idx int64 0 61 | prompt stringlengths 868 2.58k | answers listlengths 1 1 | gt stringlengths 1 53 |
|---|---|---|---|
28 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"A $90^\\circ$ rotation around the origin in the counter-clockwise direction is equivalent to multiplying the complex number by $i$. So, if we have a complex number $z = 7 + 2i$, the resulting complex number after a $90^\\circ$ rotation will be $i \\cdot (7 + 2i)$. Let's calculate this step by step.\n\n1. Distribut... | -2 + 7i |
9 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To convert the binary number \\(10101001110_2\\) to its base eight (octal) equivalent, we can follow these steps:\n\n1. **Group the binary number into sets of three digits, starting from the right.** Since each octal digit represents three binary digits, this makes the conversion straightforward.\n2. **Convert eac... | 2516_8 |
19 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to find the inverse function values \\( f^{-1}(0) \\) and \\( f^{-1}(6) \\) for the piecewise function \\( f(x) \\):\n\n\\[\nf(x) = \\left\\{\n\\begin{array}{cl}\n3-x & \\text{if } x \\leq 3, \\\\\n-x^3 + 2x^2 + 3x & \\text{if } x > 3.\n\\end{array}\n\\right.\n\\]\n\nWe'll follow thes... | 0 |
39 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the time it takes for both gears to align their marks back to the north position simultaneously. This is equivalent to finding the least common multiple (LCM) of the periods of the two gears.\n\n1. The first gear turns \\(33\\frac{1}{3}\\) times per minute, which is \\(\\frac... | 36 |
33 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the value of \\( y \\), we need to use the formula for the projection of one vector onto another. The projection of vector \\(\\mathbf{a}\\) onto vector \\(\\mathbf{b}\\) is given by:\n\\[\n\\text{proj}_{\\mathbf{b}} \\mathbf{a} = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\mathbf{b} \\cdot \\mathbf{b}} \\mat... | -4 |
58 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to use the properties of the interior angles of a convex polygon. The sum of the interior angles of a polygon with \\( n \\) sides is given by the formula:\n\n\\[ \\text{Sum of interior angles} = 180(n-2) \\]\n\nIf the sum of all but one of the interior angles is \\( 1070^\\circ \\),... | 8 |
36 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the given problem, we need to find all solutions to the equation:\n\\[\n\\sin \\left( \\tan^{-1} (x) + \\cot^{-1} \\left( \\frac{1}{x} \\right) \\right) = \\frac{1}{3}.\n\\]\n\nFirst, let's recall a useful trigonometric identity involving the inverse tangent and inverse cotangent functions. Specifically, ... | 3 \pm 2 \sqrt{2} |
51 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the smallest possible value of $\\left| m - n \\right|$, we need to solve the equation $3m + 4n = 100$ for integers $m$ and $n$.\n\nFirst, we can express $m$ in terms of $n$:\n\\[ 3m = 100 - 4n \\]\n\\[ m = \\frac{100 - 4n}{3} \\]\n\nFor $m$ to be an integer, $100 - 4n$ must be divisible by 3. We can find ... | 3 |
24 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to convert the repeating decimal \\(0.\\overline{1331}\\) into a fraction. Let's denote \\(x = 0.\\overline{1331}\\). Then, we can write:\n\n\\[10000x = 1331.\\overline{1331}\\]\n\nSubtracting the original \\(x\\) from this equation, we get:\n\n\\[10000x - x = 1331.\\overline{1331} - ... | 1030 |
52 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the side length of the cube from which the steel sphere is made. The sphere is inscribed in the cube, meaning that the diameter of the sphere is equal to the side length of the cube. Since the radius of the sphere is 3 inches, the diameter of the sphere is \\(2 \\times 3... | 216 |
41 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to understand the properties of a parallelogram. In a parallelogram, opposite sides are equal and parallel. This means that the vector from (5, 3) to (6, 8) should be the same as the vector from (7, 4) to (x, y).\n\nThe vector from (5, 3) to (6, 8) is:\n\\[ (6 - 5, 8 - 3) = (1, 5) \\... | 17 |
61 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the maximum value of \\(|a| + |b| + |c|\\) given the constraint \\(|ax^2 + bx + c| \\le 1\\) for all \\(0 \\le x \\le 1\\).\n\nFirst, let's consider the values of the polynomial at specific points within the interval \\([0, 1]\\). We will evaluate the polynomial at \\(x = 0\\... | 17 |
