idx int64 0 61 | prompt stringlengths 868 2.58k | answers listlengths 1 1 | gt stringlengths 1 53 |
|---|---|---|---|
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 value of \\( x \\) that satisfies the equation:\n\\[\n\\tan 53^\\circ \\tan 81^\\circ \\tan x^\\circ = \\tan 53^\\circ + \\tan 81^\\circ + \\tan x^\\circ.\n\\]\n\nLet's denote:\n\\[\na = \\tan 53^\\circ, \\quad b = \\tan 81^\\circ, \\quad c = \\tan x^\\circ.\n\\]\nThe equa... | 46 |
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 find the projection of vector \\(\\mathbf{a}\\) onto vector \\(\\mathbf{b}\\), we use the formula for the projection of \\(\\mathbf{a}\\) onto \\(\\mathbf{b}\\):\n\n\\[\n\\text{proj}_{\\mathbf{b}} \\mathbf{a} = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\mathbf{b} \\cdot \\mathbf{b}} \\mathbf{b}\n\\]\n\nGiven:\n\\... | \begin{pmatrix} 16/49 \\ 48/49 \\ 24/49 \end{pmatrix} |
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 |
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 simplify the expression \\(19x + 1 - 4x - 81\\), we need to combine like terms. The like terms in this expression are the terms involving \\(x\\) and the constant terms.\n\n1. Combine the \\(x\\) terms: \\(19x - 4x = 15x\\).\n2. Combine the constant terms: \\(1 - 81 = -80\\).\n\nSo the simplified expression is ... | 15x - 80 |
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 |
35 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the smallest positive integer \\( k \\) such that the product \\( 240 \\times k \\) is a perfect cube. To do this, we need to factorize 240 and determine what additional factors are needed to make the product a perfect cube.\n\nFirst, we factorize 240 into its prime factors:\... | 900 |
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 problem. However, a common approach for such symmetric problems is to assume \\(x = y = z\\) to sim... | 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-... | [
"To solve this problem, we need to use the principle of inclusion and exclusion. Here's the step-by-step reasoning:\n\n1. We know that there are 10 boxes in total.\n2. Five boxes contain pencils.\n3. Four boxes contain pens.\n4. Two boxes contain both pens and pencils.\n\nUsing the principle of inclusion and exclus... | 3 |
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 largest $x$-value at which the graphs of $f(x)$ and $g(x)$ intersect, we need to solve the equation $f(x) = g(x)$. Let's start by simplifying the expressions for $f(x)$ and $g(x)$:\n\n\\[\nf(x) = e^{3x^2 - |\\lfloor x \\rfloor|!} + \\binom{22 + 735235|\\lfloor x \\rfloor|}{2356} + \\phi(|\\lfloor x \\r... | 6 |
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 |
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 perform two reflections: first over the vertical line $x=1$ and then over the horizontal line $y=2$.\n\n1. **Reflection over the vertical line $x=1$:**\n - The point $(0,0)$ is reflected over the line $x=1$. The reflection of a point $(x,y)$ over the line $x=a$ is given by $(2a ... | (2,4) |
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 problem, we need to find \\(z^{85} + \\frac{1}{z^{85}}\\) given that \\(z + \\frac{1}{z} = \\frac{1 + \\sqrt{5}}{2}\\).\n\nFirst, let's denote \\(z + \\frac{1}{z} = a\\), where \\(a = \\frac{1 + \\sqrt{5}}{2}\\). We can use the identity for powers of \\(z + \\frac{1}{z}\\):\n\n\\[z^n + \\frac{1}{z^n} ... | -2 |
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 this problem, we need to find a number of students \\( n \\) between 30 and 50 such that \\( n \\) is not divisible by any integer \\( k \\) (where \\( 2 \\leq k \\leq n-1 \\)), and \\( n+2 \\) is also not divisible by any integer \\( k \\) (where \\( 2 \\leq k \\leq n+1 \\)).\n\nThis means \\( n \\) and ... | 41 |
10 | <|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 radius \\( r \\) of a sphere where the volume and surface area are numerically equal.\n\n1. The formula for the volume \\( V \\) of a sphere is:\n \\[\n V = \\frac{4}{3} \\pi r^3\n \\]\n\n2. The formula for the surface area \\( A \\) of a sphere is:\n \\[\n A = ... | 3 |
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 |
