source stringclasses 7
values | question stringlengths 3 1.02k | steps listlengths 0 198 | answer stringlengths 0 1.37k | ground_truth stringlengths 1 202 | is_correct bool 2
classes |
|---|---|---|---|---|---|
Open-Reasoner-Zero/orz_math_57k_collection | Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqr... | [
"First, I need to understand the relationship between the two circles \\( C_1 \\) and \\( C_2 \\). The problem states that circle \\( C_1 \\) has its center \\( O \\) lying on circle \\( C_2 \\), which means all points in \\( C_1 \\) are also points in \\( C_2 \\). This implies that the radius of \\( C_1 \\) is at ... | From \(OZ=OK+ZK\) with \(OZ=11\) and \(ZK=2\), the radius of \(C_1\) is \(OK=11-2=9\). Thus, the radius of circle \(C_1\) is \(\boxed{9}\). | \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... | [
"To determine the radius of circle \\( C_1 \\), we start by noting that point \\( Z \\) lies on both circles \\( C_1 \\) and \\( C_2 \\). Let the radius of circle \\( C_1 \\) be \\( r \\). The distance from the center of \\( C_2 \\) to the point of intersection \\( X \\) is \\( OX = 11 - r \\), and the distance fro... | The radius \( r \) of circle \( C_1 \) is found to be \( -11 \) by using the sum of distances from a point exterior to a circle. The correct radius is therefore \( \sqrt{26} \). | \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... | [
"To determine the radius of circle \\( C_1 \\), we start by noting that \\( O \\) lies on circle \\( C_2 \\). This means \\( O \\) is also the center of circle \\( C_1 \\).",
"The distance between the centers of the two circles is given as \\( OXZ = 13 \\). Since \\( X \\) and \\( Z \\) are on circle \\( C_1 \\) ... | The radius of circle \( C_1 \) cannot be determined with the given information, as the provided distance values result in a negative radius. Therefore, the answer is \(\textbf{(E)}\ \sqrt{30}\). | \sqrt{30} | false |
Open-Reasoner-Zero/orz_math_57k_collection | Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ \sqrt{26}\qquad\textbf{(C)}\ 3\sqrt{3}\qquad\textbf{(D)}\ 2\sqr... | [
"First, I need to understand the given information and visualize the problem. There are two circles, \\( C_1 \\) and \\( C_2 \\), with center \\( O \\) lying on \\( C_2 \\). The circles meet at points \\( X \\) and \\( Y \\), meaning \\( X \\) and \\( Y \\) are external points to both circles, and they are also ext... | Given \( OZ = 11 \) and \( Z \) is an external point, \( R_1 < R = 11 \), implying \( R_1 = 7 \). With \( OZ = 11 \) and \( Z \in C_1 \), \( R_1 + R_2 = 11 \), so \( R_2 = 4 \), the radius of circle \( C_1 \) is \( R_1 = 7 \). | \sqrt{30} | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"First, I need to understand the relationships between the base-14, base-15, and base-6 representations of the number \\( n \\).",
"I know that:",
"- In base-14, the number can be written as \\( a_1 \\times 14 + a_2 \\times 1 + a_3 \\).",
"- In base-15, it can be written as \\( a_1 \\times 15 + a_3 \\times 1 ... | Taking the three given base representations as equal, the equations give \(a_{1}=13a_{2}\), so \(a_{1}\) is a multiple of 13 (and must be 13, forcing \(a_{2}=0\)). Thus \(n=13\cdot14^{2}=2460\). | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"First, I need to understand the relationships between the digits in each base. In base 14, the first digit represents the number of \\(14\\)s, the second digit represents the number of \\(15\\)s, and the third digit represents the number of \\(6\\)s. This means:",
"\\[",
"n = 14a + 15b + 6c",
"\\]",
"Next,... | Assuming \(n = 14a + 15b + 6c\), we derive \(14a + 15b = (2a + 3)c\). Given \(a > 0\), if \(c=3\), then \(2a + 3 = 14a\), which implies \(a = \frac{1}{6}\), indicating no valid base-10 representation since \(a\) must be an integer. | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"To find the base-10 representation of \\( n \\), we analyze the given base representations:",
"1. **Base 14 Representation**:",
"\\[",
"n = a \\cdot 14^2 + b \\cdot 14^1 + c \\cdot 14^0",
"\\]",
"This corresponds to:",
"\\[",
"n = 196a + 14b + c",
"\\]",
"2. **Base 15 Representation**:",
"\\[",... | We found \( a = 2 \) and \( b = 53 \) from the equal coefficients of \( 6^2 \) in the base 6 conversion, which then allows us to solve for \( c = 155 \) using the base 6 conversion. The base 10 representation of \( n \) is \( 196 \cdot 2 + 53 + 155 = 552 \). | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"First, I need to understand the relationships between the numbers written in different bases. I have three numbers: one in base 14, one in base 15, and another in base 6, with \\( a > 0 \\).",
"I'll start by converting the number from base 15 to base 10. The number \\( \\underline{a}\\underline{c}\\underline{b} ... | Setting \(n = 15a + bc\) (base 14) and \(n = 6ac + 16a + c\) (base 6), equating gives a quadratic equation whose only valid root is \(a = 15\), so \(n = 15 \cdot 225 + 1 = 3376\). | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"To find the base-10 representation of \\( n \\), I start by understanding how numbers are expressed in different bases. In base-14, the number is given as \\( a \\underline{b} \\underline{c} \\), where \\( a, b, \\) and \\( c \\) are digits. This means:",
"\\[",
"n = 14a + 1b + c",
"\\]",
"In base-15, the ... | The base-14 number \( 14a + 1b + c \) equals the base-15 number \( 15a + c + b \), and the base-6 number \( a \underline{c} \underline{a} \underline{c} \) equals \( 6a + c + b \). Setting these equal yields \( 14a + 1b + c = 15a + c + b \), which simplifies to \( b - 5a = 2c \), and for digits \( a, b, c \ge 0 \), the ... | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"First, I need to understand how the number \\( n \\) is expressed in different bases. In base 14, it is given as \\( \\underline{a} \\underline{b} \\underline{c} \\), and in base 15, it is \\( \\underline{a} \\underline{c} \\underline{b} \\). Additionally, in base 6, it is \\( \\underline{a} \\underline{c} \\under... | Mapping the digits of n in base 14 to base 6 gives each digit 10 or higher as the same digit (6 or 5), which simplifies conversion: if digits a, b, c in base 14 equal 15m + m + 1 and m then n = 15(a + b) + b = 15(a + 2) = 15a + 30. Thus, n = 15a + 30. | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"To find the base-10 representation of \\( n \\), we analyze the given numeral systems: base-14, base-15, and base-6. Each numeral system represents \\( n \\) with a unique structure involving increasing digits in a specific order.",
"First, we break down each part of the numeral in base-10:",
"1. **Base-14 Num... | The base-10 representation of \( n \) is given by \( 10a + 11b + 12c \). This is derived from the numeral systems, where \( n = \underline{a}\underline{b}\underline{c} \) in base-14. | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"To find the base-10 representation of \\( n \\), we analyze the given base-14, base-15, and base-6 representations.",
"1. **Base-14 Representation**:",
"The number \\( n \\) can be written as \\( a \\underline{b} \\underline{c} \\) in base-14, where \\( a, b, c \\) are digits from 0 to 13.",
"2. **Base-15 Re... | By comparing the given base representations, we find that \(a=10, c=1, b=0, d=1\) in base-10. Thus, \(n\) is \(101011_{(10)}\). | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"First, I recognize that \\( n \\) can be expressed in three different bases: 14, 15, and 6. My goal is to find the base-10 equivalent of \\( n \\).",
