source stringclasses 7
values | question stringlengths 3 1.02k | steps listlengths 1 168 | answer stringlengths 0 2.93k | 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 length a 2-digit integer a base 10. I'm supposed to find the length of AB, which is an 2-digit integer that has a 2-digit perpendicular stroke going from H to O. It's also asked that the distance from their intersection point H to the center O is a positive rational... | The length of AB is equal to the length of the perpendicular chord CD, which is 20. This length is 20, and including the 2-digit 2-digit integer a2, it's guaranteed to be 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 $AB$. | [
"To determine the length of \\( AB \\), we start by noting that the diameter \\( a \\) of a circle is a 2-digit integer \\( 10 \\times k + 2 \\) for some integer \\( k \\).",
"The perpendicular chord \\( CD \\) between points \\( A (length \\( a \\)) \\) and \\( B (length \\( a + 2 \\)) of the circle is given by ... | The length of the perpendicular chord \( CD \) is \( \lambda = 2k \) for the 2-digit integer \( k \). The diameter \( a \) of the circle equals \( 10k + 2 \), and thus \( k = 18 \), thus the length of \( AB \) is \( a \cdot k = 2 \cdot 18 = 36 \). | 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 length a 2-digit number. The question says that the length of the perpendicular chord CD is a positive rational number. The point H is at point H's intersection with the center O in the circle, which means it's also a line. I need to find the length of AB.",
... | The length of the perpendicular chord CD is the distance from H to O, i.e., an integer. Thus AB = CD = 4, since the circumference O equals CD. | 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 perpendicular chord \\( CD \\) in the circle with diameter \\( AB \\) in base ten. In a circle of diameter is 2 lines, there are 2 positive integers representing the diameters, one more than the diameter, so there are 4 lines.",
"Using the division method, I'll divide... | The diameter \( AB \) is \( 2 \) units, making \( AC = 0.5 \) units. The perpendicular chord \( CD \) is $$ \frac{1}{2} \) or \( 2 \) units, and solving \( AB^2 + CD^2 = 4 \) yields \( AB = 4 \) base ten 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's see. The problem is about finding the length of span AB, the diagonal BC, and their perpendicular distance. Hmm, I remember something about the Pythagorean theorem in geometry, but wait, span AB is a base distance from H to O. So their length can be found in a right triangle bisector theorem.",
"Firs... | The Pythagorean theorem for a 30-60-90 triangle gives the ratio \(a/2\), so \(AB = \sqrt{AB^2 + \text{BC}^2 + AB^2}\). | 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 $AB$. | [
"To solve the problem, we start by understanding the structure of a circle with a diameter of \\( a \\)-digit length \\( a \\)-digit integer (base ten). When reversing the digits of a circle, each digit represents a power of 10, where the digit \\( a-1\\) is on the rightmost side and \\( a \\) is on the left.",
"... | The length of the perpendicular chord \( AB \) between the point \( H \) and the center \( O \) of a circle of diameter \( a \)-digit length \( a \)-digit length \( a \)-digit position is given by \( AC = 2\pi \), where \( AC \geq 100 \) leading to a maximum distance of 2\pi radians. This distance is calculated using \... | 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 end lengths a 2-digit integers in base 10 that form a 2-digit perpendicular chord AB. The perpendicular chord has a distance of a 2-digit number from the center point H, and the problem says that such an end-lit address is acceptable. We need to... | The perimeter $ab$ of the circle is 2. This perimeter is equal to the sum of the two digits $a+1$, which is 1 +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 $AB$. | [
"Alright, so I have this problem here about a circle with a diameter a 2-digit integer. I need to find the length of the diameter a, the cross product CD, and the length of AB. The problem mentions the distance between their intersection point H and the center O, which is a positive rational number. Hmm, okay, let ... | The diagonals of a circle have lengths 12 or 10. In this case, \( AB = (12 + 1) - (12 - 1) = 10 \), so the length of diameter \( AB \) is \( \boxed{-10} \). | 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 $AB$. | [
"To determine the length of side \\( AB \\), we start by letting the side lengths be \\( a \\) and \\( 2a \\). Using the Pythagorean theorem, we set up the equation \\( a^2 + 2a^2 = 2ab \\) to find the integer \\( a \\).",
"Next, we square both sides to eliminate the square:",
"\\[",
"(a + 2)^2 = 2ab^2",
"\... | Setting \(a = b\) from \(\angle A = \sqrt{a^2 - 10}\), and substituting \((1)^2 + 4(1) + 10 = 24\), the side length \(AB\) is \(24\). | 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 distance from the intersection point \\( H \\) to the center \\( O \\) as \\( d \\). The perpendicular chord \\( CD \\) can be recognized as a straight line on the circle.",
"In a circle where the radius \\( r \\) is \\( 2 \\), the diameter \\( AB \\) would be half of the circumference of the ci... | Since \( AB = CD \), we have \( AB = CD - d \). Since \( 2-d \) has no rational divisors, their least common multiple is 2, making \( CD = 1 \) and thus \( AB = CD = 1 \). | 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 determine the length of \\( AB \\), I start by converting the 2-digit length in diameter into a base ten number. The diameter is \\( 2 \\times 10^2 = 200 \\), so \\( a \\) could be expressed as \\( 200 \\times 10^2 - 20^2 = 20000 - 400 = 19960 \\).",
"Next, I calculate the length of \\( CD \\). For the perpen... | The diameter \( a=200 \times 10^2 = 19960 \) can be expressed as \( 19960 \times (10^2 - 1) = 19960(10^2 - 1) \), leading to \( CD = 19960(10^2 - 1)(10^2 - 2^2) = \frac{363{,}600\ldots}{3.1416} \approx 1000.574 \). The perpendicular chord length \( ABC \) is approximately the diagonal length \( DE = AC = 5 \times CE \)... | 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 identifying the points \\( H \\) and \\( O \\) such that \\( 20000 - AB \\) is the vertical distance from the center \\( O_c \\) to the centered line \\( AB \\).",
"A vertical distance of \\( 20000 - 2000 \\) units corresponds to the digits in \\( AB \\), which ... | The length of \( AB \) is 20000-2000 = 20001-21 units, and the perimeter is \( 20000-2000+20001 = 20000-21 = 20000-22 \). Thus, the length of \( AB \) is \(\boxed{20001-22}\). | 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 $p + q$. | [
"Okay, so I have this problem where there's a set, which is all 1000-element subsets of {1, 2, ..., 2015}. I need to find the arithmetic mean of all the least elements in this set. The problem says that there are 4096 elements in the world, and I need to find the arithmetic mean of 4096 elements, inclusive. But wai... | The arithmetic mean is calculated as the sum of the smallest 5999 elements in the set \(\{n, 2n, \ldots, 2015\}\) factorial, which is \(\binom{1000}{5} = 5999\). The sum of the least elements, which are from the set's composition \(\binom{4096}{5}\), is 100. | 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 determine the smallest element in a 1000-element subset of {1, 2, ..., 2015}. The smallest occurrence of this least element is at position 1, where it is 1. Therefore, we need to find all 1000-element subsets from which this smallest element, which is 1, has to be selected.",
"Th... | One can eliminate the smallest element (1 from 1000 subsets of order 1000) while more elements can be chosen from the remaining subsets (1000 subsets of order 1000), giving sum S = 100,001,000. | 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 each 1000-element subset contains three elements: 1, 2, 3, ..., 2014. To find the minimum value of \\( p \\), I use the addition Inequality:",
"\\[",
"3 \\leq 1 + 2 + 3 + \\dots + 2014 \\leq 2015",
"\\]",
"Simplifying this, I get:",
"\\[",
"1 \\leq \\dots \\leq 2000",
"\\]",
... | The minimum value of \( p \) is taken by considering the additive logarithm covering the five non-negative integers from \( 1, 2, \dots, 2000 \). The minimum \( f \) is calculated to be \( 1 - 0.655 \approx 0.345 \), making \( p \) \( 4 \). | 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 need to find the least common multiple (LCM) of the numbers 1, 2, 3, ..., 2015. To do this, I'll start by determining the prime factors of each number.",
"- The prime factorization of 1 is simply 1.",
"- For 2, it is \\(2^1\\).",
"- The prime factorization of 3 is \\(3\\).",
"- The prime factoriza... | The length of the smallest complete set is 100 elements. The LCM of 1, 2, 3, ..., 2015 is 252, leading to the smallest total for the least elements being 5, the sum being \( 252 + 5 = 257 \). | 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 the least elements from each of the 1000-element subsets of the set \\(\\{1, 2, 3, \\dots, 2015\\}\\), we need to consider how each subset chooses its least element.",
"1. **Total Number of Subsets**: The total number of subsets of degree 1000 is calculated as:",
"\\[",
"\\bino... | The total number of subsets is $\binom{2015}{1000} = 1,026,075$. The arithmetic mean of the least elements in each subset is $\frac{p+q}{2} = \frac{1}{0} = 0$, which means we get $p=q=1026$. | 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 \\( p \\) of the least elements from each of all the subsets of the set \\(\\{1, 2, \\dots, 2015\\}\\), we need to consider only the subsets that have at least two elements and any number less than 2015.",
"1. **Total Suits**: The total number of subsets \\( n \\) when using 1000 elem... | The arithmetic mean of all subsets of \(\{1, 2, \dots, 2015\}\) is 1, calculated by considering subsets of 3 elements and \(H=\{1\}\), resulting in \((3\times2)+1\). Thus, \(p+q=3\). | 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 $p + q$. | [
"First, I need to determine the least common multiple (LCM) of 1 through 99 with the numbers 2 through 2014. Since both 1 and 2 are multiples of 2, the LCM is simply their product:",
"\\[",
"\\text{LCM}(1, 2, 2014) = 2^{41}",
"\\] The LCM is 2^{41}.",
"Next, I'll find the largest power of 2 that is less tha... | The LCM of 1, 2, ..., 2014 is 2^{44}, so \(p+q = (2^5)(2^42) = 105\). | 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 from each 1000-element subset that include the least element from each subset, we can follow these steps:",
"1. **Understand the Repeating Breakdown**:",
"- Each subset \\( n \\times (n+1) \\times (n+2) \\) has 1000 elements starting from 1.",
"- The... | The sum of the arithmetic mean of the least elements from each 1000-element subset is $1000 \times 10^{10}$, which equals $1000,000$. Thus, $p + q = 1000,000$. | 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 selected from each of the 1000-element subsets of the set {1, 2, ..., 2015}, we can follow these steps:",
"1. **Determine the Number of Subsets:**",
"- Each subset contains 2 elements, so the total number of subsets is \\(2^10 = 1024\\).",
"- Since we are... | The sum of the least elements, which are the smallest mentoring filters about the first 2015 elements of the set, is \(2r + \sqrt{r}\), where \(1 < r < 2015\). This expression simplifies to \(\sqrt{2015}\), leading to an arithmetic mean of \(1 + \sqrt{2015}\) and thus \(p + q = 7\). | 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 need to find the least common multiple (LCM) of the numbers 1, 2, 6, 7, 8, 9. The prime factors of each number are 2, 3, and 5. Since these are not coprime, their LCM is the product of the highest powers of all prime factors present in any multiple of the original number.",
"Calculating the LCM:",
"- ... | The solution iteratively computes the LCM (the largest even number among the 1000-element subsets), yielding three sets of 448. The sum of these five elements is \(119.25\), and \(p + q = 560\). | 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 all subsets of the set {1, 2, ..., 2015}. The question is asking me to find the arithmetic mean of the least elements from each of these subsets. The unique elements within each subset are all from {1, 2, ..., 2015}, and each subset has 1000 elements. Addit... | Since each subset has 2015 elements and the smallest element is 1, the arithmetic mean is 2, p = 12 and q = 4, giving p = 12 + 4 = 16 and q = 4. Therefore, p + q = 22. | 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 the least elements in each 1000-element subset chosen from the set \\(\\{1, 2, 3, \\ldots, 2015\\}\\), we can follow these steps:",
"1. **Calculate Total Elements in the Subset:**",
"\\[",
"2015 \\times 1000 = 2015000",
"\\]",
"2. **Find the Least Element in Each Subse... | The arithmetic mean of the least elements in each 1000-element subset chosen from $\{1, 2, 3, \ldots, 2015\}$ is $\frac{p}{2q}$, where $p$ and $q$ are relatively prime integers. This is found by calculating the smallest element in each subset, determining the mean for each subset, and then summing the means: $\frac{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 $p + q$. | [
"To find the arithmetic mean of the least elements from each 1000-element subset, we can use the formula for the arithmetic mean of two numbers, which is the sum of the numbers divided by two.",
"First, calculate the total sum of the elements. For 1000 elements, the sum is:",
"\\[",
"\\text{Sum} = \\frac{2015... | The sum of the smallest 1000-element subsets is $20.16$, which means the arithmetic mean of the least elements is approximately $20.16$. Since \(2015$ elements are at least three numbers of a single digit, each number of a subset must contribute to at most two possible values of the least sum. | 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 a digit sum of 17, we start by noting the digits sum to 17:",
"\\[",
"x + y = 17",
"\\]",
"This implies that \\( x + y = 16 \\).",
"Next, we list all mutually exclusive sets of digits that multiply to 16:",
"- \\( x = 2 \\) (b... | 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 \\) whose digits add up to 17, I start by examining smaller numbers.",
"First, I list the numbers in order from smallest to largest:",
"- 1",
"- 3",
"- 5",
"- 7",
"- 9",
"- 11",
"- 13",
"- 17 (this is the last digit in \\( m \\))",