13 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to use some properties of circles and triangles. Here's the step-by-step reasoning:\n\n1. **Identify the properties of the triangle and circle:**\n - Since \\(AB\\) is the diameter of the circle, the angle subtended by \\(AB\\) at any point on the circle (other than \\(A\\) and \\(... | 76^\circ |
61 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To convert the fraction $\\frac{57}{160}$ to a terminating decimal, we can perform the division of 57 by 160. Let's do this using Python to ensure accuracy.\n```python\n# Perform the division to convert the fraction to a decimal\ndecimal_value = 57 / 160\nprint(decimal_value)\n```\n```output\n0.35625\n```\nThe dec... | .35625 |
45 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the polynomial \\( g(x) \\), which is \\( f(x-1) \\), and then find the sum of its coefficients.\n\nGiven:\n\\[ f(x) = 3x^4 + 5x^2 - 9x - 2 \\]\n\nTo find \\( g(x) = f(x-1) \\), we substitute \\( x-1 \\) into \\( f(x) \\):\n\n\\[ g(x) = f(x-1) = 3(x-1)^4 + 5(x-1)^2 - 9(x-1) -... | -2 |
49 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to set up an equation based on the given information and then solve for \\( x \\).\n\nThe problem states that \"half the value of \\( 3x - 9 \\) is \\( x + 37 \\).\" This can be translated into the following equation:\n\n\\[\n\\frac{1}{2} (3x - 9) = x + 37\n\\]\n\nWe will solve this e... | 83 |
43 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find all two-digit numbers of the form \\(B6\\) that are squares of positive integers. The number \\(B6\\) can be written as \\(10B + 6\\), where \\(B\\) is a digit from 0 to 9. We need to check which of these numbers are perfect squares.\n\nWe can do this by iterating through all... | 2 |
57 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to find the value of \\( f^{-1}(f^{-1}(6)) \\). Here are the steps:\n\n1. First, we need to find \\( f^{-1}(6) \\). Since \\( f(2) = 6 \\), it follows that \\( f^{-1}(6) = 2 \\).\n2. Next, we need to find \\( f^{-1}(2) \\). Since \\( f(1) = 2 \\), it follows that \\( f^{-1}(2) = 1 \\)... | 1 |
33 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the number of ways to assign 3 people to 5 different offices, with each person getting exactly one office. This is a permutation problem where we are selecting 3 offices out of 5 and arranging 3 people in those offices.\n\nThe number of ways to assign 3 people to 5 diffe... | 60 |
54 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To convert the fraction $\\frac{3}{20}$ to a decimal, we can perform the division of 3 by 20. Here's the step-by-step process:\n\n1. Set up the division problem: 3 ÷ 20.\n2. Since 3 is less than 20, we add a decimal point and a zero to 3, making it 30.\n3. Divide 30 by 20. 20 goes into 30 once, so we write down 1 ... | 0.15 |
20 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the point the curve passes through at \\( t = 2 \\), we need to substitute \\( t = 2 \\) into the parameterized equations:\n\n\\[\nx = t^3 + 7\n\\]\n\\[\ny = -3t^2 - 6t - 5\n\\]\n\nLet's compute these step by step using Python.\n```python\n# Define the parameter t\nt = 2\n\n# Calculate x and y using the gi... | (15,-29) |
38 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To determine how far a car travels at a speed of 50 miles per hour over a time of \\(2\\frac{3}{4}\\) hours, we can use the formula for distance:\n\n\\[\n\\text{Distance} = \\text{Speed} \\times \\text{Time}\n\\]\n\nHere, the speed is 50 miles per hour and the time is \\(2\\frac{3}{4}\\) hours. First, we need to c... | 137 \frac{1}{2} |
21 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the greatest integer less than \\((\\sqrt{7} + \\sqrt{5})^6\\), we can use the binomial theorem and some algebraic manipulation. Let's denote \\(a = \\sqrt{7}\\) and \\(b = \\sqrt{5}\\). We need to find the value of \\((a + b)^6\\).\n\nThe binomial expansion of \\((a + b)^6\\) is given by:\n\\[\n(a + b)^6 ... | 13535 |
22 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to set up the equations based on the given conditions and solve for \\( x \\).\n\n1. **Initial Setup:**\n - Denali is responsible for 16 dogs.\n - Nate is responsible for 12 dogs.\n - The ratio of Denali's pay to Nate's pay is \\(\\frac{16}{12} = \\frac{4}{3}\\).\n\n2. **First ... | 5 |
45 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the intersection of two sets:\n1. The set of all prime numbers less than 30.\n2. The set of all odd numbers greater than zero.\n\nFirst, let's list the prime numbers less than 30:\n\\[ \\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29\\} \\]\n\nNext, let's list the odd numbers greater th... | 9 |
