idx int64 0 61 | prompt stringlengths 868 2.58k | answers listlengths 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 |
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-... | [
"In the diagram, two circles, each with center $D$, have radii of $1$ and $2$. The total area of the shaded region is $\\frac5{12}$ of the area of the larger circle. How many degrees are in the measure of (the smaller) $\\angle ADC$?\n[asy]\nsize(4cm);\ndefaultpen(linewidth(0.75));\nreal adc = 100;\npair d = (0, 0)... | 120 |
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 maximum value of the angle $\\angle APB$ where $A$ and $B$ are runners on a linear track, and $P$ is a point perpendicular to the track. Let's denote the speed of runner $A$ as $v$. Then the speed of runner $B$ is $3v$. Let's assume that the runners start at the same time... | 30^\circ |
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 this problem, we need to calculate the distance Jim and Martha each walk and then find the difference.\n\n1. Calculate the distance Jim walks:\n - Jim walks diagonally across the rectangular field. We can use the Pythagorean theorem to find the length of the diagonal.\n - The width of the field is 300... | 200 |
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 find the least possible sum of distinct positive integers whose product is 84, we need to factorize 84 into its prime factors and then group these factors in a way that minimizes the sum.\n\nFirst, we factorize 84:\n\\[ 84 = 2^2 \\times 3 \\times 7 \\]\n\nThe distinct positive integers whose product is 84 can b... | 14 |
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-... | [
"The set of vectors $\\mathbf{v}$ such that\n\\[\\operatorname{proj}_{\\begin{pmatrix} 2 \\\\ 1 \\end{pmatrix}} \\mathbf{v} = \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix}\\]lie on a line. Enter the equation of this line in the form \"$y = mx + b$\".\n\nTo solve this problem, we need to understand the condition under w... | y = -2x |
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 given expression, we need to simplify the numerator and the denominator step by step. The expression is:\n\n\\[\n\\sqrt{\\frac{\\frac{5}{\\sqrt{80}} + \\frac{\\sqrt{845}}{9} + \\sqrt{45}}{\\sqrt{5}}}\n\\]\n\nFirst, let's simplify each term in the numerator:\n\n1. Simplify \\(\\frac{5}{\\sqrt{80}}\\):\... | \frac{13}{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 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 |
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 determine the profit, we need to calculate the total revenue from selling the cookies, cupcakes, and brownies, then subtract the cost of baking the items.\n\n1. Calculate the revenue from selling cookies:\n - They sold 54 cookies at 3 for $1.\n - Number of sets of 3 cookies sold = 54 / 3 = 18 sets.\n - Re... | 78 |
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 find the maximum value of \\(ab^2 c^3 d^4\\) given that \\(a + b + c + d = 10\\), we can use the method of Lagrange multipliers or the AM-GM (Arithmetic Mean-Geometric Mean) inequality. Here, we will use the AM-GM inequality for simplicity.\n\nThe AM-GM inequality states that for non-negative real numbers \\(x_... | 27648 |
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 find the total number of students at Baker Middle School, we can set up a simple equation based on the information given. Let's denote the total number of students by \\( x \\). According to the problem, two-thirds of the students take music, and this number is given as 834. So we can write the equation:\n\n\\[... | 1251 |
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 \\) for which \\( 2^{24} \\) can be expressed as \\( (2^k)^n \\) for some integer \\( k \\). This means that \\( 24 \\) must be divisible by \\( n ... | 7 |
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 given problem, we need to convert the polar equation \\(r^2 \\cos 2\\theta = 4\\) into Cartesian coordinates. Recall the relationships between polar and Cartesian coordinates:\n\\[ x = r \\cos \\theta \\]\n\\[ y = r \\sin \\theta \\]\n\\[ r^2 = x^2 + y^2 \\]\n\\[ \\cos 2\\theta = \\cos^2 \\theta - \\s... | \text{(E)} |
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 the problem, we need to find an integer \\( x \\) that satisfies the inequality \\(\\frac{1}{4} < \\frac{x}{7} < \\frac{1}{3}\\). We can break this down into two separate inequalities:\n\n1. \\(\\frac{1}{4} < \\frac{x}{7}\\)\n2. \\(\\frac{x}{7} < \\frac{1}{3}\\)\n\nFirst, let's solve the first inequality ... | 2 |
