$, which simplifies to $m/2$. | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To solve the problem, we need to find all positive integers \\( a = 2^n \\cdot 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers, such that \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"1. **Understanding the Requirements**",
"- \\( a^6 \\) must not be divisible by \\( 6^a \\).",
"- Since \\(... | The positive integers $a = 2^n \cdot 3^m$ for which $a^6$ is not a divisor of $6^a$ are those where $n \in \{0, 1, 2, 3\}$ and $m \in \{0, 1, \ldots, 5\}$. The sum of such integers is $21$. | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To determine the sum of all positive integers \\( a = 2^n \\cdot 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers such that \\( a^6 \\) is not a divisor of \\( 6^a \\), we start by analyzing the prime factorization of \\( 6 \\) and the desired exponent \\( k \\) in \\( 6^a \\).",
"1. **Prime Facto... | For \( 6^a \) to be divisible by \( a^6 \), the prime factorization of \( 6^a \) must include only \( 2^6 \) and \( 3^6 \) with the minimum exponents. The sum of all such positive integers \( a = 2^n \cdot 3^m \) that are \( 2^6 \cdot 3^6 \) (not \( 2^6 \cdot 3^6 \) and not \( 2^5 \cdot 3^5 \)) is \( 2^6 + 3^6 = 793 \)... | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To solve the problem, we need to find positive integer combinations of \\( a = 2^n \\times 3^m \\) such that \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"First, express \\( 6^a \\) as \\( (2^3 \\times 3)^{a} \\), which simplifies to \\( 2^{3a} \\times 3^a \\).",
"We set up the congruence for when \\( a^6 \\... | Given that \( 6^a \equiv 1 \pmod{6} \), which simplifies to \( 2^{6a} \equiv 1 \pmod{6} \), we need to find positive integer combinations \( a = 2^n \times 3^m \) such that \( 2^{6a} \equiv 1 \pmod{6} \). This occurs when \( n=0 \) and \( m=0 \) because \( 2^{6 \times 0} \times 3^0 \equiv 2^0 \equiv 1 \pmod{6} \). | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqr... | [
"First, let's understand the given information:",
"- Circles \\( C_1 \\) and \\( C_2 \\) have centers \\( O \\) and intersect at points \\( X \\) and \\( Y \\).",
"- \\( X \\) lies on both circles, and \\( OZ = 11 \\) is the distance from the center of \\( C_1 \\) to \\( X \\).",
"- \\( OZ = 11 \\) means \\( ... | The radius of circle \( C_1 \) is determined to be \( R_1 = 5\sqrt{2} \). This value corresponds to option (D) \( 2\sqrt{7} \). | \sqrt{30} | false |
Open-Reasoner-Zero/orz_math_57k_collection | Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqr... | [
"First, I need to understand the relationship between the two circles. Since circle \\( C_1 \\) lies entirely within circle \\( C_2 \\) and their centers are aligned, they intersect at two points, \\( X \\) and \\( Y \\).",
"Next, I'll use the given information about the external distance and the sum of the radii... | With radii \(R = 2\) from \(XZ = OZ = 11\), the sum gives \(r + 2 = 2\), so \(r = 0\)—a physically invalid result and clearly not among the options. Therefore, the closest option is C, \(3\sqrt{3}\). | \sqrt{30} | false |
Open-Reasoner-Zero/orz_math_57k_collection | Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqr... | [
"First, we note that point \\( Z \\) is external to circle \\( C_1 \\) and lies on circle \\( C_2 \\). The distance from the center of circle \\( C_1 \\) to the external point \\( Z \\) is given as \\( OZ = 11 \\).",
"Next, we analyze the points \\( X \\) and \\( Y \\). Since \\( X \\) and \\( Y \\) lie on the in... | The distance from center \( O \) of circle \( C_1 \) to external point \( Z \) is \( OZ = 11 \), which is also the distance from \( O \) to the common intersection point \( Y \) of circles \( C_1 \) and \( C_2 \), making \( OY = 11 \). Given the radii \( r \) of both circles are \( r \) and \( R \) of circle \( C_2 \),... | \sqrt{30} | false |
Open-Reasoner-Zero/orz_math_57k_collection | Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqr... | [
"First, we note that point \\( Z \\) is external to circle \\( C_1 \\) and lies on circle \\( C_2 \\). The distance from the center of circle \\( C_1 \\) to the external point \\( Z \\) is given as \\( OZ = 11 \\).",
"Next, we consider the intersection points \\( X \\) and \\( Y \\) of the two circles. The tangen... | By the tangent-chord converse, \(OX = OY = 11/2\), so \(XZ = OZ + OY = 13\) and \(YZ = 7\), giving \(R^2 = 11^2 + 6.5^2 = 163.25\) and radius \(R \approx 8.66\). This matches option **(D)**. | \sqrt{30} | false |
Open-Reasoner-Zero/orz_math_57k_collection | Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqr... | [
"First, we note that point \\( Z \\) is exterior to circle \\( C_1 \\) and lies on circle \\( C_2 \\). The distance from the center of \\( C_1 \\) to the exterior of \\( C_1 \\) is given as \\( OZ = 11 \\). We can find the radius of circle \\( C_2 \\) using the Pythagorean theorem from \\( O \\) to \\( Z \\), \\( O... | The radius of circle $C_1$ is found to be 8 using the power of a point theorem and the calculated radius of circle $C_2$ as $\sqrt{11}$. This conclusion is supported by the derived relationship $OX \cdot OY = R^2$. | \sqrt{30} | false |
Open-Reasoner-Zero/orz_math_57k_collection | Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqr... | [
"We are given that circle \\( C_1 \\) is entirely inside circle \\( C_2 \\) and their centers \\( O_1 \\) and \\( O_2 \\) are aligned because \\( O \\) lies on \\( C_2 \\). They meet at points \\( X \\) and \\( Y \\). Point \\( Z \\) is outside circle \\( C_1 \\) and lies on circle \\( C_2 \\), with \\( OZ = 11 \\)... | The radius of circle \(C_2\) is \(r_2 = OZ - O_2 = 11 - 13 = -2\). Since \(O_1\) lies on \(C_2\), \(O_1 = r_1 - O_2 = 11 - (-2) = 7\), making the radius of circle \(C_1\) \(r_1 = O_1 - OZ = 7 - 11 = -4\), which is physically impossible. | \sqrt{30} | false |
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