"Next, I check each nu... | The digit 3, placed before the first digit of 17, is the least significant unit, making 3 the leading digit of \( m \). Therefore, the least positive integer with digits summing to 17 is \( 3 \). | 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 where the sum of its digits is 17, I start by trying many integers that have a digit sum of 17.",
"First, I check the smallest possible digits to see which lead to a sum of 17. The smallest positive digit is 1, but 1 + 1 = 2, which is not 17.",
"Next... | The least positive integer \( m \) that is divisible by \( 17 \) with a sum of digits of 17 is \( 16 \). This is determined by finding factors of 16 and using the digits of 16 such that their sum is 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 \\) such that it is divisible by 17 with a digit sum of 17, I start by examining the smallest possible single-digit number, which is 2. Adding this to 17 gives 19, and 19 is not divisible by 17.",
"Next, I increase 2 by 5 to 7, resulting in 17. Adding this to 19 gives 26,... | The least positive integer \( m \) that is divisible by \( 17 \) with a digit sum of 17 is 37. This is because \( 37 = 3 \times 11 \div 17 \), and \( 3 \times 11 \div 17 = 37 \). | 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 $m$. | [
"First, I need to find the least positive integer \\( m \\) that is divisible by 17. To do this, I'll start by identifying the smallest three-digit number that is divisible by 17. The smallest three-digit number is 100.",
"Next, I'll divide 100 by 17 to see how many times 17 fits into 100. The result is approxima... | The least positive integer \( m \) divisible by \( 17 \) where the digit sum is \( 17 \) is found by first ensuring the smallest three-digit number is divisible by \( 17 = 17 \). This means \( m \) must be \( 17 \), as \( 17 \div 17 = 1 \), and \( 18 \pm 8 \) are coprime, leading to \( m = 17 + 19 = 37 \). | 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 digits summing to 17, I start by recognizing that 17 is exactly 1 more than 14, which is divisible by 14.",
"Since \\( m \\) must be divisible by 17, it must also be divisible by 14. Specifically, \\( m \\) should be a multiple of both 14 ... | The least positive integer \( m \) divisible by 17 with digits summing to 17 is 17. This is because \( m = 17 \) is divisible by both 14 (divisible by 14) and 17, as \( m \) is the only number between 14 and 17 that satisfies both conditions. | 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 digits summing to 17, I start by listing all combinations of digits from 0 to 16 that sum to 17. These combinations are:",
"\\[",
"07, 09, 17, 19, 27, 29, 37, 39, 47, 49, 57, 59, 67, 69, 77, 79, 87, 89, 97, 99, 106, 116, 126, 137, 146, 1... | The least positive integer $m$ divisible by $17$ with digits summing to $17$ is $1799$. This is the only number where the combination in the required two-digit number, $090$, results in a result of $-99$. | 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 need to find the smallest \\( m \\) where the sum of its digits equals 17.",
"First, we list all possible digits combinations and their sums:",
"- 1",
"- 2",
"- 3",
"- 4",
"- 5",
"- 6",
"- 7",
"- 8",
... | The least positive integer \( m \) divisible by 17 with a digit sum of 17 is 17, as its digit sum is 17 (mod 17). Verification confirms this by checking equivalences such as \( 1 + 6 + m = 17 \) and \( 9 + m = 17 \). | 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 fewest digits that sum to 17:**",
"- The digits of 17 are 1, 7.",
"- The smallest positive integers that sum to 17 are 1 + 7 = 8 and 10 + 1 = 11.",
"- Therefore, the smallest possib... | We need the least positive integer \( m \) divisible by \( 17 \) whose digits sum to 17, with the tens digit being at least 3, thus \( m=16 + x \). Solving \( 16 + x \) must end in a digit of 6, which implies \( x \) must be \( \boxed{6} \) for \( x = 6 \), making \( m = 1666666 \). | 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 that is divisible by 17 and has a digit sum of 17, I start by listing numbers that meet this criterion.",
"First, I calculate the first divisible number that has digits that add up to 17: 89 is divisible by 17 because 8 + 9 = 17.",
"Next, I check if there are subsequent nu... | The earliest number to verify divisibility by 17 is 89, since 8 + 9 = 17. The next factor is 164, which sums to 17 (ô¼§). | 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 digits summing to 17, we can follow these steps:",
"1. **Understand the Divisibility Rule**: A number is divisible by 17 if its total sum of digits is divisible by 17.",
"2. **Identify Digits That Add Up to 17**: Let the lowest-valuation... | The digit combination \( \{c, (9c-a) \mid (a)\in \{1, ..., 9\}\} \) satisfied when \( c=204442 \) and \( a=204435 \) all have digit sums divisible by 17. Thus the digit is \( \boxed{204442} \), since \( \{204442, (9\cdot204435+a)\mid a\in \{1,\dots,9\} \} \) holds [09–20442]. | 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 can follow these steps:",
"1. **Identify the largest possible multiple of 17 less than 17**:",
"- The largest multiple of 17 less than 17 is 17 itself.",
"2. **6 can be used as the multiplier of 17 (denomina... | The least positive integer \( m \) divisible by 17 is found by evaluating \( m \) as the largest multiple of 17 less than 17, which is 17 itself. This factor checks out through \( 17 + 2 = 19 \). | 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 \\) that is divisible by 17 with a digit sum of 17, I need to identify the smallest number under 17 that meets this criterion.",
"First, I'll represent the smallest positive integer \\( n \\) such that its digits sum to 17 as \\( 10n \\). This is because adding 17 to 10 y... | The least positive integer \( m \) that is divisible by \( 17 \) is \( 673 \). This is because \( 17 \times 602 = 10994 \), and \( 17 \times 673 = 10996 \), which meets the 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 \\) that is divisible by 17 with digits summing to 17, we can follow these steps:",
"1. **Assume the sum of the digits is 17**:",
"- Let \\( p \\) be the digits of the number such that \\( p + A \\leq 17 \\). Subtracting 17 from both sides gives \\( p = A - 9 \\).",
"... | The sum \( p + A \leq 17 \) and \( A \le 1 \) implies \( A \le 9 \). The smallest possible digits \( A \) for \( p \leq 2 \leq A \leq 9 \) are 1 or 17, yielding \( m = (17 + 1)(17 - 9) + 17 = 10 \). | 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, I start by identifying all possible combinations of digits (each from 0 to 9) that add up to 17.",
"First, I examine the factors of 17. The factors are 1, 7, 11, and 17.",
"Next, I test each option for the minimum digi... | The factors of 17 are 1, 7, 11, and 17. The least positive integer divisible by 17 whose digits sum to 17 is 11. | 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 digits summing to 17, we start by considering the smallest possible number with these digits. The smallest number in this case is 10, which is divisible by 10 (since 10 ÷ 10 = 1).",
"Next, we need to ensure that this number formed by these... | The least positive integer \( m \) divisible by 17 with digits summing to 17 is found by ensuring that the sum of the digits of such a number \( m = 10 \) is divisible by 17. This condition requires the remaining digit \( 16 \) to be divisible by 17, leading to \( m = 10 \) being the solution. | 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). | [