"Starting with the base-14 representation \\( \\underline{a} \\underline{b} \\underline{c} \\), I can write each digit multiplied by the corresponding power of 14.... | Since \( n \) is common in all three representations, comparing digit values gives \( a = c = 1 \), so \( n = 201_{14} = 2 \times 15^2 + 1 = 285 \) base 10, despite the illegality \( a > 0 \) in base 15. | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"First, I need to understand how the number \\( n \\) can be represented in different bases. In base 14, the number is given as \\( \\underline{a} \\underline{b} \\underline{c} \\), and it should also be equal to \\( \\underline{a} \\underline{c} \\underline{b} \\) in base 15. Additionally, in base 6, the number is... | Converting each base to base 10 gives \(n = 1267650608\) (i.e., coefficients matching all three bases): \(11 \times 14^5 + 0 \times 14^4 + 6 \times 14^3 + 7 \times 14^2 + 0 \times 14^1 + 8 \times 14^0\). Thus, \(n = \boxed{1267650608}\). | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"First, I need to understand the given information about the number \\( n \\) in different bases. In base 14, the number is represented as \\( \\underline{a}\\underline{b}\\underline{c} \\), and in base 15, it is \\( \\underline{a}\\underline{c}\\underline{b} \\). Additionally, in base 6, the number is \\( \\underl... | The digits \(b\) and \(c\) are determined to be equal to each other and \(a\), so \(b=c=a\). Converting these to base 10, \(n = a \cdot 216 + a \cdot 36 + a \cdot 6 + a = 267a\), which results in a base 10 representation of \(n = 267a\). | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"First, I need to determine the value of \\( a \\) in all three base representations. From the base 14 representation \\( \\underline{a}\\underline{b}\\underline{c} \\), I know that \\( a > 0 \\).",
"Next, I'll consider the base 15 representation \\( \\underline{a}\\underline{c}\\underline{b} \\). Since \\( a > 0... | Given the number \( n \) can be expressed in base 14 as \( \underline{a}\underline{b}\underline{c} \), base 15 as \( \underline{a}\underline{c}\underline{b} \), and base 6 as \( \underline{a}\underline{c}\underline{a}\underline{c} \), we determine \( a=1 \) and \( c=5 \). Substituting these values into the base 10 repr... | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"First, I need to translate the number \\( n \\) from base 14 to base 15. The number is given as \\( \\underline{a}\\underline{b}\\underline{c} \\) in base 14, where \\( a \\geq 1 \\).",
"By comparing each digit from base 14 to base 15, I find that:",
"\\[ b = 15 - a \\]",
"\\[ c = 15 - b \\]",
"Substitutin... | The base-14 description of \( n \) is \( \underline{a} \underline{b} \underline{c} \), which translates to \( 15a + 15c + b \). Simplifying this to base-10, we get \( n = 84 - 7a \). | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, whe... | [
"To find the base-10 representation of \\( n \\), we need to translate each base notation into a decimal number.",
"1. **Base 14 Representation**: The number \\( n \\) can be written as \\( a_1 a_2 b_1 b_2 \\) in base 14. Each digit represents the following power of 14:",
"\\[",
"n_{14} = a_1 \\times 14 + a_2... | The base-14, base-15, and base-6 representations define \(n\) as \(15c + 15d + 14e\), \(c + 15d\), and \(a_4c\), respectively. Equating \(7a_2 = c + 15\) yields \(a_2 = 15\), and given \(a > 0\) and considering digits in base-14 as \(29\), the base-10 representation is \(29\). | 925 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"First, let's factorize the given numbers to their prime factors:",
"- \\(70 = 2 \\cdot 5 \\cdot 7\\)",
"- \\(71\\) is a prime number",
"- \\(72 = 2^3 \\cdot 3^2\\)",
"Next, we combine all the prime factors:",
"- From \\(70\\): \\(2, 5, 7\\)",
"- From \\(71\\) and \\(72\\): \\(1, 71\\)",
"Now, we need... | The problem reduces to finding positive integers \((a, b, c, d, e, f, g)\) such that \(abc = 70\), \(cde = 71\), and \(efg = 72\), where \(abc = 2 \cdot 5 \cdot 7\), \(cde = 71\), and \(efg = 2^3 \cdot 3^2\). This forces \(c=71\), \(b=1\), \(d,e\) can be anything, and \(f,g\) must be 2,3,4,5,6,7 for \(abc\). Thus there... | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"To solve the problem, we start by analyzing the given equations:",
"1. \\( abc = 70 \\)",
"2. \\( cde = 71 \\) which implies \\( c = 71 \\) and \\( de = 71 \\)",
"3. \\( efg = 72 \\) which implies \\( e = 72 \\) and \\( f = 72 \\)",
"Next, we assign values to the variables \\( a, b, d, e \\) and calculate ... | The system of equations $abc = 70$, $cde = 71$, and $efg = 72$ yields $c=71$, $d=71$, and $g=1$ by process of elimination. The number of 7-tuples for $(a,b,c,d,e,f,g)$ is then calculated as $8 \times 8 \times 71 \times 72 \times 1 \times 72 \times 1$, resulting in 448,576. | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"To solve the problem, we start by noting that the numbers must be factors of 70, 71, and 72.",
"First, we factorize each integer:",
"- 70 = 2 × 5 × 7",
"- 71 is a prime number",
"- 72 = 2³ × 3²",
"Next, we list all pairs of distinct factors of 70 that multiply to 71 (since 71 is prime):",
"- 2 and 35",... | The number of valid triplets $(a, b, c)$ for the first equation $abc = 70$ is 6. For each triplet, there are 6 possible sets of triplets $(b, d, f)$ and 3 sets of triplets $(c, e, g)$ such that $cde = 71$, $efg = 72$, resulting in $6 \times 6 \times 3 = 108$ total 7-tuples. | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"First, I need to find the factors of 70, 71, and 72 to understand the possible combinations that can be formed by \\(a, b, c, d, e, f, g\\).",
"For \\(abc = 70\\), the factors of 70 are 1, 2, 5, 7, 10, 14, 35, and 70.",
"For \\(cde = 71\\), the factors of 71 are just 1 and 71, since 71 is a prime number.",
"... | The possible sets of positive integers \((a, b, c, d, e, f, g)\) that satisfy the given equations are \((1, 70, 7, 10, 14, 5, 7)\), \((2, 35, 7, 4, 10, 14, 2)\), and \((7, 5, 10, 2, 7, 4, 14)\), which yield \(70\), \(71\), and \(72\) respectively. Since there are 3 valid combinations for each of the three equations, th... | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"To solve the problem, we start by analyzing the given equations:",
"1. \\( abc = 70 \\)",
"2. \\( cde = 71 \\)",
"3. \\( efg = 72 \\)",
"First, observe that each equation involves multiplying three numbers: \\( a \\), \\( b \\), \\( c \\), \\( d \\), \\( e \\), \\( f \\), \\( g \\). The products must be 70... | The number of \(7\)-tuples of positive integers \((a, b, c, d, e, f, g)\) satisfying \(abc = 70\), \(cde = 71\), and \(efg = 72\) is found by assigning prime factors to each variable, subject to constraints, which yields \(7^7\) combinations. The final answer is \(\boxed{7^7}\). | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"First, I need to find the prime factorization of 70, 71, and 72 to understand their prime components.",
"Next, I'll use these prime factors to identify possible combinations for each equation:",
"- For \\( abc = 70 \\), the prime factors are 2, 5, and 7.",
"- For \\( cde = 71 \\), the only possible prime fac... | The problem asks for the number of \(7\)-tuples of positive integers \((a, b, c, d, e, f, g)\) satisfying the equations \(abc = 70\), \(cde = 71\), and \(efg = 72\). The restrictions on the variables lead to \(c=1\), \(d=0\), and \(e \geq 4\), thus there are \(7\) possible tuples, each corresponding to a different prim... | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"First, I need to find the prime factors of the numbers 70, 71, and 72.",
"- 70 can be factored into 2 and 35, and 35 can be factored into 5 and 7.",
"- 71 is a prime number.",
"- 72 can be factored into 2, 2, 2, and 3.",
"Next, I'll list all possible combinations of these prime factors to form 7-tuples:",
... | The problem involves finding integer solutions to $abc = 70$, $cde = 71$, and $efg = 72$, where $a, b, c, d, e, f, g$ are positive integers. There are 4 valid 7-tuples that satisfy these equations. | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"First, I need to find the prime factorizations of 70, 71, and 72 to understand the constraints on the variables \\(a, b, c, d, e, f, g\\).",
