source stringclasses 7
values | question stringlengths 3 1.02k | steps listlengths 0 207 | answer stringlengths 0 1.46k | 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 diameter length a2-digit integer AB. The key point is that when we reverse the digits of AB, we get the length of the perpendicular chord CD. And then we need to find the length AB and determine the distance from H to O.",
"Let me try to visualize this. There's a ... | The distance from the intersection point $H$ to the center $O$ is $\boxed{2AB}$. | 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, say length a 2-digit integer. But wait, that's not what's mentioned—it's saying the diameter is length a 2-digit integer (base ten), and then reversing the digits gives the length of the perpendicular chord CD. That does sound a bit confusing. May... | The length of CE is the hypotenuse of a right triangle with legs 1 and 2, and hypotenuse 40. This hypotenuse is equal to the diameter, which 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 $AB$. | [
"First, I need to understand the problem. We have a circle with a diameter of length a two-digit integer. When the diameter is reversed, the resulting length is the length of a perpendicular chord (chord diagonal) from point H to the center O of the circle. The goal is to find the length of the diameter AB.",
"Ne... | The problem cannot be solved with the given information, as the given diameter \( a = 20 \) is not a perfect square. We cannot determine the length of diameter \( AB \). | 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 $AB$. | [
"To determine the length of \\( AB \\), we start by noting that the diameter \\( AB \\) is a two-digit integer. This means \\( AB = 10n \\), where \\( n \\) is a digit between 1 and 9 (since the diameter cannot be a decimal or fraction). We are also given that \\( AB = 2a \\), where \\( a \\) is a positive rational... | With diameter AB = 20 two-digit, reversing the digits gives CD = 10, which is half of AB. Both equal, so the answer is \(\boxed{20}\). | 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 $AB$. | [
"First, I recognize that the diameter of the circle is given as \\( a \\)2-digit integers, which means \\( 10a \\leq 289 \\). Solving for \\( a \\), I find \\( a \\leq 26 \\).",
"Next, when the diameter is reversed, the new diameter becomes \\( 26a \\). The length of the perpendicular chord \\( CD \\) can be expr... | The length of the perpendicular chord \( CD \) is given by \( \frac{26a}{2} = 13a \), and the distance from intersection point \( H \) to center \( O \) is \( \frac{338}{2} = 169 \). Since \( 10a \leq 289 \) with \( a \leq 11.5 \), the valid integers are \( a = 12 \) and \( a = 11 \), leading to lengths 156 and 133, re... | 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 $AB$. | [
"First, I need to determine the length of the diameter \\( AB \\) based on the given condition. Since the diameter is a two-digit integer, I'll consider the largest possible value for \\( a \\) while ensuring the result is a valid two-digit number.",
"Next, I'll reverse the digits of \\( a \\) to find the length ... | The length of the diameter \( AB \) is 98. The perpendicular chord \( CD \), formed by reversing its digits, is 17, which is a positive rational length and a two-digit integer. | 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 of a circle with base ten has length a two-digit integer. Then, after reversing the digits, the length of the perpendicular chord CD becomes the same length. We need to find the length of AB. Hmm, so AB is related to CD somehow. Let me break this down step by ste... | Since \( a^2 + d - 2d = 22 \) implies \( a^2 = 20 \) and \( d = 2 \) and \( a^2 + d = 44 \), the length of \( AB \) is \( \boxed{22} \). | 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 $AB$. | [
"To solve the problem, let's follow these steps:",
"1. **Given:** The diameter of the circle is \\( a2 \\)-digit integer (base ten).",
"2. **Reverse the digits:** The length of the perpendicular chord \\( CD \\) is \\( a2\\,digit\\text{-}int \\).",
"3. **Distance from \\( H \\) to \\( O \\):** Let \\( x \\) b... | The problem states that the diameter of a circle with base ten digits \(a2\)-digit integers is \(a2\)-digit integers. When the square root of this diameter is taken and its negative counterpart equals the full perpendicular chord length, the diameter is found to be \(a2\)-digit integers. | 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 \\( AB \\) be the diameter of the circle. Reversing the digits of \\( AB \\) gives \\( CA = \\text{any positive rational number} \\).",
"Since \\( CO \\) is perpendicular to \\( AB \\), \\( CO \\) is a chord that is perpendicular to the diameter \\( AB \\).",
"The radius \\( r \\) of the circle is half of ... | Since \( CA \) is the diameter, \( r=AB \), and perpendicularity condition implies \( AC=r \). Since \( AD \) is perpendicular to \( AB \), its length \( AD = \frac{AB}{2} \). | 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 $AB$. | [
"Alright, so I have this problem here about a circle with diameter \\( a \\)-digit integers and their perpendicular chord CD. I need to find the length of \\( AB \\). Hmm, okay, let me try to visualize this.",
"First, in a circle, the perpendicular from one point to another is perpendicular to the line connecting... | The length of side \( AB \) is a \( 2-digit \)-digit \( \pi \) number approximately equal to 1999, which is the hypotenuse of a right triangle with legs of lengths \( 1998 \) and \( 1997 \). Continuing this process, \( CD \) is approximately \( 450.43 \), which can be written as \( 1999 \). | 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 where I'm given a circle with diameter \\( a2 \\), where \\( a \\) is a two-digit integer. Then, I need to reverse the digits of \\( a2 \\) to get \\( cd \\), and then find the distance between the intersection point \\( H \\) and the center \\( O \\). The distance \\( x \\) from \\( H... | The distance \( x \) from point \( H \) to the center \( O \) of a circle with diameter \( 2a2 - 2a = 2a2(1 - a) \) is \( \frac{1}{2} \) if the original diameter \( 2a2 - 2a \) is a positive rational number. Therefore, the distance between \( H \) and \( O \) is \( \boxed{\frac{1}{2}} \). | 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 $AB$. | [
"To determine the length of diameter \\( AB \\), we start by letting the diameter be \\( a \\) with base ten. When its digits are reversed, the length of the perpendicular chord \\( CD \\) is \\( 2a \\).",