"To determine the number of ways to connect four towns with one road between each pair in a one-way arrangement, we need to minimize the total number of roads. Since each pair connects each town, the number of roads per connection is four, and multiplying by the number of connections gives the total number of conne... | There are 4 streets per connection, and with 4 connections, the total number of routes is \(4^4 = 256\). Thus, there are 256 ways to connect four towns with one road (and pass through others) without any restrictions. | 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 valid route designs for connecting five towns with exactly one road between each pair, we can follow these steps:",
"1. **Identify Major and Minor Keys:**",
"- The major towns are \\( M \\), \\( N \\), \\( A \\), \\( C \\), \\( D \\), which give us \\( M \\times N = 5 \\) bridges.",
... | There are $5 \times 5 = 25$ ways to choose bridges without having any major towns, and the remaining bridges can be more than 9 bridges. Dividing the total number of bridges (3150) by the number of bridges in the total way (180) gives the final answer of $25 \times 180 = 4500$. | 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 are 3 towns, and we need to connect every pair of them in a one-way road. The key here is that it's possible to get from any town to any other town, including passing through other towns. So each pair needs to join in such a way that every road is one-way.",
"Hmm. Let me visualize this. Imagine th... | The total number of households, each connected to 3 others, is \(3 \times 3 \times 3 = 27\), resulting in after-house connections of \(\frac{27!}{3!(3-2)!1}\). This computation yields \(\frac{27!}{6!3! = 81\). | 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). | [
"First, I need to understand the problem. We start by noting that there are 3 towns, each connected by a road connecting their two closest neighbors, and we want to create all possible connections with only one-way paths that still allow getting from any one town to any other.",
"To approach this, I'll consider t... | The number of ways to connect any two towns is from 3 to 3 (choices), and connecting each pair of towns and the floors from their own homes yields 3 to 2 (connections to the home outside the first household). Thus, by the trilogram, there are $3 \times 2 \times 2 = 12$ valid ways to connect the 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 me see. The problem is about linking three towns with roads, and we want to make it impossible to get from any one town to any other one once you have made all the roads one-way. Hmm.",
"First, I need to remember what John Kong and Jonathan Rafinek have been doing in theirs. Arms expressing: \"Wareizin... | A tour on a square with a cardial-shaped path will go through exactly one-dimensional (four-sided) star-shaped figure \(\left((3,7)\right)\) or \(\left((2,3)\right) \rightold square, creating four identical identical triangles. Since there are three corners on the rings, the number of mutually non-pareis in the formati... | 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). | [
"First, I need to understand how to form the first road given the connection map. Since there is exactly one road connecting each pair of towns, with no repeats, I can start by assigning the first town to the second pair, then the second town to the third pair. This efficiently assigns the first connection to the s... | The problem requires forming a one-way road from two towns to two other towns, which can only be done by alternating the use of one town in the world with the other in any two connected towns. The order is determined by first reassigning the two oldest pupils to the primary towns, then leading them to the third town, a... | 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 are five towns connected by roads, and exactly one road connects each pair of towns. That means if I had to connect any two towns, there is exactly one road there. Then, if I didn't connecting the pairs, there are no roads joining the towns, so that's the unconnected regions.",
"Hmm, so the proble... | The problem requires finding the number of six-sided rectangular faces of shapes sufficient to form directly connected pairs at the ends of the roads. This problem utilizes the concept that each pair of towns requires one specific pair, thus the number of six-sided faces necessary to form any three or four-sided struct... | 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 are five towns connected by a road span, and the roads are one-way in. I need to figure out how many one-way connections every town has to each other so that everyone can travel from any town to each other. Hmm, let me break this down step by step.",
"First, how many roads are there? If any road i... | The problem is to find the number of one-way connections, denoted by $n$, obtained by restricting all existing connections to a single pair of towns. This can be achieved by considering all possible three-circle constraints or with the assumption of two connections each to a different city, resulting in $n = 3 - 2 = 1$... | 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). | [
"First, recognize that every pair of towns will be connected by a road, and passing through others would require the two towns to alternate.",
"Since there is exactly one road connecting each pair of towns, it is impossible to drive from one town to another if the order matters.",
"To ensure the connection is o... | With exactly one road connecting each pair of towns, alternating between same-direction and opposite-direction directions is impossible since paths alternate. Thus there are **no valid connections** that fit the given conditions, and the answer is $\boxed{0}$. | 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 one-way connections that can be made between five towns connected by a system of roads, we start by considering each town individually. Since there is exactly one road connecting each pair of towns, the total number of possible one-way connections is \\(5! = 120\\).",
"However, the pro... | To achieve a one-way connection between shouts of towns, we can rotate the connections among neighboring towns, treating the diagonals as diagonals, resulting in 64 connections per town. This produces 5! ways for the first three towns (6, 24, and 18 connections) and 5! for the last two towns (4 connections each), yield... | 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 one-way connections possible in a network where each town is connected to its two neighbor, we start by noting that there is exactly one road between any two towns.",
"Since all towns connected by streets are distinct, we can consider each pair of towns as a possible connection. For ea... | Each of the \( H \) neighboring towns can be paired with any of the \( H \) streets, giving \(\frac{1}{3}H\) paths. Thus the total number of connected streets is \(\boxed{\frac{1}{3}H}\). | 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). | [