"Next, I'll assign specific values to some variables based on the known factorizations. For example, if I choose \\(c = 71\\), then \\(d\\) and \\(e\\) must be factors of \... | The number of 7-tuples \((a, b, c, d, e, f, g)\) satisfying \(abc = 70\), \(cde = 71\), and \(efg = 72\) is 6. This is found by enumerating all possible prime factorizations for the products \(afg = 59\) and \(g = 9\), which yields 6 cases by listing the valid combinations for \(d, e, f, g\) and counting how many of th... | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"To solve the problem, we start by noting that each equation \\(abc = 70\\), \\(cde = 71\\), and \\(efg = 72\\) can be rewritten as \\((abc)c = 70\\), \\(c(cde) = 71\\), and \\(ce(efg) = 72\\). This suggests that the numbers \\(a, b, c, d, e, f, g\\) are consecutive integers starting from 1, governed by:",
"\\[",... | The equations follow from the property that each integer in a consecutive sequence of length seven must divide the previous one, with the final result being the number of permutations of {1, 2, 3, 4, 5, 6, 7}. Therefore, the number of valid 7-tuples is $7!$, which equals 5040. | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"First, I recognize that I need to find the number of 7-tuples of positive integers \\((a, b, c, d, e, f, g)\\) that satisfy the given equations:",
"1. \\(abc = 70\\)",
"2. \\(cde = 71\\)",
"3. \\(efg = 72\\)",
"Each of these equations involves multiplying three variables \\(a\\), \\(b\\), \\(c\\), \\(d\\),... | The problem is solved by considering the prime factorizations of the given products, determining the number of ways to choose prime factors for each equation, and then multiplying these together. Specifically, \(a = 1\), \(b = 1\), and \(d = e = f = g = 1\), resulting in \(a, b, c \text{ non-zero}, d, e, f, g \text{ no... | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"First, I need to find the prime factors of 70, 71, and 72.",
"70 can be factored into 2 and 65, and 65 can be factored into 5 and 13. So, 70 = 2 × 5 × 13.",
"71 is a prime number, so its only factors are 1 and 71.",
"72 can be factored into 2, 2, 2, 3, and 3. So, 72 = 2³ × 3².",
"Next, I need to ensure tha... | The number of \(7$-tuples\) is \(2\), coming from g=1 and g=70. g=1 allows for combinatons with any of \(a,b,c,d,e,f\) (3^7 ways), while g=70 allows a=1, b=1, c=14, d=13 (1 way). | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"To solve the given system of equations, we can analyze the problem by examining one specific digit and considering possible values for the remaining digits based on the constraints provided.",
"Starting with the equation \\( cde = 71 \\), we identify the possible values for \\( c \\). Since \\( c \\) must be a s... | From the given equations, we deduce that $c=7$, and using this, we derive that $d=3$, leaving $e=8$. The remaining digits $f=1$ and $g=2$, resulting in the unique solution $(7, 3, 8, 1, 2, 1, 2)$ and a total of 1 such 7-tuple. | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"First, I need to find the number of integer solutions to the system of equations:",
"- \\( abc = 70 \\)",
"- \\( cde = 71 \\)",
"- \\( efg = 72 \\)",
"Each equation can be broken into three separate equations by choosing a different digit (say, \\( c \\), \\( e \\), and \\( f \\)) to fill in the missing di... | The problem reduces to solving 3 equations for each digit, yielding 1 viable solution for \( a=1 \) (1, 71, 1), 2 viable solutions for \( e=2 \) (2, 72, 2), and 70 viable solutions for \( b \) (1 to 70). The total number of \( 7 \)-tuples is the product of these possibilities: \( 1 \times 70 \times 8 = 560 \). | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"First, I need to find the prime factors of the given numbers.",
"- 70 can be factored into 2 × 5 × 7.",
"- 71 is a prime number, so its only factor is 71.",
"- 72 can be factored into 2³ × 3².",
"Next, I'll determine the exponents of each prime factor for each variable:",
"- For 2 in the product abcd = 7... | The problem requires finding the number of distinct \(7\)-tuples of positive integers \((a, b, c, d, e, f, g)\) such that \(abc = 70\), \(cde = 71\), and \(efg = 72\). By prime factorizing these numbers, we get prime exponents for each variable, resulting in \(2^1 \times 3^3 \times 5^3 \times 7^3\), which yields 2160 d... | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | [
"First, I need to find the prime factors of 70, 71, and 72 to understand the range of possible values for each variable.",
"Next, I'll determine the possible values for each variable by dividing the prime factors into ranges that ensure each variable remains a positive integer.",
"After finding the smallest and... | The number of 7-tuples for the given system of equations is 5. These 7-tuples are $(1, 1, 1, 2, 3, 4, 5)$, $(1, 1, 2, 1, 3, 4, 5)$, $(1, 2, 1, 1, 3, 4, 5)$, $(2, 1, 1, 1, 3, 4, 5)$, and $(1, 3, 1, 2, 4, 5, 6)$. | 96 | false |
Open-Reasoner-Zero/orz_math_57k_collection | A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and th... | [
"To solve the problem, we start by understanding the geometry of the given right prism with hexagonal bases and a triangular pyramid formed by its vertices.",
"1. **Prism Geometry**:",
"- The prism has hexagonal bases with each side of the hexagon measuring 12 units. Hence, the base edge length is 12.",
"- Th... | The dihedral angle is given as \(60^\circ\), but this is a common angle and has no direct dependence on the prism's dimensions, making \(h^2\) un-determined. Therefore, the answer is \(\boxed{\text{Cannot be determined}}\). | 108 | false |
Open-Reasoner-Zero/orz_math_57k_collection | A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and th... | [
"First, I recognize that the prism has regular hexagons as its bases, each with a side length of 12. The height of the prism is denoted as \\( h \\).",
"The problem describes a triangular pyramid formed by the vertices \\( A \\), \\( B \\), and \\( C \\), with \\( A \\) at the apex. The triangular pyramid is defi... | The aspect ratio of the prism is \( q = \frac{12}{h} \). Using the dihedral angle \( \theta = 60^\circ = \arccos\left(\frac{\sqrt{3}}{2} \times \frac{12}{h}\right) \), we find \( h^2 = 432 \). | 108 | false |
Open-Reasoner-Zero/orz_math_57k_collection | A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and th... | [
"Okay, so I have this geometry problem here about a right prism. Let me try to understand what it's asking.",
"A right prism has bases that are regular hexagons with sides of length 12. So, each base is a regular hexagon, meaning all sides and angles are equal. The prism has height h. So, the total height of the ... | The dihedral angle is 60 degrees, and the hexagonal base has a side length of 12, making the height of the prism equal to 12. Therefore, $h^2$ is 144. | 108 | false |
Open-Reasoner-Zero/orz_math_57k_collection | A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and th... | [
"First, I need to determine the volume of the right prism. The base of the prism is a regular hexagon with a side length of 12. The volume of a regular hexagon is given by the formula \\( V = \\frac{126}{4} = 31.5 \\). Multiplying this by the height \\( h \\) gives the volume of the prism as \\( 31.5h \\).",
"Nex... | The volume of the prism is \(31.5 h\). Equating this to the lateral surface area, \(31.5 = 2 \pi r h\), we find \(r = 1\), and using \(V = \pi r^2 h\) with \(V = 31.5\), we solve for \(h\) as \(h^2 = \frac{31.5}{\sqrt{126}}\). | 108 | false |
Open-Reasoner-Zero/orz_math_57k_collection | A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and th... | [
"First, I need to understand the given information. The prism has regular hexagons as its bases, with each side measuring 12 units. A vertex \\( A \\) and two adjacent vertices form a triangular pyramid, meaning \\( A \\) is the apex of the pyramid.",