"We know that the sum of the digits of \\( a \\) must be such that it divides 10, as \\( a \\) has to be a 2... | The diameter AB, when reversed to CD, is the perpendicular chord CD = 20. The distance from H(the intersection point) to the center O is C/2 = 20/2 = 10, so AB = 10. | 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 diameter \\( AB \\) as \\( 2k \\) where \\( k \\) is a two-digit integer.",
"The length of the perpendicular chord \\( CD \\) is the same as the length of the diameter, which is \\( 2k \\).",
"We can compute this length by dividing \\( 2k \\) by 2:",
"\\[",
"\\frac{2k}{2} = k",
"\\]",
... | The length of the diameter \( AB \) is given by \( 2k \), where \( k \) is a two-digit integer. This length is also a positive rational number. | 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 $AB$. | [
"First, I recognize that the diameter of the circle is given as the base ten number \\( a \\). The perpendicular chord \\( CD \\) is the ratio of the corresponding sides of the triangle formed by points \\( H \\), \\( O \\), and \\( A \\).",
"Next, I calculate the sides of the triangle:",
"- \\( AH = a \\)",
... | The perpendicular chord \( CD \) has a length equal to the square of the diameter \( a \), so \( CD^2 = a^2 \). Since \( CD^2 \) is also the square of \( a+o = OH \), where \( OH = o \) and \( CH = a \), we have \( OH^2 + OH^2 = a^2 \), which means \( CD^2 = a^2 + a^2 \) or \( CD^2 = 2a^2 \), and finally \( CD = 1 \). | 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 $AB$. | [
"To solve the problem, we start by defining the vertices of the triangle as points \\( H, A, B \\) and \\( C \\). Let the length of the diagonal \\( AB \\) be \\( d \\).",
"1. **Defining Circles:**",
"- The straight-line distance from \\( H \\) to the center \\( O \\) can be expressed in terms of \\( d \\):",
... | The distance $AB$ satisfies $AB = DB$, where $D = 2b$, so $AB = 2b$. The circle's radius is $r = OD^2 - OA^2 = 4$, meaning $r = 2b = 2d$. | 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 start by noting that the set \\(\\{1, 2, 3, \\dots, 2015\\}\\) has \\(1000\\) subsets, each with \\(2015\\) elements.",
"From each subset, we need to choose the smallest element, which is \\(1\\) from the set. This means we are selecting \\(999\\) elements from the set.",
"The total nu... | The arithmetic mean of the nine-second-smallest elements is given by \(\frac{p}{\frac{p+q}{2}}\), which results in \(p = 1\). Consequently, the sum of all least elements is \(2\). | 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 $p + q$. | [
"Okay, so I have this problem about subsets of a set that has 2015 elements. Each of these subsets has 1000 elements, and I'm supposed to find the arithmetic mean of all the least elements in each subset. The arithmetic mean is given as a fraction p over q, where p and q are relatively prime integers. I need to fin... | The arithmetic mean of all the least elements was calculated as p/q by summing 2515 terms (1 subset of 2015 elements plus 500 integers starting from 1) and dividing by 1000. This resulted in 2516/1000, which simplifies to 2516. | 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 $p + q$. | [
"Okay, so I have this problem about subsets and least elements. Let me try to wrap my head around it.",
"The problem says that I need to choose the smallest element from every 1000-element subset of the set {1, 2, 3, ..., 2015}. There are 1000 subsets to choose from. So, for each element in the set, I have to pic... | The problem sets for each element's subset's smallest value, implying 0 or 1/1000 for each element. Therefore, each element's contribution to the total number of least elements, represented by \(A\), is its value \(y\) plus 5 times its smallest value \(5y\), yielding \(A + 9999(5y) = 1000\), which solves to \(A = 0.1\)... | 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 $p + q$. | [
"Okay, so I have this problem where I need to calculate the arithmetic mean of all the least elements when choosing subsets of 1000 elements from the set {1, 2, 3, ..., 2015}. Each subset has all but one element, right? So, I should consider all 1000 subsets one by one, picking the smallest one in each case.",
"L... | The arithmetic mean of all the least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Since all subsets' least element is 1, the mean is the sum of all 999 terms over the 1000 subsets, which simplifies to \( \frac{999 \times 1}{1000} = \frac{999}{1000} = \frac{1}{10} \), resulting in... | 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, calculate the total number of subsets of the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\), which is \\(2015 \\times 2014\\).",
"Since each subset includes a unique combination of the numbers from 1 to 2015, the total number of least elements in all subsets is simply the number of subsets, which is \\(2015\\) (a... | The total number of subsets \( S \) of \( \{1, 2, \ldots, 2015\} \) is \( 2015 \times 2014 \). The total chromatic number of these subsets is \( \frac{2015 \times 2014}{2} = 2521 \), which is the product of the product of pairs of chromatic numbers \( \frac{1}{8}, \frac{2}{7}, \dots, \frac{11}{10} \), yielding \( p=252... | 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 $p + q$. | [
"Okay, so I have this problem where I need to find the arithmetic mean of all the least elements from a collection of 1000-element subsets of the set {1, 2, 3, ..., 2015}. Each subset contributes one element to this collection, as it's a subset, and the least element is the smallest number in the set. So, when I sa... | p = 1011 and q = 1015, so p + q = 2026. | 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 $p + q$. | [
"Okay, so I have this problem about the arithmetic mean of subsets where I need to find the sum of my chosen elements. Let me read the problem again to make sure I understand it correctly.",
"The problem is: For all 1000-element subsets of the set {1, 2, ..., 2015}, I need to choose the least element and then fin... | The arithmetic mean of all the least elements in 1000-element subsets of a set with 2015 elements is \(\frac{1}{1000}\), and this value is \(p\). Therefore, \(p+q = 1001\). | 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 determine the arithmetic mean of all the least elements when choosing the minimum from each 1000-element subsets of the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\) and factoring the original set \\(\\{1, 2, 3, \\ldots, 2015\\}\\) as \\(pqr\\), we proceed as follows:",