44 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the given equation\n\\[\n\\frac{1}{x^2 - 1} + \\frac{2}{x^2 - 2} + \\frac{3}{x^2 - 3} + \\frac{4}{x^2 - 4} = 2010x - 4,\n\\]\nwe need to find the sum of all complex solutions. Let's break down the problem and solve it step-by-step.\n\n1. **Rewrite the Equation:**\n The equation can be written as\n \\[... | \frac{2}{1005} |
54 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the equation \\(2^{2x} = 256^\\frac{1}{2}\\), we can follow these steps:\n\n1. Simplify the right-hand side of the equation. We know that \\(256 = 2^8\\), so \\(256^\\frac{1}{2} = (2^8)^\\frac{1}{2} = 2^4\\).\n2. Now the equation becomes \\(2^{2x} = 2^4\\).\n3. Since the bases are the same, we can equate ... | 2 |
27 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find three consecutive integers whose product is 120. Let's denote the three consecutive integers by \\(n-1\\), \\(n\\), and \\(n+1\\). The product of these integers is:\n\n\\[\n(n-1) \\cdot n \\cdot (n+1) = 120\n\\]\n\nWe also know that the mean of these three integers is:\n\n\\[... | 6 |
60 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To determine the area of the set \\( S \\) of points \\((a, b)\\) such that the polynomial \\( x^4 + ax^3 - bx^2 + ax + 1 = 0 \\) has at least one real root, we need to analyze the polynomial and find the conditions on \\(a\\) and \\(b\\).\n\n1. **Polynomial Analysis:**\n The polynomial \\( x^4 + ax^3 - bx^2 + a... | \frac{1}{4} |
12 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to follow these steps:\n\n1. Find the proper divisors of 284.\n2. Calculate the sum of these proper divisors.\n3. Find the proper divisors of this sum.\n4. Calculate the sum of these proper divisors.\n\nLet's start by finding the proper divisors of 284. A proper divisor of a number \... | 284 |
26 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to understand the recursive function \\( f \\) and then compute \\( f(2015, 2) \\). Let's break down the function and compute it step by step.\n\n1. **Base Case**: If \\( i = 0 \\) and \\( 0 \\le j \\le 4 \\), then \\( f(i, j) = \\text{mod}_5(j + 1) \\).\n2. **Recursive Case 1**: If \... | 1 |
31 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to understand the properties of numbers with exactly three positive divisors. A number has exactly three positive divisors if and only if it is the square of a prime number. This is because the divisors of \\( p^2 \\) (where \\( p \\) is a prime number) are \\( 1, p, \\) and \\( p^2 ... | 121 |
26 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve $513^2 - 487^2$, we can use the difference of squares formula, which states that $a^2 - b^2 = (a - b)(a + b)$. Here, $a = 513$ and $b = 487$. So we have:\n\n\\[513^2 - 487^2 = (513 - 487)(513 + 487)\\]\n\nNow let's calculate the values inside the parentheses:\n\n\\[513 - 487 = 26\\]\n\\[513 + 487 = 1000\\... | 26000 |
44 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to find the value of $\\sin^3 18^\\circ + \\sin^2 18^\\circ$. We can use the known value of $\\sin 18^\\circ = \\frac{\\sqrt{5} - 1}{4}$.\n\nFirst, let's calculate $\\sin^2 18^\\circ$ and $\\sin^3 18^\\circ$ using this value. Then we will add them together.\n\n1. Calculate $\\sin^2 18... | \frac{1}{8} |
53 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the areas of the parallelograms formed by the given lines and use the information to determine the values of \\(a\\), \\(b\\), \\(c\\), and \\(d\\).\n\n1. **Understanding the Problem:**\n - The first parallelogram is bounded by the lines \\(y = ax + c\\), \\(y = ax + d\\), ... | 16 |
19 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the number of representatives the 8th grade should have based on the total number of representatives and the enrollments of the 7th and 8th grades.\n\nHere are the steps to solve the problem:\n\n1. Calculate the total enrollment of both grades.\n2. Determine the proporti... | 10 |
8 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the common positive factors of 14 and 42. The factors of 14 are 1, 2, 7, and 14. The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. The common factors of 14 and 42 are the numbers that appear in both lists, which are 1, 2, 7, and 14. Therefore, the number of possible number... | 4 |