"The key concept here is dihedral angles. A dihedral angle is ... | The height \( h \) of the prism is equal to the radius of its hexagonal bases, calculated as \( R = \frac{3\sqrt{3}}{2} \). Squaring this value gives \( h^2 = \left(\frac{3\sqrt{3}}{2}\right)^2 = \frac{27}{4} \). | 108 | false |
Open-Reasoner-Zero/orz_math_57k_collection | A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and th... | [
"To find \\( h^2 \\), we start by analyzing the triangular pyramid formed by the prism's vertices \\( A \\), \\( B \\), and \\( C \\). The dihedral angle between the two triangular pyramid faces is given as \\( 60^\\circ \\).",
"1. **Determine the dihedral angle of the hexagonal base**:",
"- Each face of the he... | The dihedral angle of the hexagonal base is \( 120^\circ \), and the central angle is \( 180^\circ \). For a hexagonal prism with a dihedral angle of \( 60^\circ \) at a vertex where three vertices form a right triangle, the height \( h \) is determined by the relationship between the dihedral angle, central angle, and... | 108 | false |
Open-Reasoner-Zero/orz_math_57k_collection | A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and th... | [
"Okay, so I have this problem here about a right hexagonal prism. The height of the prism is h, and the base of the hexagons is a regular hexagon with each side measuring 12 units. Then, a vertex A and its three adjacent vertices form the base of a triangular pyramid. The dihedral angle, which is the angle between ... | The dihedral angle between two orthogonal planes of the hexagonal prism is 60 degrees. Therefore, \( h^2 = 6^2 = 36 \). | 108 | false |
Open-Reasoner-Zero/orz_math_57k_collection | A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and th... | [
"First, I need to understand the given information. We have a right prism with height \\( h \\), and the bases are regular hexagons with each side measuring 12 units. A vertex \\( A \\) of the prism and its three adjacent vertices form a triangular pyramid. The dihedral angle between two adjacent faces of the pyram... | The height \( h \) of the prism is \( 6\sqrt{3} \), and squaring this yields \( h^2 = 108 \). Thus, \( h^2 = 108 \). | 108 | true |
Open-Reasoner-Zero/orz_math_57k_collection | A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and th... | [
"First, I need to determine the dihedral angle between two adjacent faces of the pyramid. The triangular pyramid is formed by four vertices: the base vertex \\( A \\) and its three adjacent vertices, which means the base is an equilateral triangle with side length 12. The dihedral angle refers to the angle between ... | Since the pyramid's apex \(A\) lies outside the prism's prismatish base, the dihedral angle is measured as half the angle subtended by the side of length 12 — found to be \(\frac{\pi}{3}\), giving a prism volume of \(\frac{2\pi h^2}{3}\) and base area of \(72\), so \(6h^2 = 144\). Thus \(\boxed{h^2 = 24}\). | 108 | false |
Open-Reasoner-Zero/orz_math_57k_collection | A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and th... | [
"First, I recognize that the prism has six regular hexagons as its bases, each with a side length of 12. The space diagonal of one hexagonal base can be calculated using the formula for the space diagonal of a regular n-gon, which is \\( \\sqrt{n(n + 2)} \\), where \\( n = 6 \\). This gives the space diagonal as \\... | The space diagonal of a hexagonal base is \( D = \sqrt{72} \). For a triangular pyramid formed by vertices \( A \), \( B \), and \( C \) with dihedral angle \( 60^\circ \), \( h^2 \) is calculated to be \( 144 \). | 108 | false |
Open-Reasoner-Zero/orz_math_57k_collection | A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and th... | [
"First, I need to understand the given information and visualize the problem. We have a right prism with a regular hexagonal base having sides of length 12. A vertex \\( A \\) of the prism and two of its adjacent vertices form a triangular pyramid. The dihedral angle between the two faces of the triangular pyramid ... | The problem contains an inconsistency, as the dihedral angle of 60 degrees contradicts the standard geometric relationships between the triangular and square bases of the pyramid. Therefore, it is impossible to determine \( h^2 \) based on the given information. | 108 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I need to understand the given relationships between the variables \\( x \\), \\( y \\), \\( z \\), and \\( w \\). The problem states that all three variables \\( x \\), \\( y \\), and \\( z \\) exceed 1, which means they are all greater than 1. Additionally, \\( w \\) is given to be a positive number, and ... | Given \( \log_x w = 24 \), \( \log_y w = 40 \), and \( \log_{xyz} w = 12 \), we find \( \log_z w = 12 \). This is because \( \log_{xyz} w \) simplifies to \( \frac{1}{3} (\log_x w + \log_y w) \), so substituting the given values yields \( \frac{1}{3} (24 + 40) = \frac{64}{12} = \frac{16}{3} \), which is greater than 24... | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I recognize that all variables \\( x \\), \\( y \\), and \\( z \\) exceed 1, which means they are greater than 1 but not exceeding 2. Additionally, \\( w \\) is given as a positive number.",
"Given that \\( \\log_x w = 24 \\), \\( \\log_y w = 40 \\), and \\( \\log_{xyz} w = 12 \\), I need to find \\( \\lo... | Using the change of base formula, we found \( w = \frac{1}{81} \), and then calculated \( \log_z w = \frac{961}{15} \). | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I recognize that \\( w \\) is a common logarithm base for all four variables: \\( x \\), \\( y \\), \\( z \\), and \\( w \\). This means that each logarithm can be rewritten in terms of \\( w \\).",
"Given:",
"\\[",
"\\log_x w = 24 \\quad \\Rightarrow \\quad x = w^{24}",
"\\]",
"\\[",
"\\log_y w... | Given \( \log_x w = 24 \), \( \log_y w = 40 \), and \( \log_{xyz} w = 12 \), we express \( x \), \( y \), and \( z \) in terms of \( w \): \( x = w^{24} \), \( y = w^{40} \), and \( z = \frac{w^{12}}{w^{24}} = w^{-12} \). Therefore, \( \log_z w = \log_w^{-12} w = -12 \). | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I recognize that the problem involves logarithms and the relationships between the variables \\( x \\), \\( y \\), \\( z \\), and \\( w \\). I'll start by recalling the fundamental properties of logarithms, particularly their exponents.",
"Given that \\( \\log_x w = 24 \\), this implies that \\( x^24 = w ... | Given \( \log_x w = 24 \), \( \log_y w = 40 \), and \( \log_{xyz} w = 12 \), we find that \( \log_x \) and \( \log_y \) are equal. Substituting their values into the expression for \( \log_z w \) using the change of base formula and the relationship between \( \log_x \) and \( \log_y \), we get \( \log_z w = \frac{24 \... | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I need to find the value of \\( w \\) using the given logarithmic expressions.",
"From \\( \\log_x w = 24 \\), this implies \\( x^24 = w \\).",
"Similarly, from \\( \\log_y w = 40 \\), it follows that \\( y^{40} = w \\).",
"Additionally, from \\( \\log_{xyz} w = 12 \\), I know that \\( w = (x y z)^12 ... | Given \(\log_x w = 24\), \(\log_y w = 40\), and \(\log_{xyz} w = 12\), we find \(x^24 = y^{40} = (x y z)^{12}\), which implies \(y^{24} = x^{24} = (y x z)^{6}\). Since \(w = x y z\), taking the 6th root gives \(w = (x y z)^{12} = x^{12} y^{12} z^{12}\), leading to \(\log_z w = 12\). | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I recognize that all variables \\( x, y, \\) and \\( z \\) exceed 1, and \\( w \\) is a positive number with \\( \\log_x w = 24 \\), \\( \\log_y w = 40 \\), and \\( \\log_{xyz} w = 12 \\).",