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 the 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 evaluate the polynomial at key points \\(x = 0\\), \\(x = \\frac{1}{2}\\), and \\(x = 1\\):\n\n1. At \\(x = 0\\), we have \\(|c| \\le 1\\... | 17 |
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 the congruence \\(14u \\equiv 46 \\pmod{100}\\), we need to find the integer solutions for \\(u\\). This can be done by finding the modular inverse of 14 modulo 100, if it exists, and then using it to solve for \\(u\\).\n\nFirst, we need to check if 14 and 100 are coprime, which is a requirement for the m... | 64 |
18 | <|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 distance between the origin and a point on the graph of \\( y = \\frac{1}{\\sqrt{2}}(x^2 - 3) \\), we need to minimize the distance function \\( D = \\sqrt{x^2 + y^2} \\).\n\nGiven \\( y = \\frac{1}{\\sqrt{2}}(x^2 - 3) \\), we can substitute \\( y \\) into the distance function:\n\\[ D = \\sqr... | 12 |
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 need to use the properties of trapezoids and triangles. Here's the step-by-step reasoning:\n\n1. **Understand the Geometry:**\n - \\(AD \\parallel BC\\), so \\(ABCD\\) is a trapezoid.\n - \\(AC \\perp CD\\), so \\(\\angle ACD = 90^\\circ\\).\n - \\(AC\\) bisects \\(\\angle BAD\\), s... | 28 |
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 equation \\(\\frac{\\sqrt{2x}}{\\sqrt{3x-1}} = \\frac{3}{2}\\), we can follow these steps:\n\n1. Square both sides to eliminate the square roots.\n2. Simplify the resulting equation.\n3. Solve for \\(x\\).\n\nLet's implement this in Python using sympy:\n\n```python\nimport sympy as sp\n\n# Define the ... | \frac9{19} |
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-... | [
"The largest eight-digit base 2 integer is \\(11111111_2\\). To convert this to base 10, we can use the formula for binary to decimal conversion. The binary number \\(11111111_2\\) can be expressed as:\n\n\\[1 \\cdot 2^7 + 1 \\cdot 2^6 + 1 \\cdot 2^5 + 1 \\cdot 2^4 + 1 \\cdot 2^3 + 1 \\cdot 2^2 + 1 \\cdot 2^1 + 1 \... | 255 |
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 solve this problem, let's denote the number of coins in each of the original seven bags as \\( x \\). Therefore, the total number of coins you originally had is \\( 7x \\). After finding the bag of 53 coins, you have a total of \\( 7x + 53 \\) coins. You redistribute these coins equally among eight bags, so the... | 203 |
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-... | [
"The problem is to find the value of \\(1^{(2^{235423523})}\\).\n\nFirst, let's understand the properties of exponents. Any number raised to any power is still that number if the base is 1. In other words, \\(1^n = 1\\) for any integer \\(n\\).\n\nSo, \\(1^{(2^{235423523})} = 1\\).\n\nThe final answer is \\(\\boxed... | 1 |
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 this problem, we need to use the properties of similar triangles and the given information about the areas.\n\n1. Since $\\overline{AB} \\parallel \\overline{DE}$, triangles $ABC$ and $CDE$ are similar by the AA similarity criterion (Angle-Angle).\n2. The ratio of the corresponding sides of similar triang... | 54 |
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 the given expression \\((3x-2)(4x+1)-(3x-2)4x+1\\) when \\(x=4\\), we can break it down into smaller steps. Let's first simplify the expression and then substitute \\(x=4\\).\n\nThe expression can be broken down as follows:\n1. Expand \\((3x-2)(4x+1)\\).\n2. Expand \\((3x-2)4x\\).\n3. Substitute these exp... | 11 |
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 find the number of quadratic equations of the form \\(x^2 + ax + b = 0\\) such that whenever \\(c\\) is a root of the equation, \\(c^2 - 2\\) is also a root of the equation.\n\nLet's break down the problem:\n\n1. **Understanding the Condition:**\n If \\(c\\) is a root, then \\(c... | 6 |
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 the inequality \\(\\frac{|x^2 - 81|}{x^2 - 36x} < 0\\), we need to analyze the expression and determine the values of \\(x\\) that make the inequality true. Here's the step-by-step reasoning:\n\n1. **Analyze the Numerator:**\n The numerator is \\(|x^2 - 81|\\). The absolute value function \\(|x^2 - 81|\... | (0,9) \cup (9,36) |