"First, I need to understand the problem: there are three towns, and each town is connected only by roads. The roads are one-way, which means each town directly connects to the town directly above it, and the roads are not shared or moved.",
"I want to find the number of ways to connect three towns without passin... | To connect three towns such that no town is connected to the one above, there are 3 valid two-way paths of lengths 1, 2, 3, and 4 each, totaling 27. Thus, there are 27 ways to connect the three 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). | [
"First, I need to understand the structure of the roads. There are exactly one road connecting each pair of towns, which means that for any two towns, the roads to their nearest neighbors are the only connections.",
"Next, I should consider the structure of the roads. A tree-shaped network is formed by connecting... | In a tree-shaped network with one road connecting each pair of towns, each return route starting with the nearest neighbors of the other twigs must start at the last street. This can be calculated using the combination formula $\binom{k}{r}$, resulting in $\binom{5}{3} = 10$ possible sequences. | 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 are five towns connected by this connection system. It says exactly one road connects each pair of towns. So, if I start walking from town A to town B, I have two options. Since there are five towns, the number of ways to arrange all pairs would be 5. Let me write that down:",
"Number of ways = 5"... | The problem has three conditions: No return to a previously chosen fellow town; At least two rural (with four such villages); and No people (the five pairs mutually exchange). With four towns on the airfield and 5 distinct pairs, there are 5 ways, as the options are distinct, and each as a trip, making 3 trips each, wh... | 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 the problem, we start with the given equation:",
"\\[",
"z + |z| = 2 + 8i",
"\\]",
"We are asked to find \\( |z|^{2} \\), which means we need to find the square of the complex number \\( z \\).",
"First, let's rearrange the given equation:",
"\\[",
"z + 2 = 8i",
"\\]",
"Solving for \\( z... | Given \( z + |z| = 2 + 8i \), we solve for \( z \) to get \( z = 8i - 2 \), which leads to \( |z|^2 = (8i - 2)^2 = 16 \). The final answer is $\boxed{\textbf{(B)}\ 100}$. | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"First, I need to find the modulus of \\( z \\) using the given equation \\( z + |z| = 2 + 8i \\).",
"I know that in complex numbers, the modulus \\( |z| \\) is equal to the distance from \\( z \\) to the origin, which can be positive or negative. Therefore, to find \\( |z| \\), I can take the absolute value of t... | Given \( z + |z| = 2 + 8i \), we find \( |z| = 2z^2 = 100 \). Therefore, \( |z|^2 = |z|^2 = |z|^2 = 2z^2 = 100 \). | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"First, we are given the equation \\( z + |z| = 2 + 8i \\). This implies that \\( |z| = -2 + 4i \\).",
"Next, we need to find the modulus \\( |z|^2 \\) using the fact that \\( |z| = \\sqrt{a^2 + b^2} \\) where \\( a \\) and \\( b \\) are complex numbers. Substituting \\( a = -2 \\) and \\( b = 4 \\), we get:",
... | Given \( z + |z| = 2 + 8i \), we find \( |z| = -2 + 4i \). Substituting \( b=4 \) into the modulus formula \( |z|^2 = (\sqrt{a^2 + b^2})^2 \) gives \( |z|^2 = (\sqrt{(-2)^2 + 4^2})^2 = 20 \), so \( |z| = -2 \). | 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 $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"To solve for \\( |z|^2 \\) given the equation \\( z + |z| = 2 + 8i \\), we start by calculating \\( |z| \\).",
"Given \\( z = a + bi \\), where \\( a \\) and \\( b \\) are real numbers and \\( |z| = \\sqrt{a^2 + b^2} \\), we can express \\( z \\) in terms of this imaginary unit.",
"By squaring the imaginary un... | Given $z + |z| = 2 + 8i$, we can express $|z|$ as $|z|^2 = 100$. Substituting $|z| = \sqrt{a^2 + b^2}$, we find $|z|^2$ to be $(2+8)^{-1} = 100$. | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"First, I need to find the value of \\( |z| \\) from the given equation \\( z + |z| = 2 + 8i \\).",
"Using the difference equation:",
"\\[",
"z + |z| = 2 + 8i \\implies (z + |z|) + |z| = 2 + 8i",
"\\]",
"Substituting \\( z \\):",
"\\[",
"(z + |z|) + |z| = 2 + 8i \\implies |z| + |z| = (2 + 8i) + (2 + 8... | Given \( z + |z| = 2 + 8i \), we find \( |z| = 6 \). Consequently, \( |z|^{2} = \sqrt{6^{2}} = 6 \). | 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 system of equations:",
"1. \\( z + |z| = 2 + 8i \\)",
"2. \\( |z| = \\sqrt{a^2 + b^2} \\) with \\( a = -1 \\) and \\( b = 8 \\)",
"First, we square both sides of the first equation to make use of the property of square identities:",
"\\[",
"(z + |z|)^... | From \(z = a + 2bi\), squaring gives \(z^2 = \pm 16\); the roots actually lie as \(\frac{-\bignum \pm 10.6667}{3}\), and their sum \(\sqrt{16} \approx 4\) produces an integer \(|z| \approx 27\), so \(|z|^{2} \approx 27 \times 101 = \boxed{27777}\). | 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.",
"According to the equation \\( z + |z| = 2 + 8i \\), substituting \\( z \\) gives:",
"\\[",
"a + (a + bi) + |a + bi| = 2 + 8i",
"\\]",
"Simplifying this, we get:",
"\\[",
"2a + 2bi... | From \(z+|z|=2+8i\), we get \(2a+2bi=1\), so \(|z|^2=1\). The answer is \(\boxed{\textbf{(B)\ 100}}\). | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"First, I recognize that the complex number \\( z \\) is given as \\( z + |z| = 2 + 8i \\). To find \\( |z| \\), I can subtract \\( 2 + 8i \\) from both sides of the equation:",
"\\[",
"|z| = 2 + 8i - (2 + 8i)",
"\\]",
"Simplifying further:",
"\\[",
"|z| = 2 - 6i",
"\\]",
"Next, I need to find \\( |... | We found \( |z| = 2 - 6i \), and \( |z|^2 = \sqrt{(2 - 6i)^2} = -8 \). The final answer is \( \textbf{(A)}\ 68 \). | 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 $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"Let \\( z = a + bi \\), where \\( a \\) and \\( b \\) are real numbers. According to the problem, \\( z + |z| = 2 + 8i \\). This implies two things:",
"1. \\( |z| = 8 \\) because \\( |z| + |z| = 8 \\).",
"2. \\( z + |z| = 2 + 8i \\), so \\( z - 2 = 8 \\).",
"Solving the second equation gives \\( z = 6 \\).",... | Since \( z = 6 \), which satisfies \( z + |z| = 2 + 8i \), we get \( |zan| = 8 \). Therefore, \( |z|^{2} = 6^{2} = 36 \). | 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 z, and I need to find the modulus of its square and then then square it. The problem says that z + |z| = 2 + 8i, and z is defined as a + bi where a is the real part and b is the imaginary part. Hmm, let's break this down step by step.",
"First, I need to find |z| since |z| is ... | The modulus of the square of $|z|$ is 2102. | 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 for \\( z \\):",
"\\[ z + |z| = 2 + 8i \\]",
"1. **Substitute \\( z \\) and \\( |z| \\) into the equation:**",
"\\[ z + |z| = 2 + 8i \\]",
"2. **Use the expression for \\( |z| \\):**",
"\\[ |z|^{2} = |z|^{2} - 2 |z| \\times |z| \\]",
"Simplifying:"... | Given \( z + |z| = 2 + 8i \), we find \( |z|^{2} = 2^2 - 2(|z|)^{2} = 4 - 2(8) = -4 \). Then, \( |z|^{2} = (-4)^{2} = 16 \), so \( |z| = \sqrt{16} = 4 \). | 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 $|z| = \sqrt{a^{2} + b^{2}}$.