"1. **Selection of Least Element**:",
"- The... | The number of ways to pick a subset of 999 elements from \(\{1, 2, \ldots, 2015\}\) with minimum elements forms prizes, characterized by two double-factor pair sets and one independent set. Summing these sets, \(4 \times p + 4 \times q\), results in \(16p + 8q\), which, factoring as \(16(p+q) - 8q\), yields \(p+q=16\). | 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 calculating the total number of 1000-element subsets of the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\). This is a straightforward calculation, resulting in \\(2^{1000}\\) subsets.",
"Next, we determine the number of subsets whose least elements are \\(1\\) and \\(1000\\). The remain... | The problem requires finding the number of 1000-element subsets with least elements 1 and 1000 such that the remaining elements are chosen from \(\{2,\dots,2014\}\) in such a way that each subset contains at least one 1. Solving the equation \(p = \binom{2014}{2} - \binom{2014}{2013}\) for positive integers yields \(p=... | 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 noting the following:",
"- The sum of the digits is 17.",
"- The number must be divisible by both 17 and 9.",
"1. **Divisibility by 9 (Sum of digits must be a multiple of 9)**:",
"- The sum of the... | The least positive integer \( m \) divisible by \( 17 \) whose digits sum to \( 17 \) is found by firstly verifying that \( m \equiv 0 \pmod{17} \). The sum of the digits of \( 1077 \) is \( 17 \), and the number is divisible by both \( 9 \) (\( \text{sum of digits} \geq 887 \)) and \( 17 \) (i.e., \( m \equiv 0 \pmod{... | 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 \\) that is divisible by 17 whose digits sum to 17, I start by understanding the divisibility rule for 17, which states that a number is divisible by 17 if its digits sum to a multiple of 17.",
"First, I aim to find the smallest number \\( n \\) such that \\( n \\) plu... | The smallest positive integer \( m \) divisible by 17 whose digits sum to 17 is \( m = 17 + 16 \), resulting in 33. This satisfies \( 17 + 33 = 50 \), which is divisible by 17, as required by the divisibility rule for 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 noting that the sum of the digits must be 17, and the number \\( m \\) must start with the smallest digit for a divisibility rule.",
"The smallest possible digit for the first place is 1. Thus, we choose \\( m = 1111... | The smallest positive integer \( m \) divisible by \( 17 \) with a digit sum of \( 17 \) is \( 1111 \). This is because \( 1111 \div 17 \) is approximately \( 62.73 \), which is not an 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 least positive integer \\( m \\) that is divisible by 17 and whose digits sum to 17, I need to identify the smallest number with this property.",
"First, I'll list the smallest two-digit numbers: 17, 16, 15, 14, 13, 12, 11, and so on.",
"Next, I'll calculate the sum of the digits for each two-digit... | The least positive integer \( m \) divisible by 17 whose digits sum to 17 is 14. This is determined by listing the odd two-digit numbers ending in 7 that sum to 17, which are 14. | 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 number \\( m \\) that is divisible by 17 and has digits that sum to 17, I start by setting the tens digit to 17. This gives me the equation \\( 17 + x + y = 17 \\), which simplifies to \\( x + y = 0 \\). The smallest possible value for \\( x \\) and \\( y \\) are both 1, so the number is 171."... | The smallest positive integer \( m \) divisible by 17 whose digits sum to 17 is 171. This is found by setting the tens digit to 17, totaling 0, and then updating the units digit to maintain the sum of 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 has digits summing to 17, I start by looking for the smallest number with these properties.",
"I begin with the smallest possible number, which is 17. The sum of its digits is 17, so I check divisibility by 17. Since 17 divided by 17 is 1, ... | The least positive integer \(m\) divisible by \(17\) whose digits sum to \(17\) is \(17\). This is because the sum of its digits, \(1+7=8\), is not divisble by \(17\) (or any perfect square), and the structure \(17n+0\) does not allow for a complete divisor. | 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 noting the components of the digits sum:",
"\\[",
"17 + 17 + 1 + 1 = 40",
"\\]",
"This sum lies between \\( 40 \\) and \\( 44 \\). Approximating the digits as \\( 4 \\), \\( 4 \\), and \\( 4 \\)... | The smallest positive integer \( m \) divisible by \( 17 \) whose digits sum to \( 17 \) is \( 4 \). This is found by recognizing that digit \( 4 \) is the smallest integer \( m \) such that \( m \equiv 4 \pmod{17} \). | 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 need to consider the smallest possible number between 100 and 999 with a digit sum of 17.",
"First, we check if 179 meets the criteria. The sum of the digits is \\( 1 + 7 + 9 = 17 \\). Since 179 is divisible by 17... | The least positive integer divisible by 17 whose digits sum to 17 is 179. This is because 179 is the smallest number with a sum of digits equal to 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, follow these steps:",
"1. **Identify the Range of Possible Numbers**:",
"- The sum of the digits of a number divisible by 17 must be a multiple of 17.",
"- Calculate multiples of 17 until the maximum sum of digits is reached... | The least positive integer \( m \) divisible by \( 17 \) whose digits sum to \( 17 \) is \( 20 \). \( 20 \) is the smallest number with a digit sum of \( 17 \) 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 smallest number \\( m \\) that consists of the digits 1, 7, and sums up to 17, we need to determine the smallest number composed of these digits that equals 17.",
"First, we list the digits from 1 to 17 and look for the smallest number where the digit sum adds up to 17.",
"- The smallest number sta... | The smallest positive integer \( m \) divisible by \( 17 \) whose digits sum to \( 17 \) is \( 17 \), as \( 17 = 1 \times 17 + 1 \). | 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 whose digits sum to 17, we need to identify the smallest set of digits that adds up to 17.",
"1. **Identify the digits that sum to 17:**",
"- The smallest set of digits that add up to 17 is 1, 7, and 6, which can be formed as 17 in the tens p... | The digits 1, 7, and 6 sum to 17, but 17 itself needs more than one digit, so replace it with 6 to get 169. Since 169 is divisible by 17 with no remainder (169 ÷ 17 = 9), the least such number is \(\boxed{169}\). | 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 \\) that is divisible by 17 and has digits that sum to 17, we can follow these steps:",
"1. **Identify the range of possible digit sums:**",
"- The sum of the digits of \\( m \\) is 17.",
"- Lowest digit sum: 1 (can't be 17)",
"- Midpoint digit sum: 16 (possible ... | The smallest positive integer \( m \) grouped in a way that \( m \equiv 17 \pmod{17} \) by its digit sum condition must satisfy \( m \equiv 16 \pmod{17} \). The smallest multiple of 17 with this property is 100, which is \( 100 \equiv 17 \pmod{17} \). | 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 has digits summing to 17, we start by identifying the possible sets of 3 digits (\\( d_1, d_2, d_3 \\)) whose sum is 17. The valid combinations are:",
"- \\( d_1 + d_2 + d_3 = 17 \\)",
"- \\( 17 - d_1 = 17 - d_2 \\)",
"- \\( 17 - d_1 - ... | The number \( m \) is the smallest positive integer divisible by 17 that has digits that sum to 17, which are \( 3876 \). Thus, \( m = 567 \). | 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 recognizing that the sum of the digits is 17, so \\( m - 17 = 17 \\). Solving for \\( m \\), we get \\( m = 17 + 1 = 18 \\).",
"We then check the sum of digits for numbers ending with 8, such as 18, 28,... | The least positive integer \( m \) divisible by \( 17 \) whose digits sum to \( 17 \) is \( m=18 \) because \( 17+1=18 \). | 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 understanding the divisibility rule for 17. A number is divisible by 17 if and only if the sum of its digits is a multiple of 17.",
"First, I examine the sum of the digits of 17, which is 1. Since the sum ... | The sum of the digits of the number must be a multiple of 17. Adding 1 to 17 gives a digit sum of 18, which is a multiple of 17 (18/17 = 1), confirming \( m=18 \). | 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 with a digit sum of 17, we start by noting that the sum of the digits of 17 is 17 itself. Since we need a number with a digit sum of 17, the number must be divisible by 17.",
"Next, we need to determine the smallest such number. We begin by che... | The least positive integer divisible by 17 whose digits sum to 17 is 5, as 1 has a sum of 1 and is divisible by 17. This is confirmed by checking subsequent numbers like 24 and 38 for digit sums of 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 five towns connected by roads, and there's exactly one road connecting each pair of towns. I need to find the number of ways to make all these roads one-way, meaning from each starting town, you can only go to that town, and the same for the opposite towns. The only restriction i... | Since there are 10 towns each connected to 10 other towns by exactly one road, the total number of handshakes is 2 raised to the power of 10, which equals 10. Therefore, there are 10 ways to route all roads one-way. | 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 there's exactly one road connecting each pair. We need to find the number of ways to make all roads one-way, while keeping the road system still valid for reaching each town.",
"Hmm, so a one-way road means each road's direction changes whe... | The number of undirected graph $G$ formed by all one-way roads from the five towns is $n=5$. Every undirected path between any pair of sizes 1, 1, 2, 2, 2 will intersect uniquely, so such $G$ exists. | 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 about five towns connected by roads. Each town has exactly one road connecting it to every other town. The question is asking for the number of ways to make all these roads one-way, and I remember something about combinations with repetition in combinatorics—specifically the stars an... | The problem can be solved using the stars and bars theorem with n=5 roads and k=5 acceptions for pairing, yielding a total of (5+5-1 choose 5-1) = (8 choose 1) = 120 roads. The total number of ways to form one-way roads with the specified conditions is 1440. | 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. That's three roads between four towns, and then five more roads connecting four of the five towns. The key point is that there's exactly one road between each pair of towns. I need to find the number of ways to make all these roads one-... | Choices of two towns uniquely position each cycle segment to the optimal endpoint (no loop while efficiently traversing every town). The total is **4** valid ways. | 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 there's exactly one road connecting each pair of towns. We need to find the number of ways to make all roads one-way, while still having the original roads function correctly and allow going from any town to any other via those roads. Hmm, so... | The problem can be solved by rerouting roads so that the original directions are thwarted, requiring reversing known direct routes to create zigzag paths. Such paths exist for any number of roads, and by preserving the connection structure, the distance for any cashier path is infinite. | 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 there's exactly one road between each pair of towns. I need to find the number of ways to make all these roads one-way, while keeping every town connected and passing through at least one road. Hmm, interesting. Let me try to wrap my h... | The problem requires finding ways to map roads such that the two roads serve as one-way when barring opposite ends and having two distinct arbitrarily oriented directions. Using the maximum of 2 roads serving as opposing ends yields 2 paths, and with just 2 opposing ends, the number of valid one-way roads is determined... | 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, and there's a system of roads connecting all the towns. Each road is one-way only. I need to find out the number of ways to set this up so that all roads are one-way, and still be able to go from any town to the other towns using those roads. Hmm, interestin... | The number of ways one-way roads can be arranged within a complete graph with non-negative integers $n+1$ towns is the number of orderings of the towns, which is $n(n-1)$. This is derived from the number of path graphs with $\binom{n}{2} = n(n-1)$ vertices, as each road connects a specified number of towns. | 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. So there are five towns connected by roads, and each road connects two towns. There's exactly one road between any pair of towns. But we need to make all roads one-way, while still allowing the return trip. Hmm. So the problem is about graph theory. Maybe using paths or something?",
"First, let ... | The number of ways to use roads of one-way length such that all roads have one-way and there is a cycle inside the graph is $(2 \times 4) = 8$. Thus, there are 8 such cycles. | 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). | [
"To determine the number of ways to connect five towns using exactly one road for each pair, we start by avoiding the potential blocked paths. A blocked path occurs when a road is used twice, which can happen if a town is connected to itself or its two adjacent towns. Using a path to replace one of these blocked pa... | To achieve one-way roads with no blocked paths, we remove blocked paths between non-adjacent towns to form short flat networks. Then, triangular roads with transversal gaps are used to form the roads, ensuring a flow of roads, culminating in a total of 6 roads. | 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 town can be connected to any other town, and there's exactly one road between each pair of towns. The goal is to make all these roads one-way in such a way that it's still possible to get from any town to any other town using the road... | There are infinitely many routes from any town to any other town. The path is valid as long as the pair of towns satisfies the one-way condition. | 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 rule is that roads can be made one-way, which means from any town, you can go to any other town but you can't go back there. The question is asking how many different ways I can make all the ro... | The number of directed paths from any vertex to any other vertex is \(2^{n-1} - 1\). For \(n=5\), this gives \(2^{5-1} - 1 = 31\) directed paths. | 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 about five towns connected by roads. Each town has exactly one road connecting it to every other town. They want to know the number of ways to make all these roads one-way and still be able to go back where you started without having to retrace any roads. Hmm, that sounds a bit like pe... | The number of ways to make the roads one-way and still allow for some repeating routes is 10. Using the formula for permutations with restrictions, P = C(n, 0), where n is the number of remaining settlements after fixing towns' positions, we calculate P = C(5, 0) = 10. | 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 $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"To solve for the modulus squared of the complex number \\( z \\) given that \\( z + |z| = 2 + 8i \\), we start by isolating \\( |z| \\) in the equation.",
"First, rewrite the equation as:",
"\\[",
"z + |z| = 2 + 8i",
"\\]",
"Subtract \\( z \\) from both sides to solve for \\( |z| \\):",
"\\[",
"|z| =... | Given $z + |z| = 2 + 8i$, we found $|z| = \sqrt{\sqrt{2^2 + 8^2}} = \sqrt{\sqrt{68}} = 2\sqrt{68}$. Therefore, $|z|^2 = (2\sqrt{68})^2 = 4 \cdot 68 = 272$. | 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 need to find the modulus of \\( z \\) given the equation \\( z + |z| = 2 + 8i \\). Using the formula for the modulus of a complex number \\( z = a + bi \\), which is \\( |z| = \\sqrt{a^{2} + b^{2}} \\), I substitute \\( a = 2 \\) and \\( b = 8 \\) to get:",
"\\[",
"|z| = \\sqrt{2^{2} + 8^{2}} = \\sqrt... | Given \( z + |z| = 2 + 8i \), we first find \( |z| = \sqrt{2^2 + 8^2} = \sqrt{68} \). Then, \( |z|^2 = |z|^2 = (\sqrt{68})^2 = 68 \). | 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 $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"First, we start with the given equation:",
"\\[ z + |z| = 2 + 8i \\]",
"We need to find \\( |z|^2 \\). To do this, we can use the property of complex numbers:",
"\\[ |z + z_0| = |z + a| \\]",
"where \\( z_0 = 2 + 8i \\). Therefore,",
"\\[ |z| = \\sqrt{2^2 + 8^2} = \\sqrt{4 + 64} = \\sqrt{68} \\]",
"Nex... | Given \( z + |z| = 2 + 8i \) and the property \( |z + z_0| = |z| \), we find \( |z| = \sqrt{2^2 + 8^2} = \sqrt{68} \). Squaring this modulus, \( |z|^2 = (\sqrt{68})^2 = 68 \). | 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 $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"To find \\(|z|^{2}\\), which is the square of the modulus of \\(z\\), I start with the given equation \\(z + |z| = 2 + 8i\\).",
"First, I isolate \\(|z|\\) by subtracting \\(z\\) from both sides:",
"\\[",
"|z| = \\sqrt{2^2 + 8^2} = \\sqrt{4 + 64} = \\sqrt{68}",
"\\]",
"Next, I square the modulus of \\(|z... | Given \( z + |z| = 2 + 8i \), we first solve for \( |z| \) to get \( |z| = \sqrt{2^2 + 8^2} = \sqrt{68} \). Then, \( |z|^2 = |z|^2 = (\sqrt{68})^2 = 68 \). | 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 $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"First, let \\( z = a + bi \\). Then, we can write \\( |z| \\) as \\( \\sqrt{a^{2} + b^{2}} \\).",
"Substituting into the equation \\( z + |z| = 2 + 8i \\), we get:",
"\\[",
"(a + bi) + \\sqrt{ab + 4} = 2 + 8i",
"\\]",
"Subtracting \\( (a + bi) \\) and \\( 2 \\) from both sides, we obtain:",
"\\[",
"\... | Since \(z=a+bi\) with \((a+bi)+2=-2\) gives \(ab=0\), it follows that \(z=0\), thus \(|z|^2=0\) after squaring and taking the square root. The answer is \(\boxed{\textbf{(C)}\ 169}\). | 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 determine \\( |z|^2 \\), we start with the given equation:",
"\\[ z + |z| = 2 + 8i \\]",
"Substituting \\( z = a + bi \\) from the complex number form:",
"\\[ a + bi + |z| = 2 + 8i \\]",
"This can be rewritten as:",
"\\[ (a + bi) + (|z| + 2)i = 2 + 8i \\]",
"Equating the real and imaginary parts:",
... | Given \( z + |z| = 2 + 8i \), we substitute \( z = a + bi \), leading to \( (a + bi) + (|z| + 2)i = 2 + 8i \), which simplifies to \( a + bi = 2 \) and \( bi = 8 \) so \( b=8 \). Thus, \( |z| = \sqrt{a^2 + 8^2} = \sqrt{a^2 + 64} = a^2 + 64 \), so \( |z|^2 = (a^2 + 64)^2 = a^2 + 64^2 = 72 \). | 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 equation \\( z + |z| = 2 + 8i \\) implies that \\( z \\) and its conjugate \\( \\overline{z} \\) add up to 2 + 8i. Therefore, we can write:",
"\\[",
"z + \\overline{z} = (a + bi) + (a... | From \(z + \overline{z} = 2 + 8i\), we get \(z = \overline{z} = 8i\), so \(|z| = \sqrt{0^2 + 8^2} = 8\). Thus \(|z|^2 = 8^2 = \boxed{64}\). | 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 the given equation \\( z + |z| = 2 + 8i \\) involves a complex number \\( z \\) and its modulus \\( |z| \\).",
"To find \\( |z| \\), I can subtract \\( z \\) from the left side of the equation:",
"\\[ |z| = 2 + 8i - z \\]",
"Simplifying this, I get:",
"\\[ |z| = 2 - z \\]",
"Next,... | The given equation \( z + |z| = 2 + 8i \) simplifies to \( |z| = 2 - z \). Therefore, \( |z|^2 = 2 - z^2 \), and since \( z^2 = 2 - z^2 \), \( |z|^2 = 4 - 2 = 2 \). | 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: If z satisfies z + |z| = 2 + 8i, what is |z|²? The options are A) 68, B) 100, C) 169, D) 208, E) 289. Hmm, let me see how to approach this.",
"First, I remember that if z + |z| = 2 + 8i, maybe I can find |z| first. I think the modulus of a complex number a + bi is the square ro... | The complex number $z = 10i$ has a modulus $|z| = \sqrt{10^2} = 10$. Therefore, $|z|^2 = |z|^2 = \sqrt{10^2} = 10$. | 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, complex numbers can be a bit tricky, but let me see if I can figure this out step by step.",
"First, I remember that in complex numbers, any complex number z = a + bi can be written in the form z = a + bi,... | The modulus squared of the complex number \(z\) is 0, corresponding to option D. | 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: if z equals a plus bi, then z plus the modulus of z equals 2 plus 8i. I need to find the modulus squared of z, which is written as |z|². The options are from A to E, with different numbers. Hmm, let me think through this step by step.",
"First, I remember that for any complex n... | We found a modified complex number \(z = a + bi\) such that \(z + |z| = 2 + 8i\). The modulus squared of \(z = a + bi\) is \(|z|^2\) which simplifies to \(68 - 4a - 16b\), corresponding to option (A). | 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. It says that if z plus the modulus of z equals 2 plus 8i, then what is the square of the modulus of z, which is written as |z|². The answer choices are A) 68, B) 100, C) 169, D) 208, and E) 289.",
"Alright, let me start by recalling what the modulus of a complex ... | The modulus of a complex number \( z = a + bi \) is \( |z| = \sqrt{a^2 + b^2} \). Solving \( z + |z| = 2 + 8i \) for \( |z| \) gives \( |z| = 2 - 8i \), so \( |z|^2 = (2 - 8i)^2 = \sqrt{2^2 + 8^2} = 8.236 \). | 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 \\]",
"First, we isolate \\( |z| \\):",
"\\[ |z| = 2 + 8i - z \\]",
"\\[ |z| = 2 - 8 + 8i \\]",
"\\[ |z| = 2 + 8i \\]",
"Next, we square \\( |z| \\):",
"\\[ |z|^2 = (2 + 8i)^2 \\]",
"\\[ |z|^2 = 2^2 + 2 \\cdot 2 \\cdot ... | Given \( z + |z| = 2 + 8i \), we find \( |z| = 2 + 8i \). Squaring this real and imaginary part, \( |z|^2 = (2 + 8i)^2 = 6 - 8 \), which simplifies to \(\boxed{\textbf{(A)}\ 68}\). | 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. The equation given is z + |z| = 2 + 8i. I need to find |z|². Hmm, complex numbers, right? Let me recall how they work.",
"A complex number z can be written as a + bi, where a is the real part and b is the imaginary part. The modulus |z| is the distance ... | Given $z + |z| = 2 + 8i$, we first express this as $(a+bi) + \sqrt{a^2+b^2}$, which quickly becomes $a^2 + 2a + b^2 - (a+b)^2 - 2ab$. Simplifying further yields $2a^2b^2$. | 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:",
"$$",
"z + |z| = 2 + 8i",
"$$",
"We solve for $ z $:",
"$$",
"z = |z| - 2 + ai",
"$$",
"$$",
"|z| = \\sqrt{2^2 + 8^2} = \\sqrt{4 + 64} = \\sqrt{68}",
"$$",
"$$",
"z = \\sqrt{68} - 2ai",
"$$",
"Computing $|z|^2$:",
"$$",
"|z|^2 = (\\sqrt{68})^2 - 2... | Given $z + |z| = 2 + 8i$, we find $|z| = \sqrt{2^2 + 8^2} = \sqrt{68}$. Therefore, $|z|^2 = (\sqrt{68})^2 - 2^2 = 68 - 4 = 64$. | 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... | [
"First, I need to determine the total number of ways to paint the six sections of the ring without any restrictions. Since each section can be painted with one of four colors and no two adjacent sections can share the same color, I will use the combination formula to calculate the number of possible pairings.",
"... | The total number of ways to paint the six sections is $\binom{4}{2} = 6$. The number of ways to paint adjacent sections with the same color is 4, so the number of ways to choose not to paint adjacent colors is $6 - 4 = 2$. | 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, I need to consider the constraints that no two adjacent sections can be the same color. This means that the sections surrounding each other must be unique within the ring.",
"First, I'll list the four available colors and consider the possibl... | There are 6 ways to choose to paint the sections, which can be approached using a multiset optimization. Dividing the 6 sections into four distinct color classes and one multiset yields 6 ways to paint them, calculated as \(\binom{6}{5} = 6\). | 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 small sections of a ring with four distinct paint colors such that no two adjacent sections can be the same color, we can use a systematic approach.",
"1. **Total Number of Ways Without Restriction:**",
"- Each section can independently be col... | The problem requires finding the number of ways to choose five sections such that no consecutive sections with the same color can be adjacent, which is equivalent to choosing how many distinct five-section subsets can be formed from six sections. Using combinations, this is $\binom{6}{5} = 6$, and the total number of w... | 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... | [
"First, I need to determine the total number of sections and the available color options. There are six sections, and there are four colors available, each suitable for a different section.",
"Next, I should consider the constraint that no two adjacent sections can be painted with the same color. This means that ... | The 6 sections can be painted uniquely in \(4^6=6561\) ways, and the 4 adjacent sections with different colors give 4 combinations. Summing these possibilities gives a total of \(4 + 4 + 4 = 12\) 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... | [
"First, we need to paint each section with a different color, making a ring with six sections, where no two adjacent sections can share the same color. We can choose any paint color for the middle circle (Section 2) since adjacent sections cannot share colors.",
"Next, we consider the nine options for the surroun... | The middle section can be painted with any of the 4 available colors. The remaining 5 adjacent sections must have different colors, resulting in \(4 \times 5 = 20\) total 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... | [
"First, I need to determine the total number of sections in the ring, which is 6.",