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 one pencil by \\( p \\) and the cost of a jumbo eraser by \\( e \\).\n\nFrom the problem statement, we have the following equations:\n1. \\( 3p + e = 1.24 \\)\n2. \\( 5p + e = 1.82 \\)\n\nWe can solve these equations step b... | 29 |
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 quotient when $x^9 + 1$ is divided by $x - 1$, we can use polynomial long division or synthetic division. However, a more efficient method is to use the polynomial remainder theorem and the factor theorem.\n\nAccording to the factor theorem, if we divide a polynomial $f(x)$ by $x - c$, the remainder of... | x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 |
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 remainder when \\( f(x) = x^{10} + 5x^9 - 8x^8 + 7x^7 - x^6 - 12x^5 + 4x^4 - 8x^3 + 12x^2 - 5x - 5 \\) is divided by \\( x^2 - 1 \\), we can use the fact that \\( x^2 \\equiv 1 \\pmod{x^2 - 1} \\). This means we can replace every \\( x^2 \\) with 1 in the polynomial \\( f(x) \\).\n\nLet's rewrite \\( f... | -13x+3 |
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 for \\(2\\frac{3}{4}\\) hours, we can use the formula for distance:\n\n\\[ \\text{Distance} = \\text{Speed} \\times \\text{Time} \\]\n\nHere, the speed is 50 miles per hour and the time is \\(2\\frac{3}{4}\\) hours. First, we need to convert the mi... | 137 \frac{1}{2} |
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-... | [
"The data in the table below shows the percent of bus riders in a survey of Central H.S. students; 300 males and 300 females in each grade were surveyed. For which grade is the number of its male bus riders closest to $135\\%$ of the number of its female bus riders? \\begin{tabular}{|l|c|c|}\\hline\nGrade & Males &... | 12 |
40 | <|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 determining how many ways there are to put 5 indistinguishable balls into 2 indistinguishable boxes, we need to consider the possible distributions of the balls. Since the balls and boxes are indistinguishable, the order of the distribution does not matter. The possible distributions are:\n... | 3 |
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 follow these steps:\n\n1. Perform the division of 57 by 160.\n2. Since the denominator 160 can be factored into prime factors as $160 = 2^5 \\times 5$, the fraction $\\frac{57}{160}$ will have a terminating decimal representation.\n\nLet's ... | .35625 |
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-... | [
"Let's solve the problem step-by-step.\n\n1. **Identify the quadratic polynomial with integer coefficients:**\n Since \\(4 - \\sqrt{11}\\) is a root and the coefficients are integers, the other root must be \\(4 + \\sqrt{11}\\). The quadratic polynomial can be written as:\n \\[\n p(x) = a(x - (4 - \\sqrt{11})... | \frac{10}{11} |
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 determine the degree of the polynomial \\( f(x) \\), we need to analyze the given equation:\n\n\\[\n(x-1) \\cdot f(x) = 3x^4 + x^3 - 25x^2 + 38x - 17.\n\\]\n\nFirst, let's denote the polynomial on the right-hand side as \\( P(x) \\):\n\n\\[\nP(x) = 3x^4 + x^3 - 25x^2 + 38x - 17.\n\\]\n\nThe degree of \\( P(x) \... | 3 |
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 follow these steps:\n\n1. Calculate how much money Kathy will withdraw in US dollars.\n2. Convert half of that amount to British pounds.\n3. Convert the other half to euros.\n4. Find the difference between the amounts in euros and pounds.\n\nLet's break it down:\n\n1. Kathy has $3... | 22 |
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-... | [
"In cylindrical coordinates $(r, \\theta, z)$, the equation $z = c$ represents a plane parallel to the $xy$-plane at a constant height $c$.\n\nTo confirm this, let's break it down:\n\n1. **Understanding Cylindrical Coordinates:**\n - $r$ is the radial distance from the $z$-axis.\n - $\\theta$ is the angle in th... | \text{(C)} |
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 use the given area formula and relate it to the standard area formula of a triangle. The area of a triangle can also be expressed using the formula:\n\n\\[ \\text{Area} = \\frac{1}{2}bc \\sin A \\]\n\nGiven that the area of triangle \\(ABC\\) is \\(a^2 - (b - c)^2\\), we can equate... | \frac{8}{15} |
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 complex number $-4 - 5i$ 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 $-4 - 5i$... | 1 - 12i |