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-... | [
"Given the conditions \\( |a| = |b| = |c| = |d| = 1 \\) and \\( a + b + c + d = 0 \\), we need to find the maximum value of \\( |(a + b)(a + c)(a + d)(b + c)(b + d)(c + d)| \\).\n\nFirst, let's consider the geometric interpretation. Since \\( |a| = |b| = |c| = |d| = 1 \\), the complex numbers \\( a, b, c, \\) and \... | 0 |
61 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the maximum value of \\(|a| + |b| + |c|\\) given the constraint \\(|ax^2 + bx + c| \\le 1\\) for all \\(0 \\le x \\le 1\\).\n\nFirst, let's consider the values of the polynomial at specific points within the interval \\([0, 1]\\). We will evaluate the polynomial at \\(x = 0\\... | 17 |
34 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine how much milk is left in the bucket after Sam walks 3 miles, given that the bucket leaks such that for each mile he walks, there is $\\frac{2}{3}$ as much milk left as at the beginning of the mile.\n\nWe can break this down into the following steps:\n\n1. Start with 2 ga... | \frac{16}{27} |
35 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem \\( t(g(16)) \\), we need to follow these steps:\n\n1. First, we need to find \\( g(16) \\). Given that \\( g(x) = \\sqrt{x} \\), we can substitute \\( x = 16 \\) into the function.\n2. Next, we need to find \\( t(g(16)) \\). Given that \\( t(x) = 3 - g(x) \\), we can substitute \\( x = g(16) ... | 1 |
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 the problem, we need to add 313.9 and 12.6. Here are the steps:\n\n1. Write down the numbers one below the other, aligning the decimal points:\n \\[\n \\begin{array}{r}\n 313.9 \\\\\n + 12.6 \\\\\n \\end{array}\n \\]\n\n2. Add the numbers starting from the rightmost digit (the tenths place)... | 326.5 |
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} |
24 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to convert the repeating decimal \\(0.\\overline{1331}\\) into a fraction. Let's denote \\(x = 0.\\overline{1331}\\). Then, we can write:\n\n\\[10000x = 1331.\\overline{1331}\\]\n\nSubtracting \\(x = 0.\\overline{1331}\\) from this equation, we get:\n\n\\[10000x - x = 1331.\\overline{... | 1030 |
56 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to find the sum of the arithmetic series \\(1 + 3 + 5 + 7 + 9 + \\dots + 195 + 197 + 199\\) and then find the remainder when this sum is divided by 7.\n\n1. First, we need to determine the number of terms in the series. The series is an arithmetic sequence where the first term \\(a = ... | 4 |
28 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To divide by a fraction, you can multiply by its reciprocal. The reciprocal of $\\frac{2}{3}$ is $\\frac{3}{2}$. So, to divide $10$ by $\\frac{2}{3}$, you should multiply $10$ by $\\frac{3}{2}$. Let's calculate this using Python to ensure accuracy.\n```python\n# Define the number and the reciprocal of the fraction... | \frac{3}{2} |
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 and then evaluate the sum of the values of these factors at \\(x = 1\\).\n\nFirst, we can use a substitution to simplify the polynomial. Let \\(y = x^4\\). Then the polynomial becomes \\(y^2 + ... | 10 |
34 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the given equation\n\n\\[\n\\frac{(x + 1)(x - 3)}{5(x + 2)(x - 4)} + \\frac{(x + 3)(x - 5)}{9(x + 4)(x - 6)} - \\frac{2(x + 5)(x - 7)}{13(x + 6)(x - 8)} = \\frac{92}{585},\n\\]\n\nwe will first define the equation in Python and then use a numerical solver to find the roots. We'll use the `sympy` library f... | 1 \pm \sqrt{19} |
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 the problem, we need to compute the cross product \\((2 \\mathbf{b} - \\mathbf{a}) \\times (3 \\mathbf{c} + \\mathbf{a})\\). We can use the distributive property of the cross product to simplify this expression:\n\n\\[\n(2 \\mathbf{b} - \\mathbf{a}) \\times (3 \\mathbf{c} + \\mathbf{a}) = (2 \\mathbf{b} \... | \begin{pmatrix} -18 \\ -49 \\ 96 \end{pmatrix} |
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 find the value of the product $abc$, we can follow these steps:\n\n1. We have three equations:\n \\[\n \\begin{cases}\n a + b = 8 \\\\\n b + c = -3 \\\\\n a + c = -5\n \\end{cases}\n \\]\n\n2. We can solve these equations to find the values of $a$, $b$, and $c$. First, we can add all three equatio... | -120 |
0 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To determine for how many positive integers \\( n > 1 \\) it is true that \\( 2^{24} \\) is a perfect \\( n^{\\text{th}} \\) power, we need to find the values of \\( n \\) such that \\( 2^{24} \\) can be expressed as \\( (2^k)^n \\) for some integer \\( k \\). This means that \\( 24 \\) must be divisible by \\( n ... | 7 |
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 the problem, we need to find the roots of the polynomial equation $x^4 + 2x^3 + 2 = 0$ and then determine the set $\\{\\alpha_1\\alpha_2 + \\alpha_3\\alpha_4, \\alpha_1\\alpha_3 + \\alpha_2\\alpha_4, \\alpha_1\\alpha_4 + \\alpha_2\\alpha_3\\}$.\n\nFirst, we will find the roots of the polynomial equation. ... | \{1\pm\sqrt{5},-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 this problem, we need to consider the fact that the balls are indistinguishable and the boxes are also indistinguishable. This means that the order of the balls in the boxes does not matter, and swapping the boxes does not create a new arrangement.\n\nWe can approach this problem by considering the possib... | 3 |
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 convert the point $(0,3)$ from rectangular coordinates to polar coordinates, we need to find the values of $r$ and $\\theta$.\n\n1. **Calculate \\( r \\):**\n The formula for \\( r \\) is given by:\n \\[\n r = \\sqrt{x^2 + y^2}\n \\]\n where \\( x = 0 \\) and \\( y = 3 \\).\n\n2. **Calculate \\( \\the... | \left( 3, \frac{\pi}{2} \right) |
0 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to compute the matrix powers \\(\\mathbf{A}^{27}\\), \\(\\mathbf{A}^{31}\\), and \\(\\mathbf{A}^{40}\\), and then sum them up. We can use the properties of matrix exponentiation and diagonalization to simplify the computation.\n\nFirst, let's find the eigenvalues and eigenvectors of ... | \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} |
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 |
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-... | [
"In the diagram, \\(D\\) and \\(E\\) are the midpoints of \\(\\overline{AB}\\) and \\(\\overline{BC}\\) respectively. Determine the area of quadrilateral \\(DBEF\\).\n\nFirst, let's identify the coordinates of the points involved:\n- \\(A(0, 6)\\)\n- \\(B(0, 0)\\)\n- \\(C(8, 0)\\)\n- \\(D\\) is the midpoint of \\(... | 8 |
57 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to determine the value of \\( f^{-1}(f^{-1}(6)) \\). Here are the steps:\n\n1. Identify the value of \\( f^{-1}(6) \\). Since \\( f(2) = 6 \\), it follows that \\( f^{-1}(6) = 2 \\).\n2. Next, identify the value of \\( f^{-1}(2) \\). Since \\( f(1) = 2 \\), it follows that \\( f^{-1}(... | 1 |
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, we need to simplify the expression \\(\\tan 100^\\circ + 4 \\sin 100^\\circ\\). We can use trigonometric identities and properties to simplify this expression.\n\nFirst, let's express \\(\\tan 100^\\circ\\) and \\(\\sin 100^\\circ\\) in terms of simpler angles. Note that \\(100^\\circ = 90^\\... | -\sqrt{3} |
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 $f(x)$ is divided by $x^2-1$, we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial $f(x)$ by a linear polynomial $x-a$ is $f(a)$. However, in this case, we are dividing by a quadratic polynomial, so we need to use a different a... | -13x+3 |
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 8 has an inverse modulo \\(p\\). The integer 8 has an inverse modulo \\(p\\) if and only if \\(\\gcd(8, p) = 1\\). Since \\(p\\) is a prime number, \\(\\gcd(8, p) = 1\\) unless \\(... | 2 |
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 this problem, we need to determine the total area of the shaded regions in the given rectangle. The rectangle is composed of twelve 1 by 1 squares, and the shaded regions are triangles.\n\nLet's break down the problem:\n\n1. **Identify the triangles:**\n - There are two triangles in the given figure.\n ... | 10 |
29 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the simplest radical form of $\\sqrt{53}$, we need to check if 53 can be factored into a perfect square and another integer. If it can, we can simplify the square root. Otherwise, $\\sqrt{53}$ is already in its simplest radical form.\n\nLet's check the factors of 53. The number 53 is a prime number, meanin... | \sqrt{53} |
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 use some geometric properties and formulas. Here's the step-by-step approach:\n\n1. **Understand the Geometry:**\n - The centroid \\(G\\) of a triangle \\(ABC\\) divides each median into a ratio of 2:1.\n - The orthocenter \\(H\\) is the point where the altitudes of the triang... | 3R^2 |
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 Remainder Theorem and the Factor Theorem.\n\nAccording to the Remainder Theorem, when a polynomial $f(x)$ is divided by $x - c$, the remainder of this... | x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 |
35 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the inverse of the function \\( f(x) = \\frac{2}{x+1} \\) and then evaluate it at \\( \\frac{1}{5} \\), we can follow these steps:\n\n1. Start by expressing \\( x \\) in terms of \\( y \\) from the equation \\( y = \\frac{2}{x+1} \\).\n2. Solve for \\( x \\) to find the inverse function \\( f^{-1}(y) \\).\... | 9 |
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 this problem, we need to determine the minimum compound interest rate \\( r \\) that the savings account must provide so that the worker has at least \\$66,200 by the end of the third year. The worker deposits \\$20,000 at the end of each year, and the interest is compounded annually.\n\nThe formula for t... | 10 |
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 |
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 amount in euros and the amount in pounds.\n\nLet's break it down:\n\n1.... | 22 |
26 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to understand the recursive function \\( f \\) and then compute \\( f(2015, 2) \\). Let's break down the function and compute the value step by step.\n\n1. **Base Case**: If \\( i = 0 \\) and \\( 0 \\le j \\le 4 \\), then \\( f(i, j) = \\text{mod}_5(j + 1) \\).\n2. **Recursive Case 1*... | 1 |
22 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to analyze the function \\(z(x, y) = (2x - y)^2 - 2y^2 - 3y\\). Archimedes wants to minimize \\(z\\) while Brahmagupta wants to maximize \\(z\\). We need to find the optimal value of \\(x\\) that minimizes \\(z\\) given that Brahmagupta will play optimally to maximize \\(z\\).\n\nFir... | -\frac{3}{8} |
54 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to follow these steps:\n\n1. **Determine the side length of the equilateral triangle**:\n - The area of an equilateral triangle with side length \\(a\\) is given by:\n \\[\n \\text{Area} = \\frac{\\sqrt{3}}{4} a^2\n \\]\n - Given that the area is \\(16\\sqrt{3}\\), we... | 6\sqrt{2} |
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 given information about the sums of the sequence and its powers. Let's denote the sequence by \\(a_1, a_2, \\ldots, a_n\\).\n\nGiven:\n\\[\n\\sum_{i=1}^n a_i = 96,\n\\]\n\\[\n\\sum_{i=1}^n a_i^2 = 144,\n\\]\n\\[\n\\sum_{i=1}^n a_i^3 = 216.\n\\]\n\nWe can use the identity f... | 64 |
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 solve this problem, let's follow these steps:\n\n1. **Define the original square:**\n - Let the side length of the original square be \\( s \\).\n - The area of the original square is \\( s^2 \\).\n\n2. **Define the dimensions of the resulting rectangle:**\n - After decreasing two opposite sides by 40%, t... | 10 |
44 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to first understand the recurrence relation given:\n\n\\[ x_n = x_{n-1} - x_{n-2} + x_{n-3} - x_{n-4} \\]\n\nWe are given the initial values:\n\\[ x_1 = 211, \\]\n\\[ x_2 = 375, \\]\n\\[ x_3 = 420, \\]\n\\[ x_4 = 523. \\]\n\nWe need to find the values of \\( x_{531} \\), \\( x_{753} ... | 898 |
33 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the value of \\( y \\), we need to use the formula for the projection of one vector onto another. The projection of vector \\(\\mathbf{a}\\) onto vector \\(\\mathbf{b}\\) is given by:\n\\[\n\\text{proj}_{\\mathbf{b}} \\mathbf{a} = \\left( \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\mathbf{b} \\cdot \\mathbf{b... | -4 |
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\nLe... | 504 |
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 |
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 the given expression \\( x\\left(x(1+x)+2x\\right)-3(x^2-x+2) \\), we will follow these steps:\n\n1. Distribute \\( x \\) inside the parentheses.\n2. Simplify the expression inside the parentheses.\n3. Distribute the simplified expression.\n4. Combine like terms.\n\nLet's perform these steps using Python ... | x^3+3x-6 |
54 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to use the properties of parallel lines and angles. Here's the step-by-step reasoning:\n\n1. Since \\(PT\\) is parallel to \\(QR\\), the angle \\(\\angle PQR\\) is equal to the angle \\(\\angle QPT\\) (alternate interior angles).\n\n2. The angles \\(\\angle QPT\\) and \\(\\angle PQR\\... | 116 |
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 range of the function \\( f(x) = (\\arccos x)^2 + (\\arccos x)^2 \\), we need to analyze the behavior of the function over its domain. The domain of both \\(\\arccos x\\) and \\(\\arcsin x\\) is \\([-1, 1]\\). \n\nFirst, let's recall the relationship between \\(\\arccos x\\) and \\(\\arcsin x\\):\n\\[\... | \left[ \frac{\pi^2}{8}, \frac{5 \pi^2}{4} \right] |
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 find the maximum value of the expression \\(\\frac{x - y}{x^4 + y^4 + 6}\\), we can use calculus and algebraic manipulation. However, a more insightful approach involves understanding the behavior of the function.\n\nFirst, let's consider the expression \\(\\frac{x - y}{x^4 + y^4 + 6}\\). The denominator \\(x^4... | \frac{1}{4} |
23 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to use the relationship between the volume of water being added to the tank and the rate at which the water level rises. Here are the steps to solve the problem:\n\n1. **Understand the problem:**\n - The volume of water being added to the tank is \\(20\\pi\\) cubic meters per hour.... | \sqrt{5} |
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 inequality. Here, we will use the AM-GM inequality for simplicity.\n\nThe AM-GM inequality states that for non-negative real numbers \\(x_1, x_2, \\ldots, x_n\\),\n\\[\n\\... | 27648 |
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 the problem, we need to evaluate the function \\(f(n)\\) for three different values: \\(\\frac{\\pi}{3}\\), \\(\\sqrt{45}\\), and \\(8^{2/3}\\). We will then sum the results.\n\n1. Evaluate \\(f\\left(\\frac{\\pi}{3}\\right)\\):\n - Since \\(\\frac{\\pi}{3} \\approx 1.047\\) and \\(1.047 < 4\\), we use ... | 12 |
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 find the median number of students in 2005 for the eight counties, we need to follow these steps:\n\n1. Extract the number of students for each county in 2005.\n2. Sort the list of students in ascending order.\n3. Find the median of the sorted list.\n\nLet's extract the data for 2005 and perform the necessary c... | 29 |
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-... | [
"In the diagram, \\(AD = BD = CD\\) and \\(\\angle BCA = 40^\\circ\\). We need to find the measure of \\(\\angle BAC\\).\n\nFirst, let's analyze the given information and the relationships in the diagram:\n\n1. Since \\(AD = BD = CD\\), point \\(D\\) is equidistant from points \\(A\\), \\(B\\), and \\(C\\). This me... | 90^\circ |
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 this problem, we need to find the side length of an equilateral triangle inscribed in the parabola \\(x^2 = 8y\\) with one vertex at the origin (0, 0). Let's denote the other two vertices of the equilateral triangle as \\((x_1, y_1)\\) and \\((x_2, y_2)\\).\n\nSince the triangle is equilateral and one of ... | 16 \sqrt{3} |
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 this problem, we need to find the measure of the angle \\(\\angle ADC\\) given that the total area of the shaded region is \\(\\frac{5}{12}\\) of the area of the larger circle.\n\n1. **Calculate the area of the larger circle**:\n The radius of the larger circle is 2, so its area is \\( \\pi \\times 2^2 ... | 120 |
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 quotient when \\(x^6 - 3\\) is divided by \\(x + 1\\), we can use polynomial long division. Here's the step-by-step process:\n\n1. Divide the leading term of the dividend (\\(x^6\\)) by the leading term of the divisor (\\(x\\)) to get the first term of the quotient (\\(x^5\\)).\n2. Multiply the entire ... | x^5 - x^4 + x^3 - x^2 + x - 1 |
13 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the probability that two randomly chosen chords of a circle intersect. The key insight here is that the chords $AB$ and $CD$ intersect if and only if the points $A$, $B$, $C$, and $D$ are arranged in such a way that $C$ and $D$ are on opposite sides of the line segment $... | \frac{1}{3} |
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-... | [
"To solve this problem, we need to use the information given about the polynomial \\(P(x)\\) and its remainders when divided by certain factors. We'll start by expressing \\(P(x)\\) in terms of its remainders and then use the given conditions to find the coefficients of the remainders.\n\n1. Since \\(P(x)\\) is a m... | 15 |
35 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to follow these steps:\n\n1. Calculate the square root of 20.\n2. Apply the ceiling function to the result of the square root.\n3. Square the result of the ceiling function.\n\nLet's do this step by step using Python code.\n```python\nimport math\n\n# Step 1: Calculate the square root... | 25 |
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 find the value of \\( x \\), we need to use the definition of the arithmetic mean. The arithmetic mean of a set of numbers is the sum of the numbers divided by the count of the numbers. In this case, the arithmetic mean of 7, 2, \\( x \\), and 10 is 9. This can be expressed as:\n\n\\[\n\\frac{7 + 2 + x + 10}{4}... | 17 |
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 $\\triangle ABC$ and $\\triangle CDE$ are similar by the AA similarity criterion (Angle-Angle).\n2. The ratio of the corresponding s... | 54 |
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 approach:\n\n1. **Analyze the Numerator:**\n The numerator is \\(|x^2 - 81|\\). The absolute value function \\(|x^2 - 81|\\... | (0,9) \cup (9,36) |
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 |
38 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to compute the binomial coefficient $\\dbinom{31}{28}$. The binomial coefficient $\\dbinom{n}{k}$ is given by the formula:\n\n\\[\n\\dbinom{n}{k} = \\frac{n!}{k!(n-k)!}\n\\]\n\nIn this case, $n = 31$ and $k = 28$. So we have:\n\n\\[\n\\dbinom{31}{28} = \\frac{31!}{28!(31-28)!} = \\fra... | 4495 |
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 shortest distance that can be traveled starting from any point \\(A\\), \\(B\\), \\(C\\), or \\(D\\) and visiting the other three points exactly once, we need to calculate the total distance for all possible routes and then find the minimum.\n\nWe can represent the distances between the points as follo... | 13 |
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 will use trigonometric identities and properties of the cosine function. Let's break it down step by... | \frac{1}{16} |
20 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem $(1+2i)6-3i$, we need to follow the order of operations, which in this case is multiplication and then subtraction.\n\n1. First, we multiply $(1+2i)$ by $6$:\n \\[\n (1+2i) \\cdot 6 = 6 + 12i\n \\]\n\n2. Next, we subtract $3i$ from the result:\n \\[\n 6 + 12i - 3i = 6 + 9i\n \\]\n\... | 6+9i |
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 solve the given problem, we need to simplify the expression for \\(\\tan \\theta\\). The given expression is:\n\n\\[\n\\tan \\theta = \\frac{\\cos 5^\\circ \\cos 20^\\circ + \\cos 35^\\circ \\cos 50^\\circ - \\sin 5^\\circ \\sin 20^\\circ - \\sin 35^\\circ \\sin 50^\\circ}{\\sin 5^\\circ \\cos 20^\\circ - \\sin... | 120^\circ |
46 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the shortest distance from a point to a line, we can use the formula for the distance from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$:\n\n\\[ \\text{Distance} = \\frac{|Ax_1 + By_1 + C|}{\\sqrt{A^2 + B^2}} \\]\n\nFirst, we need to rewrite the line equation $y = 2x - 2$ in the standard form $Ax + By +... | 2\sqrt{5} |
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 \\(\\frac{6r^2 - 19r - 7}{2r - 7} = 4r - 3\\), we can start by eliminating the denominator by multiplying both sides of the equation by \\(2r - 7\\). This gives us:\n\n\\[6r^2 - 19r - 7 = (4r - 3)(2r - 7).\\]\n\nNext, we expand the right-hand side:\n\n\\[6r^2 - 19r - 7 = 8r^2 - 28r - 6r + 21.... | 4 |
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 find the modular inverse of $997$ modulo $1000$, we need to find an integer $x$ such that:\n\n\\[997x \\equiv 1 \\pmod{1000}\\]\n\nThis is equivalent to finding $x$ such that:\n\n\\[997x = 1000k + 1\\]\n\nfor some integer $k$. This can be solved using the Extended Euclidean Algorithm, which finds solutions to e... | 333 |
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 use the property of a magic square where the sum of the numbers in each row, each column, and the two main diagonals are equal. Let's denote the common sum by \\( S \\).\n\nGiven the magic square:\n\\[\n\\begin{array}{|c|c|c|}\n\\hline\nn-3 & 3 & n+1 \\\\\n\\hline\nn+2 & 2n-9 & 1 ... | 7 |
43 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we will use modular arithmetic and properties of exponents. Here are the steps:\n\n1. Simplify the bases modulo 11:\n - $129 \\equiv 3 \\pmod{11}$\n - $96 \\equiv 5 \\pmod{11}$\n\n2. Rewrite the expression using these simplified bases:\n - $129^{34} + 96^{38} \\equiv 3^{34} + 5^{38} \\... | 9 |
3 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To convert the decimal number 555 to base 5, we need to repeatedly divide the number by 5 and keep track of the remainders. Here is the step-by-step process:\n\n1. Divide 555 by 5 to get the quotient and remainder.\n2. Divide the quotient from step 1 by 5 to get the next quotient and remainder.\n3. Continue this p... | 4210_{5} |
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 correctly, we need to find the value of $\\frac{-3+4i}{1+2i}$. We can do this by multiplying the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of $1+2i$ is $1-2i$. So we have:\n\n\\[\n\\frac{-3+4i}{1+2i} \\cdot \\frac{1-2i}{1-2i} = \\frac{(-3+... | 1+2i |
54 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the equation of the circle after reflecting each point of the circle \\(x^2 + y^2 = 25\\) in the point \\((4,1)\\).\n\n1. **Reflection of a Point:**\n The reflection of a point \\((x, y)\\) in the point \\((4, 1)\\) is given by:\n \\[\n (x', y') = (2 \\cdot 4 - x, 2 \\c... | (1,-16,-4,43) |
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 as \\( p \\) dollars and the cost of a jumbo eraser as \\( e \\) dollars. We are given the following two equations:\n\n1. \\( 3p + e = 1.24 \\)\n2. \\( 5p + e = 1.82 \\)\n\nWe can solve these equations to find th... | 29 |
26 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to set up a system of equations based on the information given:\n\n1. Let \\( x \\) be the number of questions Frank answered correctly.\n2. Let \\( y \\) be the number of questions Frank answered incorrectly.\n3. Since Frank answered 80 questions, the number of questions he did not ... | 56 |
52 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the side length of the cube from which the steel sphere is made. The sphere is inscribed in the cube, meaning that the diameter of the sphere is equal to the side length of the cube. Since the radius of the sphere is 3 inches, the diameter of the sphere is \\(2 \\times 3... | 216 |
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 problem, we need to evaluate the function \\( f(x) = 2^x \\) iteratively and then find the square root of the result. Here's the step-by-step process:\n\n1. Start with \\( x = 1 \\).\n2. