source stringclasses 7
values | question stringlengths 3 1.02k | steps listlengths 0 317 | answer stringlengths 0 2.56k | 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, so I've got this problem here about a circle with a diameter AB that has a length that's a 2-digit integer. Reversing the digits gives me the length of a perpendicular chord CD. The distance from the center H to the chord CD is a positive rational number. Hmm, I need to figure out what the length of AB is.",... | 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 that has a length which is a 2-digit integer. The problem mentions that if you reverse the digits of this length, you get the length of another chord CD. Then, it says the distance from the center H to the chord's intersection point O is a positive rat... | 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 try to figure out this problem. Hmm. So we have a circle with diameter AB, and its length is a 2-digit integer. Then, reversing the digits gives the length of the perpendicular chord CD, and the distance between intersection point H of the diameter and the chord CD is a positive rational number. We nee... | The length of the perpendicular chord \( | 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 try to figure out this problem step by step. First, the problem says that the diameter AB has a length which is a two-digit integer in base ten. So AB can be represented as 10n + r where n is the tens digit and r is the units digit, both integers from 0 to 9. Wait, but typically, in two-digit numbers i... | 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 here about a circle with diameter AB. The length of this diameter is a two-digit integer, and reversing the digits gives a chord CD with a length that's also a two-digit integer. There's also something about the distance from the intersection point H to the center O being a positive ra... | 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 diameter AB, and the length of this diameter is a two-digit integer. Then, if I reverse the digits of that diameter, I get the length of another perpendicular chord, CD, which is also a two-digit integer. The distance from the center O to the intersection point H of... | 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 $AB$. | [
"Okay, so I have this problem about a circle with a chord AB, and its diameter has a length related to this circle's radius. The problem says that if we reverse the digits of the diameter's length, we get a different chord CD. The distance from the center O to the intersection point H of the chord CD is a positive ... | 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 try to figure out this problem. So, we have a circle with diameter AB, and the length of this diameter is a 2-digit integer in base 10. When we reverse the digits of this diameter, we get the length of a perpendicular chord CD. The distance from the center O to the chord CD is also a positive rational ... | 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 $AB$. | [
"Okay, let me try to figure out this problem. So, we have a circle with diameter AB. The diameter has a length that's a two-digit integer in base ten. Then, reversing the digits gives the length of the perpendicular chord CD. The distance from the center O to the chord CD is a positive rational number, and we need ... | 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, so I have this problem where there are five towns connected by roads, and each road connects exactly two towns. The task is to find the number of ways to make all these roads one-way, ensuring that it's still possible to travel between any two towns, even if the route takes a detour through other towns. Hmm,... | 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've got this problem here: there are five towns connected by roads, all one-way connections. The question is asking how many ways there are to make all these roads one-way such that it's still possible to travel from any town to any other town. Hmm, that sounds a bit tricky, but let me try to break it do... | The number of ways to make roads one-way such that each town can reach any other town is two. This is | 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). | [
"Alright, so I have this problem where there are five towns connected by roads, and each road connects exactly two towns. The question is asking me to find the number of ways to make all the roads one-way such that it's still possible to get from any town to any other town, and also the roads can be used in both di... | 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 here about five towns connected by roads, exactly one road between each pair. I need to figure out 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, but also avoiding any towns that might get in the way. Hmm, prett... | 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 exactly once. The task is to find the number of ways to make all the roads one-way such that it's still possible to get from any town to any other town, even if you might pass through other towns along t... | 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 four-dimensional Rubik's cubes, right? Because there are four towns and their roads form a qu Brainf***-uhm? Wait, maybe that's a typo. Let me think again. The question says \"Find the number of ways there are to make all the roads one-way in such a way that it is still possib... | 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've got this problem here about five towns connected by roads. Each pair of towns has exactly one road connecting them. The question is asking how many ways there are to make all the roads one-way, such that it's still possible to travel between any two towns, even if you take a detour through other town... | 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 road connects two towns. The task is to figure out the number of ways to make all these roads one-way such that you can still travel between any two towns, even if you take a detour through other towns if needed.",
"Alright, let... | 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 $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"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 |z| squared. Hmm, let me figure this out step by step.",
"First, I remember that for any complex number z, the absolute value, denoted as |z|, is the distance from the origin t... | 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 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 $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"Okay, so I have this problem here: A complex number \\( z \\) satisfies \\( z + |z| = 2 + 8i \\). I need to find \\( |z|^2 \\). Hmm, let me think about how to approach this.",
"First, I remember that for any complex number \\( z = a + bi \\), the modulus \\( |z| \\) is calculated as \\( \\sqrt{a^2 + b^2} \\). So... | 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. It says that z plus the absolute value of z equals 2 plus 8i. And I need to find the square of the absolute value of z, which is |z| squared. Hmm, let me break this down step by step.",
"First, I remember that for any complex number z, which is usually ... | 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 $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"Okay, so I have this complex number problem here. Let me try to figure it out step by step. The problem says:",
"\"The complex number \\( z \\) satisfies \\( z + |z| = 2 + 8i \\). What is \\( |z|^2 \\)?\"",
"Hmm, alright. I remember that a complex number is usually written as \\( z = a + bi \\), where \\( a \\... | 289 | false |
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