$\textbf{(A)}\ 68\qquad \textbf{(B)}\ 100\qquad \textbf{(C)}\ 169\qquad \textbf{(D)}\ 208\qquad \textbf{(E)}\ 289$ | [
"Let's solve the problem step by step.",
"**Given:**",
"\\[",
"z + |z| = 2 + 8i",
"\\]",
"\\[",
"z = a + bi",
"\\]",
"**Step 1: Plug \\( z \\) into the equation to find \\( |z| \\):**",
"\\[",
"a + bi + 8i = 2 + 8i",
"\\]",
"Compare the real parts and the imaginary parts:",
"\\[",
"Ab = ... | With \( z = a+bi \) with real \( ab = 2 \) and imaginary \( bi = 8 \), we get \( |z| = 4 \). Thus \( |z|^{2} = 4^2 = \boxed{CC} \). | 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 are given the equation:",
"$$",
"z + |z| = 2 + 8i",
"$$",
"We are also asked to find $ |z|^{2} $. Here, $ z = a + bi $, where $ a $ and $ b $ are real numbers, so $ |z | $ must be $ \\sqrt{a^2 + b^2} $, which expands to $ |z |^2 $.",
"### Step 1: Plug in $ a $ and $ b $ into the equation",
"$$",
"... | Given $z = a + bi$ and $z + |z| = 2 + 8i$, we find that $|z|^{2} = |z|^(-2) = 2 \cdot 8i^2 = 2 \cdot -8 = -16$. However, this argument is not valid as the condition $|z|^{2} = 2 + 8i$ does not hold within a rectangular complex number, implying the statement is off by 1. | 289 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To determine the number of ways to paint the six sections on the circle, I need to consider the constraints given: no two adjacent sections can be the same color, and no two adjacent surrounding sections can be painted with the same color.",
"First, for each section, the color can be assigned to a separate group... | For each painted section, the number of primary colors is \( D_1(4,3) = 12 \), and for each of the seven colors specified for the inner section, the total number of partitions of the circle is \( D_1(7,3) \times 2 = 42 \). Since color is allocated in ascending order, there are \( 42 \ATED = 42 \) possible partitions. | 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 choose a solid color for six small sections such that no two adjacent sections share the same color and no line segments share the same color, we can follow these steps:",
"1. **Identify Cubic Sections**:",
"- Three directions: (Cube East, Blue), (Cube Blue, Green), (Cube Blu... | There are 6 ways to paint the cubic sections, and for each cubic section, there are 24 ways to paint the rectangular sections, resulting in a total of \( 6 \times 24 = 144 \) 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, I need to visualize how the circles are assigned colors, which helps in determining the vertical spacing between them.",
"Next, I'll categorize the circles into those that are adjacent or not adjacent based on their vertical positions and the color they represent.",
"Then, I'll determine the number of c... | The problem is resolved by considering all connected circles (with corresponding right angles) and connected segments (with shared edges), and then combining these into slices as patterns of colors. This results in \(23916\) total ways to paint the areas. | 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 understand the problem. There is a ring with six sections, and I need to paint each section a different color, with no two adjacent sections being the same color. I have four paint colors available and need to choose four colors.",
"Since no two adjacent sections can share the same color, I can ... | The number of ways to choose four colors out of four available from overlapping sections is \(1 \times 11 = 11\). This is because there are 11 different corner color combinations where adjacent sections share color, resulting in \(4 \times 11 = 44\) possible primary sections. | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"First, I need to determine the colors to use for the four given sections. Since no two adjacent sections can be painted with the same color, I'll only use colors that reduce the red of the circle.",
"Next, I'll analyze the remaining section drawn to see if it can be colored with either the same color or a differ... | The circle is painted with red, either directly or through two adjacent shaded sections, yielding 1 way to choose red; each of the remaining three rows must choose a color of white. Thus the total number of colored sections is \(1 + 1 = \boxed{2}\). | 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 understand the problem. I have a central point, surrounded by six small sections. I need to choose how to paint each of these sections, considering no two adjacent sections can have the same color. Since no two adjacent sections have the same color, I can paint only one section of each type on top... | There are 36 ways to paint the sections with no two adjacent sections having the same color. This is determined by choosing two sticky points at each location for horizontal painting and the other two at the opposite sticky points for vertical painting. | 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 region that does not share the same color with adjacent sections. This region is the region \\((-4, 0)\\).",
"Next, determine how many ways to choose colors for each region except the union region (if it overlaps with the union).",
"Calculate the number of colors to choose from for the unio... | The figure demonstrates that certain regions are excluded due to overlapping colors, leaving 1 region with multiple colors (i.e., the empty region.) This leaves 4 colored regions, each with 5 possible colors, for a total of $4 \times 5 = 20$ 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, I need to determine how many ways I can choose two sections that do not share the same color. The sections are arranged along a straight line with four sides, four corners, and five sides.",
"I'll consider the four corners of the circle, and there are 4 corners in a circle. Since I cannot paint the same c... | There are 16 unique color pairings, and 3 orthogonal orientation rotations keep adjacent sections distinct, resulting in 3 × 16 = 48 total color pairings. The total number of ways to paint the spaces is 48. | 732 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | [
"To solve this problem, we start by considering the given circle and circle portion. We need to determine the number of ways to paint the section W on the circle and the corresponding section T on the circle. Since no two adjacent sections can be painted with the same color, we look at the framework, ensuring that ... | With four horizontal segments and four vertical segments of the circle, there are 9 color combinations, 3 loop combinations, and 128 ways to plan cross-legs. The total number of ways to paint W, and specifically T, is 128. | 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 four relatively distant non-adjacent sections using colors that avoid adjacent sections in a given way.",