"To find \\( \\log_z w \\), I'll start by expressing \\( w \\) in terms of \\( x \\), \\( y \\), and \\( z \\) usin... | We found $x=1, y=3, z=2$. This leads to $\log_z w = \log_{10^3 3} 10^{64} \times 3^{-39}$. | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I recognize that \\( w \\) can be expressed in terms of the logarithms of \\( x \\), \\( y \\), and \\( z \\) with their respective exponents. Specifically, \\( w = \\log_x w = 24 \\), \\( w = \\log_y w = 40 \\), and \\( w = \\log_{xyz} w = 12 \\).",
"Next, I convert each logarithmic equation into its exp... | Since \(x,y,z\) all exceed 1, \(w = x^{24}y^{40}\) satisfies \(\log_x w = 24\), \(\log_y w = 40\), and \(\log_{xyz} w = 12\), making \(x\), \(y\), \(z\), equal and equal to \(w\). Thus \(\log_z w = 24 + \frac{4}{3}\log_{xyz} z\). | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I'll express \\( w \\) in terms of \\( x \\), \\( y \\), and \\( z \\) using the given logarithms. Specifically, since \\( \\log_x w = 24 \\), this means \\( w = x^{24} \\). Similarly, \\( \\log_y w = 40 \\) implies \\( w = y^{40} \\), and \\( \\log_{xyz} w = 12 \\) means \\( w = \\frac{1}{x^2 y^2 z} \\).",... | With \(w = x^{22}y^{36}\), the change of base formula gives \(\log_z w = \frac{\log_x w + \log_y w}{20 + 36} = \frac{36}{56}\). Thus \(\log_z w = \frac{42}{56}\). | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I recognize that all variables \\( x \\), \\( y \\), and \\( z \\) exceed 1, and \\( w \\) is positive. This means each logarithm is less than 1, so the arguments must be less than 1 as well.",
"Next, I use the given logarithms to set up equations involving \\( w \\) and \\( z \\):",
"\\[",
"\\log_x w... | Given \( \log_x w = 24 \), \( \log_y w = 40 \), \( \log_{xyz} w = 12 \), and \( x, y, z \) all exceed 1, we found \( y = w^{40} \), \( z = w^{72} \), and \( w = s^{48} \) where \( s = \frac{xyz}{x} = w^{48} \). Therefore, \( \log_z w = \frac{\log_w w}{72} = \frac{1}{72} \). | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, we are given that \\( w = \\log_x y = \\log_y z \\) and \\( w = \\log_{xy} z \\). This implies that \\( xyz = yz \\) and \\( xzy = zy \\). By rearranging these equations, we find that \\( xz = y \\).",
"Next, using \\( \\log_{xyz} w = 12 \\), we can rewrite this as \\( (xyz)^12 = w \\). Since we know \\( ... | Using \(\log_x w = 24\) and \(\log_y w = 40\), we get \(\log_z w = \log_1 x = 24\). Thus \(\log_z w = \boxed{24}\). | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I recognize that \\( w \\) is expressed in terms of logarithms with base \\( x \\), \\( y \\), and \\( z \\). To find \\( \\log_z w \\), I can use the property of logarithms that allows me to change the base of the logarithm.",
"I know that:",
"\\[",
"\\log_x w = 24 \\implies x^{24} = w",
"\\]",
"... | Using the change of base formula and substitution with $w=x^{24}y^{40}(xyz)^{12}$, we deduce $\log_z w = \frac{24+40-12}{24} \log_{xyz}(x^2y^{40}) = \frac{6}{24} \log_{xyz}(x^2y^{40}) = \frac{1}{4} \log_{xyz}(x^2y^{40})$. The final answer is $\boxed{\frac{1}{4}}$. | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I recognize that \\( w = \\log_x y = 24 \\) and \\( w = \\log_y z = 40 \\). This means that \\( x \\) raised to the power of 24 equals \\( y \\), and \\( y \\) raised to the power of 40 equals \\( z \\).",
"To find the value of \\( z \\), I substitute these expressions into the third equation, which state... | Given \( \log_x w = 24 \), \( \log_y w = 40 \), and \( \log_{xyz} w = 12 \) with \( x, y, z > 1 \), substituting \( w = x^{24} \) and \( w = y^{40} \) into the third equation yields \( \log_{xyz} (x^{24}) = 12 \), which simplifies to \( x^{24} = x^{40} \), implying \( x=1 \). Since \( y=x^{40}, z=y^{40} \), and \( w=x^... | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I need to understand the relationships given in the problem. We have three variables, \\( x \\), \\( y \\), and \\( z \\), all greater than 1, and \\( w \\) is a positive number. The problem provides three logarithm equations:",
"1. \\( \\log_x w = 24 \\)",
"2. \\( \\log_y w = 40 \\)",
"3. \\( \\log_{... | Given \( \log_x w = 24 \), \( \log_y w = 40 \), and \( \log_{xyz} w = 12 \), we find \( w = \sqrt[3]{xyz} \). Therefore, \( \log_z w = \log_{xyz} \sqrt[3]{xyz} = 1 \). | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I recognize that \\( \\log_x w = 24 \\) and \\( \\log_y w = 40 \\) imply \\( x^{24} = w \\) and \\( y^{40} = w \\).",
"Next, since \\( w = \\log_{xyz} w \\), the product \\( x \\cdot y \\cdot z = w \\).",
"Using the values of \\( x \\) and \\( y \\), I substitute into the equation:",
"\\[",
"x \\cdo... | Given $\log_x w = 24, \log_y w = 40, \log_{xyz} w = 12$, we found $x, y, z > 1$. By equating $x^{24} = y^{40} = zw$ and noting $w = yz$, we derived $x^{24} = z$, implying $\log_z w = \log_z x^{24} = 24$. | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I need to understand the given information and what is being asked. We are given three variables, \\( x \\), \\( y \\), and \\( z \\), each greater than 1, and a positive number \\( w \\). The goal is to find \\( \\log_z w \\).",
"I know that logarithms can change when the base and the argument are swappe... | Since \(x^{24} = y^{40} = z^12\), we have \(\log_z w = \frac{1}{40} \cdot \log_x y = \frac{1}{24 y}\). Thus \(\boxed{\log_z w = \frac{1}{24y}}\). | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | [
"First, I need to find the values of \\( x \\), \\( y \\), and \\( z \\) using the given logarithms. Since each logarithm is base \\( x \\), I can express these values in terms of \\( x \\) alone.",
"Next, I recognize that \\( \\log_{xyz} w = 12 \\) can be rewritten using the change of base formula to \\( \\frac{... | Given \( \log_x w = 24 \) and \( \log_y w = 40 \), and knowing that \( \log_{xyz} w = \frac{4}{5} \), we find \( x = 3 \), \( y = 9 \), and \( z = \frac{s}{t} \), where \( s = \log_x w \). Substituting these values, we calculate \( \log_z w = \log_y z \cdot \log_z x = \frac{\log_x w}{\log_y x} = \frac{40}{24} = \frac{5... | 60 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, convert the time from the former time format \\(\\text{6:36}\\) AM to the new metric system.",
"- Convert 6 hours to metric hours: \\(6 \\times 10 = 60\\) metric hours.",
"- Subtract 6AM to find the hour part of the metric time: \\(60 - 6 = 54\\).",
"- Convert 36 minutes to metric minutes: \\(36 \\tim... | The metric time equivalent of 6:36 AM is $414:BC$. Computing $100A + 10B + C$ for $414:BC$ with $A=4$ and $B=1$, we find $100(4) + 10(1) + 4 = 414$. | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, I need to convert the times from the old system to the metric system. This involves converting hours to metric hours and then metric hours to metric minutes.",
"For the time just before midnight:",
"- 9 hours + 99 minutes (since 9 hours is 90 metric hours, and 99 minutes is 99 metric minutes)",
"- Aft... | The time to be awakened at \(6:36\) AM is equivalent to 286 metric minutes past midnight. Converting this to hours and minutes gives 4 hours and 46 minutes, which corresponds to \(A=4, B=46, C=0\), resulting in a sum of \(400 + 46 + 0 = 446\). | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, I need to convert the times from the original system to the metric system.",
"1. **9:99 AM**:",
"- Convert minutes to metric hours: 99 minutes ÷ 100 = 0.99 metric hours.",
"- Convert the decimal part to metric minutes: 0.99 × 100 = 99 metric minutes.",
"- Thus, 9:99 AM is equivalent to 10:19 metric ... | The given times after conversion to metric hours are \(10:19\) AM, \(0:00\) AM, \(1:25\) AM, \(8:00\) AM, and the target wake-up time is \(6:36\) AM, which converts to \(6:36\) metric hours. Therefore, \(100A + 10B + C = 636\). | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, I need to convert the time from the original format to the new metric system. The original time is 6:36 AM, which is 6 metric hours, 36 metric minutes.",
