idx int64 0 61 | prompt stringlengths 868 2.58k | answers sequencelengths 1 1 | gt stringlengths 1 53 |
|---|---|---|---|
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 simplify $\\sqrt{242}$, we need to find the prime factorization of 242 and then simplify the square root based on the factors.\n\n1. Find the prime factorization of 242.\n2. Simplify the square root using the prime factors.\n\nLet's start by finding the prime factorization of 242. We can do this by dividing 242... | 11\sqrt2 |
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 \\(0.\\overline{7} - 0.\\overline{4} + 0.\\overline{2}\\), we first need to convert the repeating decimals into fractions. Let's start by converting each repeating decimal into its fractional form.\n\n1. Convert \\(0.\\overline{7}\\) to a fraction:\n Let \\(x = 0.\\overline{7}\\). Then, \\(1... | \frac 59 |
4 | <|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 even and odd numbers and how they relate to the sum of two numbers. The first six prime numbers are 2, 3, 5, 7, 11, and 13. Among these, only 2 is even, and the rest are odd.\n\nThe sum of two numbers is even if and only if both numbers are even or bot... | \frac{13}{18} |
17 | <|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-... | [
"On a particular map, $3$ inches on the map equates to $10$ miles in real life. If you know that the real life distance between two buildings on the map is $53.25$ miles, what would the distance between the buildings be (in inches) on the map, expressed as a fraction?\n\nTo solve this problem, we need to determine ... | \frac{639}{40} |
2 | <|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 constants \\(a\\), \\(b\\), \\(c\\), and \\(d\\) such that\n\\[\n(\\sin x)^7 = a \\sin 7x + b \\sin 5x + c \\sin 3x + d \\sin x\n\\]\nfor all angles \\(x\\), we can use the trigonometric identity for \\(\\sin^7 x\\). We can use the multiple angle formulas and the binomial theorem to express \\(\\sin^7 ... | \frac{35}{64} |
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 find the midpoint of a line segment between two points \\((x_1, y_1)\\) and \\((x_2, y_2)\\), we use the midpoint formula:\n\n\\[\n\\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right)\n\\]\n\nGiven the points \\((-5, 5)\\) and \\((3, 7)\\), we can substitute these values into the formula:\n\n\\[\n\\left(... | (-1,6) |
48 | <|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 an arithmetic sequence. In an arithmetic sequence, the difference between consecutive terms is constant. Let's denote the first term of the sequence by $a$ and the common difference by $d$. The $n$-th term of the arithmetic sequence can be expressed as:\n\n\\... | 350 |
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 find the area of triangle \\(OAB\\), we can use the cross product of vectors \\(\\overrightarrow{OA}\\) and \\(\\overrightarrow{OB}\\). The magnitude of the cross product of two vectors gives the area of the parallelogram formed by those vectors. The area of the triangle is half of that.\n\nGiven:\n\\[\n\\overr... | 3 \sqrt{5} |
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 solve this problem, we need to determine the number of ways to choose 4 upper class soldiers from 5 and 8 lower class soldiers from 10. This is a combinatorial problem where we use the binomial coefficient, often referred to as \"n choose k\".\n\nThe binomial coefficient is given by:\n\\[\n\\binom{n}{k} = \\fra... | 225 |
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 determine the area of the gray piece in the Chinese tangram. Here are the steps to solve it:\n\n1. **Determine the side length of the square**: Since the area of the square \\(ABCD\\) is 16 square inches, the side length of the square is \\(\\sqrt{16} = 4\\) inches.\n2. **Identify ... | 2 |
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-... | [
"The given polynomial is:\n\\[P(z) = z^8 - z^7 + z^6 - z^5 + z^4 - z^3 + z^2 - z + 1 = 0.\\]\nNotice that this polynomial can be written as:\n\\[P(z) = \\frac{z^9 + 1}{z + 1}.\\]\nThis means that the roots of \\(P(z)\\) are the ninth roots of \\(-1\\) excluding the root \\(z = -1\\).\n\nThe ninth roots of \\(-1\\) ... | 8 \pi |