37 | <|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 sum of the digits in the terminating decimal representation of the fraction $\\frac{4321}{5^7 \\cdot 2^8}$, we can follow these steps:\n\n1. Simplify the fraction by converting the denominator to a power of 10.\n2. Convert the fraction to its decimal form.\n3. Sum the digits of the resulting decimal nu... | 14 |
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 use the formula for the sum of the first \\( n \\) terms of a geometric series. The formula is given by:\n\n\\[ S_n = a \\frac{1 - r^n}{1 - r} \\]\n\nwhere:\n- \\( S_n \\) is the sum of the first \\( n \\) terms,\n- \\( a \\) is the first term of the series,\n- \\( r \\) is the co... | 8 |
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 this problem, we need to determine the number of ways to choose 3 boys out of 7 and 2 girls out of 4, where the order of selection does not matter. This is a combinatorial problem, and we can use the binomial coefficient formula to find the solution.\n\nThe binomial coefficient \\(\\binom{n}{k}\\) represe... | 210 |
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 all the integer roots of the polynomial equation \\(x^4 + 5x^3 + 9x^2 - x - 14 = 0\\), we can use the Rational Root Theorem. The Rational Root Theorem states that any rational solution, expressed in its lowest terms, can be written as \\( \\frac{p}{q} \\), where \\( p \\) is a factor of the constant term (... | -2,1 |
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 need to use the relationship between the greatest common divisor (GCD) and the least common multiple (LCM) of two numbers. Specifically, for any two integers \\(a\\) and \\(b\\), the following relationship holds:\n\n\\[\n\\gcd(a, b) \\times \\mathop{\\text{lcm}}[a, b] = a \\times b\n\\]\n... | 8 |
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 find the integer value of \\( x \\) in the arithmetic sequence \\( 3^2, x, 3^4 \\), we need to ensure that the difference between consecutive terms is constant. \n\nGiven the sequence: \n\\[ 3^2, x, 3^4 \\]\n\n1. Calculate the first term:\n\\[ a_1 = 3^2 = 9 \\]\n\n2. Calculate the third term:\n\\[ a_3 = 3^4 = 8... | 45 |
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 the problem, we need to understand the congruence \\(8x \\equiv 1 \\pmod{p}\\). This congruence has a solution if and only if \\(\\gcd(8, p) = 1\\). Since \\(p\\) is a prime number, \\(\\gcd(8, p) = 1\\) unless \\(p\\) is a divisor of 8. The prime divisors of 8 are 2. Therefore, the only prime \\(p\\) for... | 2 |
40 | <|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 \\(\\tan 100^\\circ + 4 \\sin 100^\\circ\\), we can use trigonometric identities and properties. Let's break it down:\n\n1. Recall that \\(\\tan 100^\\circ = \\tan (180^\\circ - 80^\\circ) = -\\tan 80^\\circ\\).\n2. So the expression becomes \\(-\\tan 80^\\circ + 4 \\sin 100^\\circ\\).\n3. ... | -\sqrt{3} |
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 $-4 < 2(x - 1) < 8$ and express it in the form $a < x < b$, we need to follow these steps:\n\n1. Divide all parts of the inequality by 2.\n2. Simplify the resulting inequality to find the values of $a$ and $b$.\n3. Add the values of $a$ and $b$ to get the final answer.\n\nLet's solve this s... | 4 |
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 these 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\\[\n\\frac{(n-1) +... | 6 |
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 solve for \\( x \\), \\( y \\), and \\( z \\) given the equations \\( xy = 24 \\), \\( xz = 48 \\), and \\( yz = 72 \\), we can follow these steps:\n\n1. Multiply all three equations together:\n \\[\n (xy) \\cdot (xz) \\cdot (yz) = 24 \\cdot 48 \\cdot 72\n \\]\n This gives:\n \\[\n (xyz)^2 = 24 \\cd... | 22 |
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 determine the number of positive whole-number divisors of 196, we need to follow these steps:\n\n1. **Prime Factorization**: Find the prime factorization of 196.\n2. **Divisor Function Formula**: Use the formula for finding the number of divisors based on the prime factorization. If the prime factorization of a... | 9 |
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 cube root.\n3. Compute the cube root.\n\nLet's break it down and solve it using Python and SymPy.\n\n```python\nimport s... | 4 |
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