16 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to consider the sequence of events and the probabilities associated with each.\n\n1. Alice's Turn:\n - Probability of Alice winning on her first turn: \\( P(A_1) = \\frac{1}{2} \\) (since she wins if she flips heads).\n\n2. Bob's Turn:\n - If Alice does not win on her first turn,... | \frac{2}{3} |
58 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to consider the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Given that two sides of the triangle are each 8 units long, let's denote the third side as \\(x\\). The triangle ine... | 31 |
32 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the roots of the equation \\(z^8 - z^7 + z^6 - z^5 + z^4 - z^3 + z^2 - z + 1 = 0\\). This equation can be recognized as a special polynomial related to the 18th roots of unity. Specifically, it is the 18th cyclotomic polynomial, which means its roots are the primitive 18th ro... | 8 \pi |
6 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the smallest positive perfect cube that can be written as the sum of three consecutive integers. Let's denote the three consecutive integers as \\( n-1 \\), \\( n \\), and \\( n+1 \\). The sum of these three integers is:\n\n\\[ (n-1) + n + (n+1) = 3n \\]\n\nWe need this sum t... | 27 |
23 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to use the relationship between the volume of water being added to the tank and the rate at which the water level is rising.\n\n1. **Volume of Water Added per Hour:**\n The volume of water added per hour is given as \\(20\\pi\\) cubic meters.\n\n2. **Rate of Water Level Rise:**\n ... | \sqrt{5} |
30 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to perform a $90^\\circ$ rotation around the point $-4 - 5i$ in the clockwise direction on the complex number $3$. Here are the steps to achieve this:\n\n1. Translate the complex number $3$ so that the point $-4 - 5i$ becomes the origin. This is done by subtracting $-4 - 5i$ from $3$... | 1 - 12i |
23 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To determine the remainder of \\(54 \\mod 6\\), we need to find the remainder when 54 is divided by 6. This can be done using the modulus operator in Python. Let's calculate it.\n```python\n# Calculate the remainder of 54 divided by 6\nremainder = 54 % 6\nprint(remainder)\n```\n```output\n0\n```\nThe remainder whe... | 0 |
15 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the equation \\(2^8 = 4^x\\), we can follow these steps:\n\n1. Rewrite the equation using the fact that \\(4\\) can be expressed as a power of \\(2\\). Specifically, \\(4 = 2^2\\).\n2. Substitute \\(4\\) with \\(2^2\\) in the equation, so \\(4^x = (2^2)^x\\).\n3. Use the power of a power rule, which state... | 4 |
24 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve for the product \\(abc\\), we can follow these steps:\n\n1. We have three equations:\n \\[\n a + b = 8\n \\]\n \\[\n b + c = -3\n \\]\n \\[\n a + c = -5\n \\]\n\n2. We can solve these equations to find the values of \\(a\\), \\(b\\), and \\(c\\).\n\n3. First, we can add all three equatio... | -120 |
34 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the given equation\n\n\\[\n\\frac{(x + 1)(x - 3)}{5(x + 2)(x - 4)} + \\frac{(x + 3)(x - 5)}{9(x + 4)(x - 6)} - \\frac{2(x + 5)(x - 7)}{13(x + 6)(x - 8)} = \\frac{92}{585},\n\\]\n\nwe will first define the equation in Python and then use a numerical solver to find the roots. We'll use the `fsolve` function... | 1 \pm \sqrt{19} |
15 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to calculate the volume of the cylindrical beaker and the volume of the spherical tank, and then determine how many cylindrical beakers are needed to fill the spherical tank.\n\n1. **Volume of the Cylindrical Beaker:**\n The formula for the volume of a cylinder is \\( V = \\pi r^2 ... | 4 |
26 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to understand the geometry of the set \\(S\\). The set \\(S\\) consists of two parts:\n\n1. The interior of a regular nonagon with side length 2 units.\n2. The points that are less than 1 unit away from any point on the perimeter of the nonagon.\n\nThe second part of the set \\(S\\) ... | 18+2\pi |
26 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to use the properties of parallel lines and angles. Here's the step-by-step reasoning:\n\n1. Since $\\overline{ST} \\parallel \\overline{QR}$, the angle $\\angle STR$ is equal to the angle $\\angle QRP$ because they are corresponding angles.\n2. The angle $\\angle QRP$ is the same as ... | 75^\circ |