"I'll start by considering rotating the region where adjacent sections are present. For example, one section might be rotated by 180 degrees, comple... | There are 16 ways to paint the sections where no two adjacent sections are adjacent. This spans from four relatively distant sections (4), to eight non-adjacent sections (7), and up to eight sections where no two adjacent sections are adjacent. | 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... | [
"Alright, so I have this problem here about choosing to paint a section of a ring. There are four circles, and each section should be painted the same color. The key is that no two adjacent sections can be painted with the same color. Hmm, okay let's break this down step by step.",
"First, I need to visualize wha... | There are $$6 \times 6 \times 2 = 72$$ valid color assignments where no two adjacent sections can be the same color. Therefore, the total number of ways to paint the sections without overlapping is 72. | 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 understand the problem and identify the allowed colors for each circle. The base color can be any of the six options, and any two adjacent sections can be painted with a single color.",
"Next, I'll determine which circles are used in the intersection by noting that the base color can be any of t... | There are 14 non-base colors, requiring $\binom{14}{4} = 1001$ ways to choose 4 remaining colors, and 14 ways to choose 1 color for the outward circle. The total number of ways to paint the sections is $1001 + 1446 = 2047$. | 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 identify the colors available for the circles and the gaps. Since no two adjacent sections can be painted with the same color, theophylline (WHY dots) is allowed. One of the spokes holes, at the corners of the circle, will always have WHY points. These should be painted.",
"Next, I need to deter... | The number of ways to choose colors for the outer circles and the inner circles is 8. These areas are the middle centers of the outermost outer circle and the inner inner circle, each surrounded by yellow. | 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 each color section, considering that no two adjacent sections can match a given color.",
"1. **Cycle Navigation**:",
"- When editing multiple lines at the same time, the next line can be any of the existing lines. This means that cycles in the diagram crea... | The number of ways to paint each color section is equal to the number of arcs between colored regions. With division formulas and checklists, paths that divide sections in such a way as to not have adjacent sections within each color result in circular symmetry, spiral formation, merging, hyperbolic synchronization, or... | 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^n3^m \\), where \\( n \\) and \\( m \\) are non-negative integers, we start by simplifying the expression:",
"\\[",
"a = 2^n^3 \\cdot (2^m)^2",
"\\]",
"This leads to:",
"\\[",
"a = 2^n^3 \\cdot 4^2 \\cdot (2^m)^2",
"\\]",
"Next, we recognize t... | The sum of all positive integers \( a = 2^n3^m \) for which \( a^6 \not a divisor of \( 6^a \) is \( a = 2^n^2 \cdot 8 \). Given that \( 6 = 2 \cdot 3 \), this simplifies to \( (2^n)^2 \cdot 4 \), which equals \( 4 \), and thus \( a = 2^n^2 \cdot 4 \), resulting in a sum of 16. | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"First, I need to find all positive integers \\( a \\) and \\( m \\) such that \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"I start by consider the greatest common divisor (gcd) of \\( 6 \\) and \\( 6^a \\). Since \\( 6^a = 6^{\\log_3(6)} = 2 \\times 3 \\), the gcd is \\( 2 \\times 3 = 6 \\).",
"Next, I look... | The condition that \( a^6 \) is not a divisor of \( 6^a \) is actually true because \( a^6 = 2^3 \times 3^3 \), and \( 6 \times 2 = 12 \), which makes \( 2^3 \times 3^3 \) a divisor. The only positive integer \( a \) satisfying this condition is 1, and the only positive integer \( m \) satisfies \( m = 2 \). | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To determine the sum of all positive integers \\( a = 2^n^m \\) where \\( n \\) and \\( m \\) are non-negative integers, we start by finding the parity of the term \\( 2^n^m \\). Since \\( n \\) and \\( m \\) are non-negative integers, \\( n \\) must be even (because even numbers are divisible by 2), and \\( m \\)... | Since \(2^n^m\) is a multiple of \(2^n(2^n+1)/2\) and \(4\) is divisible by both \(2\) and \(16\), all such \(a = 2^n^m\) are consecutive cubes of powers of \(2\), whose sum is 16. Therefore, \(a^6\) = 16, and the sum is \(\boxed{10000}\). | 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 \\) where \\( n \\) and \\( m \\) are non-negative integers, and \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"1. **Sum of Numbers**: The problem asks for the sum of all such \\( a \\) expressing them in terms of \\( n \\) ... | The sum \( a = 2^{n \cdot m} \) is well-defined as concatenating \( 2^n \) times times \( 2^{n \cdot m} \), where for divisibility by 2, \( a^a = 2^{n \cdot m} \). Given \( n \) and \( m \) are non-negative integers, \( a^a \) will not be divisible by any number \( 2^a \) nor any number \( 2 \times 2^n \). | 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 \\( n \\) and \\( m \\). Given \\( a = 2^n3^m \\), we substitute each value of \\( n \\) and \\( m \\) into the equation.",
"1. **Substitute \\( n = 2 \\):**",
"\\[",
"a = 2^2 \\cdot 3 \\cdot m",
"\\]",
"2. **Substitute \\( n = 3 \\):**"... | The condition for \( a^6 \) to not be a divisor of \( 6^a \) is \( m \equiv 0 \pmod{6} \). This implies that for \( a^2m \) to not be divisible by \( 6^a \), we must have \( m \equiv 0 \pmod{6} \), which is satisfied for the values of \( a \) from \( a=1 \) to \( a=23 \). | 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 \\) and \\( m \\) such that \\( a^6 \\) is not a divisor of \\( 6^a \\).",
"Since \\( 8 \\) is equal to \\( 2^3 \\times 4 \\), I can see that \\( 6^a \\) includes all powers of \\( 2 \\) from \\( (2^3) \\) up to \\( 6^3 \\). This means that \\( 6^a \\) must inclu... | Since \( 6^a \) must total the powers of 2 from \(2^3 \) to \(6^3 \), the divisors of \(6^a\) are those where \(2^3\) is the maximum power of 2, which slots within \(6^a\). For only \(2^3\) and \(6^a \uv 2^3\) ends in \(\binom{3}{1}\), there are 5 such positive integers: \(1, 2, 3, 4, 5\), so their sum is \(\boxed{15}\... | 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 $6^a$. | [
"We are given the recurrence relation:",
"$$",
"a^6 = (2^n)^3m",
"$$",
"We want to find the sum of all positive integers $ a $ such that:",
"$$",
"a^6 \\not = 2^n^3m",
"$$",
"---",
"### Step 1: Identify possible solutions",
"On the left-hand side, we get:",
"$$",
"a^6 = a^2 = a^4",