"Next, I'll consider the constraints of not being able to paint adjacent sections using the same color. This means I can paint the first and third sections separately, the second and fourth sections, and the fifth and sixth sectio... | There are $2 \times 2 = 4$ color choices for each of the 6 sections, and adjacent sections cannot share a color. Therefore, the total number of ways to paint all six sections is $4 \times 4 \times 4 \times 4 \times 4 \times 4 = 2048$. | 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... | [
"First, I need to determine the total number of ways to paint all six sections of the ring. Since there are four colors available, each section can either be painted a different color or the same color. This gives a total of 2^6 = 64 possible combinations of colors. However, I need to ensure that no two adjacent se... | The optimal arrangement of six sections across seven circles, with each circle ending at its start, results in 6 distinct colors. The sequence of colors is A, B, C, D, E, F, summing to 6 colors total with no adjacent sections painted the same. | 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... | [
"First, identify the six sections on the wall, each with a distinct width and height.",
"Next, consider the possible orientations of the ring, which can be either horizontal or vertical. This informs the choice of colors for each section, ensuring that no two adjacent sections share the same color.",
"For each ... | The number of ways to paint the ring is calculated by summing the products of each ring's width by its share of adjacent colors. This results in a total of 660 ways to paint the ring. | 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... | [
"First, I need to determine the total number of ways to paint the six sections of the ring using just four colors, ensuring that no two adjacent sections share the same color.",
"I'll start by considering the placement of different sections on the circle. The sections can be arranged in a 2x2 grid pattern to form... | The ring is arranged as a 2x2 grid of sections, and adjacent sections can't share a color. We calculate the combinations for each face separately by choosing 2 out of the 6 sections (6 ways), then multiply by the number of ways to color a face independently (1024 ways), resulting in a total of 61,440 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 this problem, we need to determine the number of ways to paint six sections of a ring using four available color options. The section has its center, and the colors alternate around the ring. We are also prohibited from painting adjacent sections with the same color.",
"First, we consider the smallest s... | The ring is divided into 4 equal sections colored alternately, with a color of 3 being shared by both (3,3) and (3,1). The total number of ways to paint the six sections is 25 because of color constraints. | 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... | [
"First, I need to determine the number of ways to paint the six sections of the ring, which are divided into two groups of four sections each. Since each section must be a different color, I can subtract one additional color from the total of four to account for the unused color.",
"Starting with 4 colors, I subt... | Subtracting one color from the initial four available (CW or CCW) reduces the possible first colors from 4 to 3, resulting in 3 × 3 × 2 = 18 distinct ways to paint the ring. The answer is 18. | 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... | [
"First, I need to determine the total number of ways to paint the six sections of the ring without any restrictions. Each section has four possible colors, so the total number of combinations is 4 raised to the power of 6, which is 4096.",
"However, there is a restriction: no two adjacent sections can be painted ... | The total number of ways to paint the six sections is $4^6 = 4096$, and the number of invalid combinations due to adjacent color restrictions is $4 \times 4 = 20$. Subtracting the invalid combinations from the total gives $4096 - 20 = 4076$ 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 sections of the ring shown, we need to consider two main cases: painting the outside sections first and painting the inside sections first.",
"**Case 1:** The outside sections are painted first.",
"- There are 6 sections in total.",
"- Each sect... | By the pigeonhole principle, at least two cases must be considered: first paint the outside sections without coloring adjacent sections (1 instance) and then paint inside sections without coloring adjacent sections (1 instance). Thus, the total number of ways is 1 + 1 = 2. | 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... | [
"First, I need to determine the total number of possible ways to paint all six sections with no two adjacent sections sharing the same color. To do this, I'll consider the available four paint colors and how they can be distributed among the six sections.",
"I'll start by considering the placement of color 1, sin... | The problem involves distributing four paint colors among six sections such that no two adjacent sections share the same color. The total number of ways to paint the ring is 39. | 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... | [
"First, I need to determine the total number of sections in the ring, which is 6.",
"Next, I note that each section can be painted in 4 different colors.",
"To find the total number of ways to paint the sections without any restrictions, I multiply the number of options for each section: 6 sections × 4 colors =... | The total number of ways to paint 6 sections with 4 colors is \(6 \times 4 = 24\) ways. Subtracting the 4 ways where adjacent sections are painted with the same color results in 20 ways to paint the sections without the restriction. | 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... | [
"First, I need to determine the total number of ways to paint the six sections of the ring without any restrictions. This can be calculated using combinations, specifically C(6,4) to choose the colors and C(12,4) to choose the sections.",
"Next, I must consider the restriction that no two adjacent sections can be... | The total number of ways to paint the six sections without restrictions is \(\binom{6}{4} \cdot \binom{12}{4} = 126 \cdot 306 = 37560\). Subtracting the \(\binom{6}{4} = 15\) ways where adjacent sections share a color results in 37554 valid ways. | 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 \\times 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers, and verify that \\( a^6 \\) is not a divisor of \\( 6^a \\), we can follow these steps:",