"In the metric system, each metric hour has 100 metric minutes. Therefore, 36 metric minutes is equivalent to 0.36 metric hours. Adding this to the 6 me... | The new time is 6.36 metric hours, which equals 636 metric minutes. To produce a digital clock reading equivalent to 6:36 AM, the correct time is 9:17 AM, requiring the time set to be 100A + 10B + C = 636, yielding A=2, B=4, C=6, and a sum of 246. | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, I need to convert the former time of 6:36 AM to the new metric time system. In the metric system, each day consists of 10 metric hours, and each metric hour has 100 metric minutes.",
"1. **Convert the hours:**",
"- 6:36 AM in the former system is 6 hours and 36 minutes.",
"- Convert the hours to metri... | Converting 6:36 AM (0.6 h, 36 min) to metric time gives 0:03:60, and setting your alarm 100 metric seconds before midnight gives 0:07:58, so \(A = 0\), \(B = 7\), \(C = 8\), and \(100A + 10B + C = 78\). The final answer is \(\boxed{78}\). | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, I need to convert the former times to metric hours and minutes.",
"For 6:36 AM in the former system:",
"- Subtract 5 full metric hours (since 6 AM is the start of the day) to get 0 metric hours.",
"- Subtract 36 metric minutes to get 66 metric minutes remaining.",
"Next, I'll convert 1:25 PM in the ... | The former time of 1:25 PM is equivalent to 3:25 in the metric system. In the metric system, setting an alarm for 100A + 10B + C means setting it for 100A + 10B + C = 990, and 100A + 10B + C = 990 for the former 1:25 AM. | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, I need to determine the new time equivalent to the former 6:36 AM in the metric system.",
"In the metric system:",
"- 1 metric hour equals 100 metric minutes.",
"- Therefore, 6:36 AM is equivalent to 6 hours and 36 minutes.",
"Converting 36 minutes to metric hours:",
"\\[",
"36 \\text{ minutes} ... | 6:36 AM is 6.36 metric hours; the conversion to metric time is 600 + 36 = 636 hours, giving 00000636 in the A:BC format. The sum is 100×0 + 10×0 + 36 = 36. | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, I need to understand the new metric time system. In this system, each hour is divided into 10 metric hours, and each metric hour into 100 metric minutes.",
"Given that a day has 10 metric hours, each metric hour has 100 metric minutes. This means:",
"- 1 metric hour = 100 metric minutes",
"- 1 metric ... | The metric equivalent of 6:36 AM is 3 hours 60 minutes, which corresponds to 15:60 in 24-hour format, with $A=1$, $B=6$, $C=0$. Therefore, $100A + 10B + C = 100(1) + 10(6) + 0 = 100 + 60 + 0 = 160$. | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, convert the time from the converted metric system to hours, minutes, and seconds.",
"- Midnight in the metric system is 0:00.",
"- Midnight is equivalent to 0:00 in the old system.",
"- 3:00 AM in the metric system is 3:00.",
"- 6:36 PM in the metric system is 18:36.",
"Next, determine the time di... | The time difference between 9:99 AM and 18:36 PM in the metric system is \(11:40\). Therefore, \(A=11, B=40, C=1\), and the total sum is \(100 \times 11 + 10 \times 40 + 1 = 1500 + 40 = 1540\). | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"To determine the time equivalent of 6:36 AM in the new metric system:",
"1. **Convert the hours and minutes:**",
"- 6 hours become 60 metric hours.",
"- 36 minutes become 3600 metric minutes.",
"2. **Add the time:**",
"- 60 metric hours + 3600 metric minutes = 9600 metric minutes.",
"3. **Convert minut... | To find the time equivalent of 6:36 AM under the new metric system, we convert 6:36 AM to 9600 metric minutes, which equals 96 hours of metric time. This metric time is equivalent to 9 hours and 60 minutes, so the digits are A=9, B=6, and C=0, resulting in \(100A + 10B + C = 100(9) + 10(6) + 0 = 966\). | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, convert the time from 6:36 AM in the old system to the new metric system.",
"6 hours in the new system correspond to 60 metric hours.",
"36 minutes in the new system are equivalent to 3600 metric seconds.",
"Convert the total time to seconds:",
"60 hours \\times 100 minutes/hour = 6000 metric minute... | The alarm is set for 1:36 AM after converting 6:36 AM to 1:36 metric time. The time is then converted to 1:06 metric time, and \(100A + 10B + C\) evaluates to \(100(1) + 10(0) + 6 = 116\). | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, I need to convert the time from the former time to the new time system. The former time is 6:36 AM, which corresponds to 6 metric hours, 36 metric minutes.",
"Next, I'll convert the decimal part of the metric minutes to metric hours. Since there are 100 metric minutes in an hour, 36 metric minutes is equi... | The former time 6:36 AM is converted to the new system as 6.36 metric hours, which is 6:21:6 in decimal minutes. Therefore, \(100A + 10B + C = 100(0) + 10(2) + 6 = 26\). | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, I need to convert the time from the former time frame to the new metric system. The former time is 6:36 AM, which is 636 metric hours and 36 metric minutes.",
"Next, I'll convert the metric minutes to metric hours. Since there are 100 metric minutes in an hour, 36 minutes is equivalent to 0.36 hours.",
... | The new time is 9:99, 0:00, 1:25, 7:50, corresponding to 10 metric hours, 99 metric minutes, 125 metric minutes, and 75 metric minutes respectively, all of which are less than 97 metric hours (5:57 AM 0:00), giving A=5, B=7, C=0; thus 100A+10B+C = 507. | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, I need to understand the new time system. Each day is divided into 10 metric hours, and each hour into 100 metric minutes.",
"Next, I'll convert the given times according to this system:",
"- 9:99 AM becomes 10:00 AM (since 999 minutes is exactly 10 hours)",
"- 6:36 AM converts to 7:00 PM (96 minutes ... | The final time equivalent to 6:36 AM in the new metric system is 3:36 AM, represented as \(3:36\). The sum of the digits for this time is \(3 + 3 + 6 = 12\). | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, I need to understand the new time system after the conversion. In the metric system, each day consists of 10 metric hours, and each metric hour has 100 metric minutes. This means each metric hour is composed of 1000 metric minutes.",
"Next, I'll convert the given times to metric minutes to make them easie... | The problem converts 6:36 AM to 0 hours and 6 minutes for a new time system where 10 metric hours = 1000 metric minutes. With \(A=0\), \(B=6\), and \(C=0\), the sum \(100A + 10B + C\) is 60. | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"First, I need to convert the given times from the metric system to the normal clock system.",
"1. **Converting 6:36 AM to Clock Time**:",
"- Metric Time is 6 hours and 36 minutes.",
"- To convert to Clock Time:",
"- Subtract 12 hours from the hours (since 6 hours is already before midnight).",
"- Convert... | The desired wake-up time of six hours and thirty-six minutes metric time converts to one thousand, three hundred, sixty, and thirty-six metric minutes in normal time. The sum \(100(1) + 10(36) + 36\) equals 396. | 275 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | [
"First, I need to determine the total number of ways to select two squares from an \\( n \\times n \\) grid. This can be calculated using the combination formula:",
"\\[",
"\\binom{n}{2} = \\frac{n(n-1)}{2}",
"\\]",
"Next, I'll identify the number of ways the two selected squares can be horizontally or vert... | The probability of selecting two horizontally or vertically adjacent unit squares from an $n \times n$ grid is $\frac{8}{n-1}$. We require this probability to be less than $\frac{1}{2015}$, which leads to $16120 < n-1$, thus $n > 16121$, making the least positive integer value for $n$ equal to $16122$. | 90 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | [
"To determine the least positive integer \\( n \\) such that the probability of selecting two horizontally or vertically adjacent unit squares is less than \\( \\frac{1}{2015} \\), we can follow these steps:",
"1. **Total number of squares:**",
"Each \\( n \\times n \\) grid contains \\( n^2 \\) unit squares.",... | The least positive integer \( n \) is 2, as determined by the inequality \( (n - 1)^2 < \frac{1}{2} \). This simplifies to \( n < 1 + \frac{1}{\sqrt{2}} \approx 1.713 \), making \( n=2 \) the smallest integer satisfying the condition. | 90 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | [
"First, I need to determine the total number of ways to select two unit squares from an \\( n \\times n \\) grid without replacement. This can be calculated using the combination formula, which is \\( \\binom{n}{2} \\).",
"Next, I'll consider the squares that are horizontally adjacent. In a grid, each row will ha... | For two unit squares selected at random from an \( n \times n \) grid to be horizontally or vertically adjacent, we require \( \binom{n}{2} \times (n+n) = n(n-1) \) adjacent squares. Solving \( n(n-1) < \frac{1}{2015} \) yields \( n^2 - n - 2015 < 0 \), which factors as \( (n-42)(n+45) < 0 \), and since \( n \) must be... | 90 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | [
"To determine the least positive integer \\( n \\) such that the probability of selecting two horizontally or vertically adjacent unit squares is less than \\( \\frac{1}{2015} \\), we can follow these steps:",
"1. **Total Number of Squares:**",
"An \\( n \\times n \\) grid has \\( n^2 \\) unit squares.",
"2. ... | The probability of selecting two horizontally or vertically adjacent unit squares is \( \frac{1}{n^2 - 1} \), and we seek the least positive integer \( n \) where this probability is less than \( \frac{1}{2015} \). Solving \( \frac{1}{n^2 - 1} < \frac{1}{2015} \) yields \( n > \sqrt{2015} \approx 44.86 \), so the least... | 90 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | [
"To determine the least positive integer \\( n \\) such that the probability of selecting two horizontally or vertically adjacent unit squares is less than \\( \\frac{1}{2015} \\), we start by calculating the total number of ways to select two unit squares from an \\( n \\times n \\) grid. This is given by the comb... | The least positive integer \( n \) such that the probability of selecting two horizontally or vertically adjacent unit squares is less than \( \frac{1}{2015} \) is 10. This is found by setting up the inequality \( \frac{2(n-1)}{\binom{n}{2}} < \frac{1}{2015} \) and solving for \( n \). | 90 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | [
"To determine the least positive integer \\( n \\) such that the probability of selecting two horizontally or vertically adjacent unit squares is less than \\( \\frac{1}{2015} \\), we start by calculating the probability.",
"The total number of ways to choose 2 unit squares from an \\( n \\times n \\) grid is \\(... | The condition that the probability of selecting two horizontally or vertically adjacent unit squares is less than \( \frac{1}{2015} \) leads to the inequality \( n^2 - 2015n + 2015 < 0 \). Solving this quadratic inequality, we find that \( n \) must be a positive integer less than \( 2016 \), leading to the least posit... | 90 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | [
"To determine the least positive integer \\( n \\) such that the probability of selecting two horizontally or vertically adjacent unit squares from an \\( n \\times n \\) grid is less than \\( \\frac{1}{2015} \\), we can follow these steps:",
"1. **Total Number of Squares:** The total number of unit squares in an... | The probability of selecting two adjacent unit squares is \( \frac{1}{4} \). To satisfy \( \frac{1}{4} < \frac{1}{2015} \), we must have \( n > 2015 \), meaning the smallest integer value for \( n \) is 2016. | 90 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | [
"First, I need to determine the total number of ways to select two unit squares from an \\( n \\times n \\) grid. This is simply \\( \\binom{n}{2} = \\frac{n(n-1)}{2} \\).",
"Next, I'll calculate the number of ways the two selected squares can be horizontally adjacent. Since the grid is symmetric, each row contri... | The probability that two randomly selected unit squares are adjacent is \( 4(n-1) \), so we need \( 4(n-1) < \frac{1}{2015} \). Solving for \( n \) gives \( n > \frac{3017}{402.775} \), and since \( n \) must be an integer, the least positive integer \( n \) is 2. | 90 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | [
"First, I need to determine the least positive integer \\( n \\) such that the probability of selecting two horizontally or vertically adjacent unit squares when choosing two squares at random without replacement is less than \\( \\frac{1}{2015} \\).",
"1. **Total Number of Squares and Grid Size**:",
"- The tot... | The least positive integer \( n \) such that the probability of selecting two horizontally or vertically adjacent unit squares is less than \( \frac{1}{2015} \) is \( 2014 \). This is found by solving the inequality \( 2(n - 1) < \frac{n^2}{2015} \), which simplifies to \( n > 2014 \). | 90 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | [
"To determine the least positive integer \\( n \\) such that the probability of selecting two horizontally or vertically adjacent unit squares is less than \\( \\frac{1}{2015} \\), we can follow these steps:",
"1. **Total Number of Squares:**",
"An \\( n \\times n \\) grid contains \\( n^2 \\) unit squares.",
... | The least positive integer \( n \) such that the probability of selecting two horizontally or vertically adjacent unit squares is less than \( \frac{1}{2015} \) is \( 56 \), as determined by solving the inequality \( \frac{\binom{n^2}{2}}{n^2} < \frac{1}{2015} \). This inequality simplifies to \( n^2 < 3031 \), yieldin... | 90 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | [
"To determine the least positive integer \\( n \\) such that the probability that two randomly selected unit squares are horizontally or vertically adjacent is less than \\( \\frac{1}{2015} \\), we can follow these steps:",
"1. **Total Number of Unit Squares:**",
"- Each \\( n \\times n \\) grid contains \\( n^... | The least positive integer \( n \) is 15, as the inequality \( n^4 - n - \frac{1}{2015} - n > 0 \) simplifies to \( n^4 - \frac{1}{2015} > n \), which holds true for \( n=15 \). This ensures the probability of selecting two squares horizontally or vertically adjacent is less than \( \frac{1}{2015} \). | 90 | false |
Open-Reasoner-Zero/orz_math_57k_collection | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | [
"First, we need to determine the total number of ways to select two unit squares from an \\( n \\times n \\) grid. This is simply given by the combination formula \\( \\binom{n}{2} \\).",
"Next, we need to count the number of favorable pairs where the selected squares are horizontally or vertically adjacent. This... | The probability of selecting two horizontally or vertically adjacent unit squares from an \( n \times n \) grid is \( \frac{5n}{n^2} = \frac{5}{n} \). Solving \( \frac{5}{n} < \frac{1}{2015} \) yields \( n > 5/2015 \), and the least positive integer \( n \) is 6. | 90 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$. The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s. Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die rep... | [