0 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To determine for how many positive integers \\( n > 1 \\) it is true that \\( 2^{24} \\) is a perfect \\( n^{\\text{th}} \\) power, we need to find the values of \\( n \\) such that \\( 2^{24} \\) can be written as \\( (2^k)^n \\) for some integer \\( k \\). This means that \\( 24 \\) must be divisible by \\( n \\... | 7 |
55 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we can set up a system of linear equations. Let's denote the cost of a pencil by \\( p \\) and the cost of a jumbo eraser by \\( e \\). The given information can be translated into the following equations:\n\n1. \\( 3p + e = 1.24 \\)\n2. \\( 5p + e = 1.82 \\)\n\nWe can solve these equations ... | 29 |
27 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the greatest common divisor (GCD) of 3339, 2961, and 1491, we can use the Euclidean algorithm. The Euclidean algorithm is an efficient method for computing the greatest common divisor of two numbers. To find the GCD of more than two numbers, we can iteratively apply the algorithm to pairs of numbers.\n\nHe... | 21 |
28 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To divide by a fraction, you can multiply by its reciprocal. The reciprocal of $\\frac{2}{3}$ is $\\frac{3}{2}$. So, to divide $10$ by $\\frac{2}{3}$, you should multiply $10$ by $\\frac{3}{2}$. Let's calculate this using Python to ensure accuracy.\n```python\n# Define the number and the reciprocal of the fraction... | \frac{3}{2} |
3 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To convert the decimal number \\(555_{10}\\) to base \\(5\\), we need to repeatedly divide the number by \\(5\\) and keep track of the remainders. The conversion process involves the following steps:\n\n1. Divide the number by \\(5\\).\n2. Record the remainder.\n3. Update the number to the quotient from the divisi... | 4210_{5} |
59 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to combine the terms in the expression $\\sqrt{2} + \\frac{1}{\\sqrt{2}} + \\sqrt{3} + \\frac{1}{\\sqrt{3}}$ into a single fraction with a common denominator.\n\nFirst, let's simplify the terms involving square roots in the denominators:\n\\[\n\\frac{1}{\\sqrt{2}} = \\frac{\\sqrt{2}}{... | 23 |
53 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to understand the geometric configuration of the circles and the triangle. The centers of the circles form a smaller triangle inside the larger triangle \\(ABC\\). Since the circles are tangent to each other and to the sides of the triangle, the centers of the circles form an equilat... | 30^\circ |
46 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the shortest distance from a point to a line, we can use the formula for the distance from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$:\n\n\\[ \\text{Distance} = \\frac{|Ax_1 + By_1 + C|}{\\sqrt{A^2 + B^2}} \\]\n\nFirst, we need to rewrite the line equation $y = 2x - 2$ in the standard form $Ax + By +... | 2\sqrt{5} |
35 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to follow these steps:\n\n1. Calculate the square root of 20.\n2. Apply the ceiling function to the result of the square root.\n3. Square the result of the ceiling function.\n\nLet's do this step by step using Python code.\n```python\nimport math\n\n# Step 1: Calculate the square root... | 25 |
5 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the units digit of $4\\$$, we need to understand the structure of the superfactorial. The superfactorial $n\\$$ is defined as a power tower of $n!$ with height $n!$. For $4\\$$, this means a power tower of $4! = 24$ with height 24. The units digit of such a large power tower can be determined by observing ... | 6 |
39 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to consider the circular permutations of 8 people around a square table. Since the table is square, it has 4 sides, and we need to place 2 people on each side. The key point is that rotating the arrangement does not create a new configuration.\n\nFirst, let's fix one person in one of... | 10080 |
28 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to use some geometric properties and theorems. Here's the step-by-step reasoning:\n\n1. **Properties of the Centroid and Medians:**\n - The centroid $G$ divides each median into a ratio of 2:1, with the longer segment being from the vertex to the centroid.\n - The line through $G... | 8 |
60 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the coefficients \\(p\\), \\(q\\), and \\(r\\) such that the polynomial \\(x^9 + px^6 + qx^3 + r\\) is divisible by \\(x^3 - 3x^2 + 4x - 1\\). This means that if we divide \\(x^9 + px^6 + qx^3 + r\\) by \\(x^3 - 3x^2 + 4x - 1\\), the remainder should be zero.\n\nLet's denote ... | (6,31,-1) |
25 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to find all integers \\( n \\) such that \\( f(n) = n \\) given the functional equation \\( f(x) + f(y) = f(x + y) - xy - 1 \\) and the condition \\( f(1) = 1 \\).\n\nLet's break it down into manageable steps:\n\n1. **Find the form of \\( f(x) \\):**\n - First, we will find \\( f(0)... | 1,-2 |
13 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"Kathy has 300 US dollars. She wants to withdraw half of it in British pounds and half of it in euros. We need to calculate how many pounds and euros she will get, and then find the difference between the number of euros and pounds.\n\n1. Calculate the amount to be withdrawn in pounds and euros:\n - Half of 300 U... | 22 |
29 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the simplest radical form of $\\sqrt{53}$, we need to determine if 53 can be factored into a product of a perfect square and another number. If it can, we can simplify the square root. If it cannot, then $\\sqrt{53}$ is already in its simplest radical form.\n\nFirst, we need to check if 53 has any perfect ... | \sqrt{53} |
42 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the inequality $-4 < 2(x - 1) < 8$, we need to isolate $x$ in the middle. Here are the steps:\n\n1. Divide all parts of the inequality by 2:\n \\[\n -2 < x - 1 < 4\n \\]\n2. Add 1 to all parts of the inequality:\n \\[\n -1 < x < 5\n \\]\nSo, the solution to the inequality is $-1 < x < 5$. This... | 4 |
13 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem of dividing \\(413_5\\) by \\(2_5\\), we can follow these steps:\n\n1. Convert the numbers from base 5 to base 10.\n2. Perform the division in base 10.\n3. Convert the quotient back to base 5.\n\nFirst, let's convert \\(413_5\\) and \\(2_5\\) to base 10.\n\nThe number \\(413_5\\) can be expand... | 204_5 |
46 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"Given that $\\omega$ is a complex number such that $\\omega^3 = 1$, we know that $\\omega$ is a cube root of unity. The cube roots of unity are $1, \\omega, \\omega^2$, where $\\omega^3 = 1$ and $\\omega \\neq 1$. Also, we have the property that $1 + \\omega + \\omega^2 = 0$.\n\nWe need to find the value of the ex... | 1 |
19 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to understand the geometric properties of the circle and the tangents.\n\n1. **Identify the properties of the tangents**: Since $HO$ and $HN$ are tangent to the circle at $O$ and $N$, the angles $\\angle HOJ$ and $\\angle HNJ$ are right angles (90 degrees each). This is because the r... | 180^\circ |
3 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the number of ways to distribute 6 cars into 3 lanes (left-turn, straight-going, and right-turn) such that each lane has at least one car. This is a combinatorial problem that can be approached using the principle of inclusion-exclusion.\n\nHere's the step-by-step breakd... | 540 |
57 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To simplify the expression \\(\\sqrt{\\sqrt[3]{\\sqrt{\\frac{1}{729}}}}\\), we can break it down into smaller steps.\n\n1. First, simplify the innermost expression \\(\\sqrt{\\frac{1}{729}}\\).\n2. Then, take the cube root of the result from step 1.\n3. Finally, take the square root of the result from step 2.\n\nL... | \frac{\sqrt{3}}{3} |
46 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the smallest positive integer \\( n \\) such that all the roots of the polynomial \\( z^4 + z^2 + 1 = 0 \\) are \\( n^{\\text{th}} \\) roots of unity. The \\( n^{\\text{th}} \\) roots of unity are the solutions to the equation \\( z^n = 1 \\).\n\nFirst, let's find the roots o... | 6 |
13 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the probability that the product of the numbers on two fair 6-sided dice is a multiple of 5. \n\nHere are the steps to solve the problem:\n\n1. **Identify the total number of outcomes**: When two dice are thrown, there are \\(6 \\times 6 = 36\\) possible outcomes.\n2. **... | \frac{11}{36} |
23 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to follow these steps:\n\n1. Substitute \\(a = 8\\) into the expression \\(\\left(16\\sqrt[3]{a^2}\\right)^{\\frac{1}{3}}\\).\n2. Simplify the expression inside the parentheses.\n3. Raise the simplified expression to the power of \\(\\frac{1}{3}\\).\n\nLet's break it down and solve it... | 4 |
29 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To determine for how many values of \\( x \\) the expression \\(\\frac{x-5}{x^2-9}\\) is not defined, we need to identify the values of \\( x \\) that make the denominator zero. The expression is not defined when the denominator is zero.\n\nThe denominator of the expression is \\( x^2 - 9 \\). We can factor this a... | 2 |
54 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the equation of the circle after reflecting each point of the circle \\(x^2 + y^2 = 25\\) in the point \\((4,1)\\).\n\n1. **Reflection of a Point:**\n The reflection of a point \\((x, y)\\) in the point \\((4, 1)\\) is given by:\n \\[\n (x', y') = (2 \\cdot 4 - x, 2 \\c... | (1,-16,-4,43) |
9 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To determine the probability that a triangle with sides of length 1, \\(x\\), and \\(y\\) exists, we need to use the triangle inequality theorem. The triangle inequality theorem states that for any triangle with sides \\(a\\), \\(b\\), and \\(c\\), the following must be true:\n\n1. \\(a + b > c\\)\n2. \\(a + c > b... | \frac{1}{2} |
52 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to consider the maximum number of intersection points that can be formed by four distinct circles in a plane. Each pair of circles can intersect at most at 2 points. The number of pairs of circles that can be formed from 4 circles is given by the combination formula \\( \\binom{4}{2}... | 12 |
58 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To determine whether the function \\( h(x) = f(f(g(f(g(f(x)))))) \\) is even, odd, or neither, we need to analyze the properties of the functions \\( f \\) and \\( g \\).\n\n1. **Properties of \\( f \\) and \\( g \\):**\n - \\( f(x) \\) is an odd function, so \\( f(-x) = -f(x) \\).\n - \\( g(x) \\) is an even ... | \text{even} |
57 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to explore different ways of adding parentheses to the expression \\(1 + 2 + 3 - 4 + 5 + 6\\) and find the minimum possible value. We can use Python to systematically evaluate all possible ways of adding parentheses and determine the minimum value.\n\nHere's the plan:\n1. Generate al... | -9 |
56 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to find the number of ordered pairs \\((a, b)\\) of integers such that \\(|a + bi| \\le 5\\). The magnitude of a complex number \\(a + bi\\) is given by \\(\\sqrt{a^2 + b^2}\\). Therefore, the condition \\(|a + bi| \\le 5\\) translates to \\(\\sqrt{a^2 + b^2} \\le 5\\), which simplifi... | 81 |
43 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the remainders of $129^{34}$ and $96^{38}$ when divided by $11$, and then add these remainders together. Finally, we will find the remainder of the sum when divided by $11$.\n\nFirst, let's simplify $129$ and $96$ modulo $11$:\n\n1. $129 \\equiv 3 \\pmod{11}$ because $129 = 1... | 9 |
1 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to evaluate the expression \\(\\lceil (3.6)^2 \\rceil - (\\lceil 3.6 \\rceil)^2\\).\n\n1. First, we calculate \\((3.6)^2\\).\n2. Then, we take the ceiling of \\((3.6)^2\\).\n3. Next, we take the ceiling of 3.6.\n4. Finally, we compute the difference between the ceiling of \\((3.6)^2\\... | -3 |
34 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine how much milk is left in the bucket after each mile Sam walks. We know that for each mile, the amount of milk left in the bucket is $\\frac{2}{3}$ of the amount at the beginning of the mile. Since the distance from the barn to Sam's house is 3 miles, we can calculate the... | \frac{16}{27} |
0 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to use the properties of the triangle and the given conditions. Here's the step-by-step reasoning:\n\n1. Since \\(AD = BD = CD\\), point \\(D\\) is equidistant from \\(A\\), \\(B\\), and \\(C\\). This means that \\(D\\) is the circumcenter of triangle \\(ABC\\).\n2. Since \\(D\\) is ... | 90^\circ |
7 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to find the value of \\(z + \\frac{1}{z} + z^2 + \\frac{1}{z^2}\\) given that \\(z^5 = 1\\) and \\(z \\neq 1\\). The complex number \\(z\\) is a fifth root of unity, which means \\(z\\) can be one of the following: \\(e^{2\\pi i k / 5}\\) for \\(k = 1, 2, 3, 4\\). Note that \\(z^5 = 1... | -1 |
45 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to set up a system of equations based on the information given:\n\n1. The average of Amy's, Ben's, and Chris's ages is 6. This means:\n \\[\n \\frac{A + B + C}{3} = 6 \\implies A + B + C = 18\n \\]\n where \\(A\\) is Amy's current age, \\(B\\) is Ben's current age, and \\(C\\... | 10 |
50 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To simplify the expression \\(\\frac{1+\\sqrt{2}}{2+\\sqrt{3}}\\), we can rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, which is \\(2 - \\sqrt{3}\\).\n\n\\[\n\\frac{1+\\sqrt{2}}{2+\\sqrt{3}} \\cdot \\frac{2-\\sqrt{3}}{2-\\sqrt{3}} = \\frac{(1+\\sq... | 13 |
16 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the smallest real number \\( x \\) in the domain of the function \\( g(x) = \\sqrt{(x-3)^2 - (x-8)^2} \\), we need to ensure that the expression inside the square root is non-negative. This means we need to solve the inequality:\n\n\\[\n(x-3)^2 - (x-8)^2 \\geq 0\n\\]\n\nLet's simplify the expression inside... | \frac{11}{2} |
38 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to find the value of the product \\(\\sin 20^\\circ \\sin 40^\\circ \\sin 60^\\circ \\sin 80^\\circ \\sin 100^\\circ \\sin 120^\\circ \\sin 140^\\circ \\sin 160^\\circ\\).\n\nWe can use the fact that \\(\\sin(180^\\circ - x) = \\sin x\\) to simplify the expression. This means \\(\\sin... | \frac{9}{256} |
45 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the area of a triangle with sides measuring 5, 5, and 6 units, we can use Heron's formula. Heron's formula states that the area of a triangle whose sides have lengths \\(a\\), \\(b\\), and \\(c\\) can be computed from the semi-perimeter \\(s\\) and the lengths of the sides. The formula is:\n\n\\[\n\\text{A... | 12 |
5 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the value of \\( b \\) when \\( a = 4 \\) for the line with the equation \\( 3x + 2y = 12 \\), we can substitute \\( a = 4 \\) into the equation and solve for \\( b \\).\n\nThe equation becomes:\n\\[ 3(4) + 2b = 12 \\]\n\nSimplifying the left side:\n\\[ 12 + 2b = 12 \\]\n\nSubtract 12 from both sides:\n\\[... | 0 |
35 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the expression \\(\\left(4\\dfrac{5}{8}\\right)^{55} \\cdot \\left(\\dfrac{8}{37}\\right)^{55}\\), we can follow these steps:\n\n1. Convert the mixed number \\(4\\dfrac{5}{8}\\) to an improper fraction.\n2. Simplify the expression by combining the fractions.\n3. Evaluate the simplified expression.\n\nLet'... | 1 |
16 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the length of \\( BH \\) in the given geometric configuration. Here's the step-by-step reasoning:\n\n1. **Identify the Coordinates:**\n - Place the square \\( CDEG \\) on the coordinate plane with \\( C \\) at the origin \\((0, 0)\\), \\( D \\) at \\((3, 0)\\), \\( E \... | 1\frac{4}{5} |
11 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the inequality \\(x^2 - 5x - 4 \\le 10\\), we first need to rewrite it in a standard form. We can do this by subtracting 10 from both sides of the inequality:\n\n\\[x^2 - 5x - 14 \\le 0\\]\n\nNext, we need to find the roots of the equation \\(x^2 - 5x - 14 = 0\\). We can factorize the quadratic equation:\... | x \in [-2,7] |
14 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the equation \\(10^x - 10 = 9990\\), we can follow these steps:\n\n1. Add 10 to both sides of the equation to isolate the term with the exponent:\n \\[\n 10^x = 10000\n \\]\n\n2. Recognize that 10000 can be written as a power of 10:\n \\[\n 10^x = 10^4\n \\]\n\n3. Since the bases are the same,... | 4 |
57 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To determine the number of different wacky personalities Tom can create for his Mr. Potato Head, we need to consider each component and how they can be combined. Here's the breakdown of the components and their possible combinations:\n\n1. **Hairstyles**: 3 options (including bald)\n2. **Eyebrows**: 2 sets of eyeb... | 64 |
6 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the equation \\(\\tan 53^\\circ \\tan 81^\\circ \\tan x^\\circ = \\tan 53^\\circ + \\tan 81^\\circ + \\tan x^\\circ\\), we can use the fact that \\(\\tan 53^\\circ\\) and \\(\\tan 81^\\circ\\) are constants. Let's denote \\(\\tan 53^\\circ = a\\) and \\(\\tan 81^\\circ = b\\). The equation then becomes:\n... | 46 |
54 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to use the properties of parallel lines and angles. Here's the step-by-step reasoning:\n\n1. Since \\(PT\\) is parallel to \\(QR\\), the angle \\(\\angle PQR\\) is equal to the angle \\(\\angle QPT\\) because they are corresponding angles.\n2. The angle \\(\\angle QPT\\) is composed o... | 116 |
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