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 equation $x = \\!\\sqrt{11-2x} + 4$, we can follow these steps:\n\n1. Isolate the square root term.\n2. Square both sides to eliminate the square root.\n3. Solve the resulting equation.\n4. Verify the solutions by substituting back into the original equation.\n\nLet's implement this in Python using sy... | x=5 |
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 find the dot product of two vectors, we multiply the corresponding components and then sum the results. Given the vectors $\\begin{pmatrix} -2 \\\\ 0 \\\\ 7 \\end{pmatrix}$ and $\\begin{pmatrix} 3 \\\\ 4 \\\\ -5 \\end{pmatrix}$, the dot product is calculated as follows:\n\n\\[\n(-2) \\cdot 3 + 0 \\cdot 4 + 7 \\... | -41 |
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 solve the equation \\(3^{2x} + 19 = 10^x\\), we can use numerical methods since it's not straightforward to solve algebraically. We'll use Python's numerical solver from the `scipy.optimize` library to find the root of the equation.\n\nFirst, we need to define the function \\(f(x) = 3^{2x} + 19 - 10^x\\). Then ... | 2 |
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 this problem, we need to determine the length of the hallway based on the robot's speed and the time it takes to travel the length of the hallway.\n\n1. First, we need to find the speed of the robot. The robot takes 2.5 hours to travel 1 kilometer. We can calculate the speed in kilometers per hour (km/h) ... | 10 |
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 the geometric properties of the circle and the semicircles. Here's the step-by-step reasoning:\n\n1. **Understand the Geometry:**\n - The circle with center \\(Q\\) has a radius of 14 inches.\n - Radii \\(AQ\\) and \\(BQ\\) form a right angle, so \\( \\angle AQB = 90^\\cir... | \frac{14}{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 |
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 base 5 numbers 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 base 5 number \\(413_5\\) can be conv... | 204_5 |
48 | <|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 reciprocal of a randomly selected positive odd integer less than 2010 gives a terminating decimal. A fraction in its simplest form gives a terminating decimal if and only if the prime factorization of its denominator (after simplifying the fracti... | 202 |
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-... | [
"Simplify\n\\[\\cos \\left( \\frac{2 \\pi}{15} \\right) \\cos \\left( \\frac{4 \\pi}{15} \\right) \\cos \\left( \\frac{8 \\pi}{15} \\right) \\cos \\left( \\frac{16 \\pi}{15} \\right).\\]\n\nTo solve this problem, we need to use some trigonometric identities and properties of the cosine function. Let's denote the an... | \frac{1}{16} |
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 the problem, we need to find the area of triangle \\( ENG \\) given that the area of triangle \\( ABC \\) is 144. Here's the step-by-step reasoning:\n\n1. **Properties of the Centroid and Parallel Lines:**\n - The centroid \\( G \\) divides each median into a ratio of 2:1, with the longer segment being ... | 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 sum of the first 12 natural numbers and then determine the remainder when this sum is divided by 4.\n\nFirst, let's calculate the sum of the first 12 natural numbers. The formula for the sum of the first \\( n \\) natural numbers is:\n\n$$ S = \\frac{n(n + 1)}{2} $$\n\nFo... | 2 |
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 factorize the polynomial \\(x^8 + 3x^4 - 4\\) into irreducible factors over the integers. Then, we will evaluate the sum of the values of these factors at \\(x = 1\\).\n\nFirst, let's factorize the polynomial \\(x^8 + 3x^4 - 4\\). We can use a substitution to make the factorization... | 10 |
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 determine the domain of the function \\( f(x) = \\frac{2-x}{\\log(2-\\log(x-2))} \\), we need to consider the constraints imposed by the logarithmic functions and the denominator not being zero.\n\n1. The expression inside the logarithm must be positive:\n \\[\n x - 2 > 0 \\implies x > 2\n \\]\n2. The exp... | (2,12) \cup (12,102) |