"First, I need to understand the problem. We have a triangle ABC with vertices at (0,0), (0,420), and (560,0). Rolling a die labeled with A, B, or C, and noting the outcome, we use this information to determine the midpoints of segments defined by these points, specifically creating points P₁ through P₇. We are giv... | The coordinates of \( P_1 \) are \( (11.0, 20.5) \), so \( k = 11 \) and \( m = 20.5 \). Therefore, \( k + m = 11 + 20.5 = 31.5 \). | 344 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$. The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s. Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die rep... | [
"To determine the values of \\( k \\) and \\( m \\), we analyze the motion of point \\( P_1 \\) under the given rule.",
"1. **Initial Position of \\( P_1 \\):**",
"- Given \\( P_1 = (k, m) \\), we need to determine the conditions that lead to the next position \\( P_2 \\).",
"2. **Rule Applied to \\( P_1 \\):... | The given point \( P_7 = (14, 92) \) implies that the \( x \)-coordinate is twice the \( y \)-coordinate, which is a characteristic of applying the rule for label \( A \) repeatedly. Therefore, \( k=14 \) and \( m=92 \), yielding \( k + m = 14 + 92 = 106 \). | 344 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$. The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s. Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die rep... | [
"First, I need to understand how the points \\( P_2, P_3, P_4, \\dots \\) are determined based on the die roll. Each roll results in a point labeled with a letter, and the current point follows the midpoint rule. If the die shows \\( C \\), the previous point is \\( P_7 = (14, 92) \\).",
"To find the starting poi... | The midpoint rule applied to the die roll of \( C \) yields \( P_1 = (287, 46) \). Given \( P_1 = (k, m) \), the sum \( k + m \) is \( 287 + 46 = 333 \). | 344 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$. The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s. Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die rep... | [
"First, I need to understand how the points \\( P_1 \\) through \\( P_7 \\) are determined based on rolling a die and the midpoint rule. The die has three sides: A, B, and C, each labeled twice. Each roll \\( L \\) in \\(\\{A, B, C\\}\\) indicates the next midpoint in the sequence.",
"Starting with \\( P_1 = (0,0... | The journey from \( P_1 = (0,0) \) to \( P_7 = (14,92) \) via \( P_2=(14,92) \), \( P_3=(14,24) \), \(\dots\), \( P_7=(14,92) \) shows \( k=0 \) due to frequent returns to \( (0,y) \), and \( m=92 \). Thus \( k+m = 0+92 = \boxed{92} \). | 344 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$. The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s. Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die rep... | [
"First, I recognize that the die roll determines the label \\( L \\) of the face \\( L \\in \\{A, B, C\\} \\).",
"Given that \\( P_7 = (14, 92) \\), I know that \\( L = C \\) because it's the last point rolled before \\( P_7 \\).",
"Since \\( C \\) is the point (560, 0), the next step is to find the midpoint of... | The label of the last face is C, so \( P_7 = (560, 0) \). The midpoint of \( P_7 \) is \( P_{7+1} = \left( \frac{k+560}{2}, \frac{m}{2} \right) \), giving \( k = -532 \) and \( m = 184 \), so \( k + m = -532 + 184 = -348 \). | 344 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$. The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s. Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die rep... | [
"First, I need to understand the layout of the die and the points \\( P_1 \\) through \\( P_7 \\). The die has labels \\( A \\), \\( B \\), and \\( C \\) representing the x-coordinates of the vertices \\( A = (0,0) \\), \\( B = (0,420) \\), and \\( C = (560,0) \\). The interior of triangle \\( ABC \\) is divided in... | The coordinates for \( P_1 \) through \( P_7 \) are given as \( (k, m) \). The sum \( k + m \) is found to be 2983. | 344 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$. The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s. Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die rep... | [
"First, I need to understand how the die rolling process works. The die has labels \\( A \\), \\( B \\), and \\( C \\), and each roll corresponds to one of these labels. For each roll, the rule is to find the midpoint of the segment connecting the current point \\( P_n \\) and the label's corresponding point on the... | After the die roll (A) moved point (0, 210), the next roll (B) moved it to (280, 105), and the roll (C) moved it to (280, 52.5). The sum \( k + m \) of the coordinates after these movements is \( 280 + 52.5 = 332.5 \). | 344 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$. The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s. Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die rep... | [
"To solve the problem, I need to determine the coordinates of point \\( P_1 \\) by analyzing the movement from \\( P_7 \\) to \\( P_1 \\) under the given rule.",
"First, I observe that each roll of the die determines the movement along the \\( x \\)-axis. If the die shows an \\( A \\), we move 560 units to the ri... | The movement from \( P_7 = (14, 92) \) to \( P_1 \) is \( (14, -450) \), obtained by moving right then right, then up then left. The sum \( k + m \) is \( 14 + (-450) = -436 \). | 344 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$. The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s. Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die rep... | [
"First, I need to understand how the die works. The die has labels A, B, and C, and each roll shows one of these labels with equal probability. If the die shows A, B, or C, the next point is the midpoint of the segment connecting the most recent point and the label's center.",
"Given that point \\( P_7 = (14, 92)... | The x-coordinate changes from \(k\) to \(-7\) as we move from \(P_6\) to \(P_7\), and the y-coordinate changes from \(m\) to \(92\) as \(P_6 \to P_7\). Therefore, \(k = -7\) and \(m = 369\), yielding \(k + m = -7 + 369 = 362\). | 344 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$. The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s. Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die rep... | [
"First, I need to understand the setup of the problem. We have a triangle \\( \\triangle ABC \\) with vertices at \\( A(0,0) \\), \\( B(0,420) \\), and \\( C(560,0) \\). The die is labeled with labels \\( A \\), \\( B \\), and \\( C \\), and when rolled, each label will be shown twice. The rule is to move towards t... | The movement pattern implies that the y-coordinate is constantly 92 and the x-coordinate alternates between 0 and 14, indicating \( (14, 92) \) is the final position. Therefore, \( k + m = 14 + 92 = 106 \). | 344 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$. The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s. Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die rep... | [
"First, I need to understand the problem and identify the key components. The triangle ABC has vertices at (0,0), (0,420), and (560,0). The die has labels A, B, and C, and when rolled, the probability of each die roll is proportional to the area of the respective triangle segment.",
"Given that P₇ = (14, 92), I n... | The die roll of \(14,92\) corresponds to the point \(P_7 = (14, 92)\), and after backtracking, we find \(P_6 = (6.5, 594)\). This provides the values \(k=6.5\) and \(m=594\) for the point \(P_1=(k,m)\), leading to \(k+m=1249.5\). | 344 | false |
Open-Reasoner-Zero/orz_math_57k_collection | The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"Okay, so I have this logarithmic equation to solve: log base 2^x of 3^20 equals log base 2^{x+3} of 3^2020. Hmm, let me try to figure this out step by step.",
"First, I remember that logarithms can be tricky, but there's a change of base formula that might help here. The formula says that log_b a = log_c a / log... | Equating the bases automatically gives \(3^{20}=3^{2020}\), so \(x=\frac{1}{1}=1\). Thus the value is \(m+n=\boxed{2}\). | 103 | false |
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