"1. **Factorize 6**: The prime factorization of 6 is \\( 2 \\times 3 \\).",
"2. **Express ... | The problem can be solved by expressing \( a = 2^n \times 3^m \) with non-negative integer \( n \) and \( m \). For each non-GCF pair, the sum of the corresponding \( a \) is 2 + 1 + 2 = 5. | 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 \\times 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers such that \\( a^6 \\) is not a divisor of \\( 6^a \\), we can follow these steps:",
"1. **Factorize 6 into its prime factors**:",
"\\[",
"6 = 2 \\times 3",
"\\]",
"2. **Expres... | The conditions \(6^a \equiv a^2 \pmod{2^n} \Rightarrow n \) must be even (\(n = \pm 2\)) and \(6^a \equiv a \pmod{3^m} \Rightarrow m = 2\) or \(m = 1\), giving \(a^6 = 2^6 \cdot 3^6 = 47944 \neq 6^a = 156792\). Therefore, the sum is \(47944 + 156792 = \boxed{63627}\). | 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, 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers such that \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"To simplify the problem, I'll express any power of \\( a \\) as a power of 2 or a power of 3. This means \\( a \\) can be written as:"... | The condition \( a^6 \not\mid 6^a \) is satisfied when \( a^2=4 \) and \( a=20 \), where \( 20-2=18 \). The sum of all such positive integers \( a = 2^n, 3^m \) is \( 20 + 1 = 18 \). | 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 \\times 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers such that \\( a^6 \\) is not a divisor of \\( 6^a \\), we can follow these steps:",
"1. **Factorize 6 into its prime components:**",
"\\[",
"6 = 2 \\times 3",
"\\]",
"2. **Exp... | The condition \( a^6 \not\equiv 1 \mod 6 \) implies \( a \equiv 1, 3 \pmod{6} \), and squares of odd \( a \) are their positive counterparts. Thus, \( (3 + a)^2 \) must be odd, leading to the sum of this quantity as \( \boxed{272} \). | 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 sum of all positive integers \\( a = 2^n \\, 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers such that \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"1. **Identify the Divisors of \\( 6^a \\)**:",
"- We want to find the divisors of \\( 6^a \\) suc... | The sum of positive integers \( a = 2^n \, 3^m \) where \( n \) and \( m \) are non-negative integers such that \( a^6 \) is not a divisor of \( 6^a \) is found by excluding powers of 2 and 3 from 6. The only remaining valid values are \( a = 2 \). | 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 this problem, we start by expressing \\( a \\) in terms of its prime factors:",
"\\[",
"a = 2^n \\cdot 3^m",
"\\]",
"Next, we write \\( 6 \\) as a power of 2 and 3:",
"\\[",
"6 = 2 \\cdot 3^1",
"\\]",
"We need \\( a^6 \\) to not be divisible by any power of 6 within the base 6 numbers:",
... | The problem requires finding positive integers \( a = 2^n \cdot 3^m \) such that \( a^6 \) is not divisible by 6, meaning \( a \) must be even. The possible pairs for \( a \) are \( (2, 2) \), \( (2, 4) \), \( (2, 6) \), \( (4, 2) \), and \( (4, 4) \), of which only \( a = 2^1 \) and \( a = 24 \) result in a sum greate... | 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 \\).",
"To do this, I'll express \\( a^6 \\) and \\( 6^a \\) in terms of their prime factors. Since \\( 6^a = 6 \\times (2 + \\frac{3}{2... | The prime factorization of $a^6 = 2^6 \times 2^6$ shows that $a^6$ is not a divisor of $6^a$ because $6 \) could not divide $8$. Since $2^6$ is not divisible by 3, it is impossible for $a^6$ to be divisible by 6, meaning the sum of positive integers $a = 2^n \cdot 3^m$ for which $a^6 \ngivot 6^a$ is 0. | 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 \\times 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers such that \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"First, we consider the prime factorization of \\( 6^a \\):",
"\\[",
"6^a = (2 \\times 3)^a = 2^{a} \\times ... | The valid exponents for \( a = 2^n \times 3^m \) such that \( a^6 \mid 6^a \) are \( n=3 \) and \( m=2 \), resulting in \( 5 \) and \( m=7 \), yielding \( 5+7=12 \). | 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, I need to find all possible positive integers \\( a = 2^n \\cdot 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers that satisfy the condition that \\( a^6 \\) is not divisible by \\( 6^a \\).",
"First, I'll consider the prime factorization of 6, which is \\( 2 \\times 3 \\). ... | The possible values for \( a \) are \( 0, 1, 3, 6 \), since \( 0 \le m \le \log_2 2 < 3 \) for divisibility by 10, and any \( a=3^k \) makes \( a^6 \equiv 1 \pmod 2 \). The sum \( 10 \). | 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 solve the problem, we need to find the set of positive integers \\( a = 2^n \\) and \\( 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers such that \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"1. **Understanding the Problem:**",
"- We need to find integers \\( a \\) for which \\( a^6 \\) is... | The condition that \( a^6 \) is not divisible by \( 6^a \) requires \( a^6 \) to be of the form \( 6^i \) for some \( i > 1 \) among \( 6 \) and \( 1296 \). For \( n \ge 3 \) with \( m \ge 3 \) and \( a = 6^i \), the only possible exponents are \( a_1 = 6, a_2 = 36, a_3 = 216 \), which sum to \( 258 \). | 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 \\times 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers such that \\( a^6 \\) is not a divisor of \\(6^a \\), we start by solving the divisibility condition for \\(6^a\\).",
"First, consider the factorization of \\(6^a\\):",
"\\[",
"6^a = 2... | The condition for \(6^a \neq 0\) requires \(3^{n+m} > 3^a = \frac{3^a}{2}\). Testing values, \(2\leq a \leq 10\) yields valid conditions when \(a=2\) or \(a=1\), and the latter is explicitly provided as \(a=1\). | 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 find the sum of all positive integers \\( a = 2^n \\times 3^m \\) where \\( n \\) and \\( m \\) are non-negative integers, such that \\( a^6 \\) is not a divisor of \\( 6^a \\), we start by recognizing that \\( 6^a \\) can be expressed as \\( 6^a = (2^2)^a \\). This simplifies to \\( 2^{2a} \\).",
"Next, we n... | For \( a^6 \) not to be a divisor of \( 6^a \), the exponents of its prime factors must be even and distinct. The valid values of \( a \) are \( 6, 16, 36, 66, 108, 297 \), and their sum is \( 795 \). | 42 | false |
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