47 | <|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 vertical asymptotes of the function \\( f(x) = \\frac{2x}{x^2 - 5x - 14} \\), we need to determine where the denominator is zero. The denominator is \\( x^2 - 5x - 14 \\), and we can factor this as \\( (x - 7)(x + 2) \\). Therefore, the vertical asymptotes occur at \\( x = 7 \\) and \\( x = -2 \\). So,... | 5 |
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 this problem, we need to understand the pattern of the sum of the first $N$ positive odd numbers. The $n$-th positive odd number can be represented as $2n - 1$. The sum of the first $N$ positive odd numbers is given by:\n\n$$\n1 + 3 + 5 + \\cdots + (2N - 1) = N^2\n$$\n\nThis is a well-known result from th... | 11 |
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 determine the time it takes for both gears to align their marks pointing due north again. This is essentially finding the least common multiple (LCM) of the times it takes for each gear to complete a full cycle.\n\n1. The first gear turns \\(33\\frac{1}{3}\\) times per minute, whi... | 36 |
48 | <|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 the sequence \\(a_n\\). The sequence is such that for every positive integer \\(k\\):\n\n1. The subsequence \\(a_{2k-1}\\), \\(a_{2k}\\), \\(a_{2k+1}\\) is geometric.\n2. The subsequence \\(a_{2k}\\), \\(a_{2k+1}\\), \\(a_{2k+2}\\) is arithmetic.\n\nGi... | 504 |
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 the problem, we need to find the minimum value of the expression\n\\[\n\\sqrt{x^2 + 400} + \\sqrt{y^2 + 900} + \\sqrt{x^2 + y^2 - 80x - 100y + 4100}\n\\]\nfor \\(0 \\le x \\le 40\\) and \\(0 \\le y \\le 50\\).\n\nFirst, let's rewrite the expression in a more interpretable form. The expression can be seen ... | 70 \sqrt{2} |
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 equation \\((3)5^2 - 4(5 - a)^2 \\div 3 = 63\\), we'll follow these steps:\n\n1. Simplify the equation.\n2. Solve for \\(a\\).\n3. Sum the values of \\(a\\) that satisfy the equation.\n\nFirst, let's rewrite the equation clearly:\n\n\\[3 \\cdot 5^2 - \\frac{4(5 - a)^2}{3} = 63\\]\n\nSimplify the left ... | 10 |
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 problem, we need to determine how many possible integer values of \\( x \\) there are such that \\( x^3 \\) is a positive factor of \\( 10! \\).\n\nFirst, we need to find the prime factorization of \\( 10! \\). The prime factorization of \\( 10! \\) can be determined by counting the powers of each pri... | 6 |
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 problem, we need to find the values of \\(A\\) and \\(B\\) such that the expression \\(10x^2 - x - 24\\) can be factored into \\((Ax - 8)(Bx + 3)\\).\n\nFirst, let's expand \\((Ax - 8)(Bx + 3)\\):\n\\[\n(Ax - 8)(Bx + 3) = ABx^2 + 3Ax - 8Bx - 24 = ABx^2 + (3A - 8B)x - 24.\n\\]\nWe need this to be equal... | 12 |
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 find the area of the trapezoid, we can use the formula for the area of a trapezoid:\n\n\\[\n\\text{Area} = \\frac{1}{2} \\times (b_1 + b_2) \\times h\n\\]\n\nwhere \\( b_1 \\) and \\( b_2 \\) are the lengths of the two parallel sides (bases), and \\( h \\) is the height of the trapezoid.\n\nFrom the given probl... | 21 |
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 find the minimum value of \\((x + y)(y + z)\\) given that \\(xyz(x + y + z) = 1\\) for positive real numbers \\(x\\), \\(y\\), and \\(z\\), we can use the method of Lagrange multipliers or explore symmetry in the variables. However, a common technique in such symmetric problems is to assume \\(x = z\\) to simpl... | 2 |
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 |
4 | <|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 maximum value of the expression \\(4(x + 7)(2 - x)\\), we can follow these steps:\n\n1. Expand the expression.\n2. Rewrite the expression in a standard quadratic form \\(ax^2 + bx + c\\).\n3. Use the vertex formula for a parabola to find the maximum value, since the parabola opens downwards (the coeffi... | 81 |
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 find the values of $\\lambda$ for which there exists an $x \\in [0,1]$ such that $f(x) \\neq x$ but $f(f(x)) = x$. Let's break it down into manageable steps:\n\n1. **Define the function $f(x)$:**\n \\[f(x) = \\lambda x(1 - x).\\]\n\n2. **Compute $f(f(x))$:**\n \\[f(f(x)) = f(\... | (3,4] |
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 this problem, we need to find a number \\( x \\) that satisfies the following conditions:\n\n1. \\( x \\equiv 1 \\pmod{2} \\)\n2. \\( x \\equiv 1 \\pmod{3} \\)\n3. \\( x \\equiv 1 \\pmod{4} \\)\n4. \\( x \\equiv 1 \\pmod{6} \\)\n5. \\( x \\equiv 0 \\pmod{7} \\)\n6. \\( x < 75 \\)\n\nFirst, let's find the ... | 49 |
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 how many representatives the 8th grade should have based on the proportional representation of their enrollment in the total enrollment.\n\nHere are the steps to solve the problem:\n\n1. Calculate the total enrollment of both grades.\n2. Determine the proportion of the 8... | 10 |
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 convert the decimal number \\(629_{10}\\) to its base 7 and base 8 representations. Then, we will compare the digits of these representations to find out how many digits are the same.\n\nLet's start by converting \\(629_{10}\\) to base 7 and base 8.\n\n### Conversion to Base 7\nTo... | 3 |
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 |
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 use the relationships between the interior and exterior angles of a polygon.\n\n1. The measure of an interior angle of a regular polygon with \\(n\\) sides is given by:\n \\[\n \\text{Interior angle} = \\frac{(n-2) \\cdot 180^\\circ}{n}\n \\]\n2. The measure of an exterior a... | 15 |
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 integer values of $m$ and $n$.\n\nFirst, we can express $n$ in terms of $m$:\n\n\\[ 4n = 100 - 3m \\]\n\n\\[ n = \\frac{100 - 3m}{4} \\]\n\nFor $n$ to be an integer, $100 - 3m$ must be divisible by 4.... | 3 |
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 find the minimum value of \\((a + b + c + d) \\left( \\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} + \\frac{1}{d} \\right)\\) for positive real numbers \\(a, b, c, d\\), we can use the method of Lagrange multipliers or the AM-HM inequality. However, a simpler approach is to use the Cauchy-Schwarz inequality in the... | 16 |
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 convert a decimal number to its base six equivalent, we need to repeatedly divide the number by 6 and keep track of the remainders. Here's the step-by-step process:\n\n1. Divide the decimal number by 6.\n2. Record the remainder.\n3. Update the decimal number to the quotient from the division.\n4. Repeat steps 1... | 4343_6 |
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 will use the principle of inclusion-exclusion for three sets. Let's define the following:\n\n- \\( C \\): the set of students taking calculus.\n- \\( P \\): the set of students taking physics.\n- \\( Ch \\): the set of students taking chemistry.\n\nWe are given the following information:\... | 110 |
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 manipulate this equation to eliminate the repeating part by multiplying both sides by 1000 (since the repeating block is 4 digits long):\n\n\\[1000x = 133... | 1030 |
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 solve this problem, we need to determine the monic polynomial \\( p(x) \\) of degree 4, given that three of its roots are 1, 2, and 3. Since it is a monic polynomial, the leading coefficient is 1. Let's denote the fourth root by \\( r \\). Then the polynomial can be written as:\n\\[ p(x) = (x - 1)(x - 2)(x - 3)... | 24 |
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 use the properties of vectors and the dot product. Let's denote the angle between vectors $\\mathbf{a}$ and $\\mathbf{b}$ as $\\theta_{ab} = 29^\\circ$, and the angle between $\\mathbf{b}$ and $\\mathbf{a} - \\mathbf{b}$ as $\\theta_{b(a-b)} = 84^\\circ$. We need to find the angle... | 55^\circ |
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