"\\Righ... | No valid exponents $n$ and $m$ for $a$ where $a^6 \neq 2^n^3m$ do not exist for $n \leq 5 \leq m \leq 8 \leq 5 \leq 8$, which yields $2a^5 \neq a^4$. The sum of all positive integers $a$ such that $a^6 = n^3 m$ for $n \leq m \leq m + 8 \leq m + 8$ is 13. | 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 $6^a$. | [
"Okay, so I have this problem where I need to find the sum of all positive integers a-powers of 23, where n is the exponent and m is the exponent, and I have the result a^6 is not a divisor of 6^a. Hmm, let me try to figure this out step by step.",
"First, I remember that n and m are non-negative integers. That m... | $a = 2$ means $n \ge 0$, and $a = 2$ implies $n \le 4$. Considering the limit requirements, the sum is $a - 2 - 2 = 4$. | 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, I need to identify all possible values of \\( n \\) and \\( m \\) that satisfy the condition \\( a^6 \\not \\div \\gcd(a^n \\cdot m) \\) for the given imperative programming language.",
"First, I'll rewrite the equation \\( a^6 \\) as \\( (2^6)a^6 = 6^6 \\). This simplifies the equation to ... | The equation \(a^3 \div gcd(a^n \cdot m) = 6^a\) requires \(a^3 \not\div 6a\), leading to \(a^3 = 5\). The 7 positive integers satisfying this are \(a = 2, 3, 4\), so their sum is \((2+3+4)a = (9+3)=12\). | 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 the problem, we need to find the positive integers \\( a \\) and \\( m \\) such that \\( a^6 \\) is not a divisor of \\( 6^a \\) for both \\( n = 2 \\) and \\( m = 2 \\).",
"1. **Step 1: Check the condition for division**",
"- \\( a = 2^n \\implies 2^n \\) is not divisible by 2 if \\( n \\) is odd (or... | For \( a^n \) not to be a divisor of \( 6^a \), \( n \) must be a multiple of 2; since every raised power is even, all exponents are even. Since \( 2^n \) is not divisible by \( 2^m \) for \( n \ge 3 = 2k \), no such \( a \) exists. | 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 \\) where \\( a^6 \\) is not a divisor of \\( 6^a \\), we can use the theorem that factors \\( 6^a \\) can never divide a number of squares of6 since they are all powers of 2. This means that there are no such positive integer \\( a \\) such that \\( a^6 \\) is divisi... | The condition \( a^6 \) is not divisible by \( 6^a \) requires that \( a^6 \) is a divisor of \( 6^a \). Given that \( a = 1 \), the only possible value for \( a \) is \( 1 \), making the sum of positive integers \( a^6 \) equal to \( 1 + 1 + 1 + 1 + 1 + 1 + 1 = 6 \). | 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 $6^a$. | [
"Okay, so I have this problem where I need to find the sum of all positive integers a and b such that a^6 is not divisible by the product of the reciprocals of n and m. Hmm, let me try to figure this out step by step.",
"First, I know that a^6 is the same as (2^5)^6. Let me calculate that. 5 times 5 is 25, and sq... | The sum of positive integers $a$ and $b$ such that $a^6 \div (2^n(2^n + 3)(2^m))$ is -1. | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To solve the problem, we start by expressing \\( a^6 \\) in terms of powers of 6. Specifically:",
"\\[ a^6 = (6^a)^6 = (6^{a*6})^6 \\]",
"Next, we simplify this expression to our common denominator:",
"\\[ a^6 = 6^{6a} \\]",
"For \\( a^6 \\) to be a divisor of \\( 6^a \\), it must be possible to express \\... | We have \(a^6 = 6^a\), and necessary conditions state that \(a^4 \equiv 1 \pmod{6}\), so \(a^{4+1} = 6^{4+1} = 6^5\) is a divisor of \(6^a\). Solving \(6^a \equiv 1 \pmod{6}\), which implies \(a^{4+1} = 6^5\), shows that the possible exponents for \(a\) are \(1\) through \(6\), leading to \(a = 1\), \(a = 4\), and \(a ... | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | [
"To solve the problem, I start by understanding the given condition: the powers of \\(2^n\\) must not be divisors of \\(6^a\\). This means that \\(a\\) and \\(1/6^n\\) must be non-negative integers, and their order \\(n\\) must be a divisor of \\(2^3\\), which is 2. Similarly, \\(2^m\\) must be a divisor of \\(6^a\... | The conditions \( n \neq 0 \) and \( 6 \neq n^2 \) imply \( n=36 \) and \( 18 \), leading to the condition \( a^6 \not= 2^3 \rightarrow a \neq 36 \) and \( 6^a \not= 12 \rightarrow 6 \neq a^2 \). The sum of these conditions is \( 6 + 6 = 12 \). | 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 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 \\), we need to consider the properties of the exponents \\( n \\) and \\( m \\).",
"1. Define the exponents: \\( n = 3 \\) and \\( m = 2 \\).",
... | The sum of all positive integers \( a = 2^n3^m \) for which \( a^6 \notis \) a divisor of \( 6^a \) is found by combining the conditions \( \langle n \rangle = 2^n \) and \( m \) with the divisor condition leading to \( 2^n \equiv 2 \pmod{3} \). This implies the sum equals the product of the remainders. | 42 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqr... | [
"First, let's identify the distances from the center of circle \\( C_1 \\) to either point \\( E \\) or \\( Z \\) on circle \\( C_2 \\). Since \\( O \\) lies on circle \\( C_2 \\) and lies on circle \\( C_1 \\), we can define the radius of circle \\( C_1 \\) as the distance from \\( O \\) to \\( E \\) or from \\( O... | The radius of circle \( C_1 \) is the distance from the center \( O \) to the point \( E \), which is the distance from \( O \) to the distance from \( O \) to the point \( Z \), and between \( O_Z = 11 \) and \( O_X = 6 \) of \( C_2 \). Assuming \( O_Z \) is the center of \( C_1 \) and its distance from \( O_X \) to \... | \sqrt{30} | false |
Open-Reasoner-Zero/orz_math_57k_collection | Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqr... | [
"First, I need to determine the radius of the external circle \\( C_2 \\) centered at \\( O \\) with diameters \\( OZ \\) and \\( YZ = 7 \\). The distance from \\( OZ \\) to the center \\( Y \\) is \\( OP = 7 \\), so \\( OP = 14 \\) units. Since the radii are proportional to the increases in \\( OP \\), I can calcu... | Since \(C_1\'s ring is centered at \(O\), \(OZ = OP/14 = \boxed{\frac{5\pi}{14}}\), and with radius \(OZ = 11\) as the diameter of \(C_1\). The radius \(OZ\) occurs as the inscribed angle at the center. | \sqrt{30} | false |
Open-Reasoner-Zero/orz_math_57k_collection | Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqr... | [
"First, we know that the circle \\( O_1 \\) lies on circle \\( O_2 \\). Given:",
"\\[",
"OZ = 13 \\quad \\text{and} \\quad OZ = 11",
"\\]",
"Since \\( O_1 \\) lies on \\( O_2 \\), the distance from \\( O_1 \\) to \\( X \\) is \\( 13 \\). Similarly, the distance from \\( O_1 \\) to \\( Y \\) is \\( 11 \\). T... | Given \( O_2 \), the distance to \( O_1 \) is \( P = 13 \), with \( P-O = 13 \). Since \( O_1 \) lies on \( O_2 \), \( 13P + P^2 = 13^2 + (11-P)^2 \), which leads to \( 21P = 224 \) and thus \( P = 12 \). | \sqrt{30} | false |
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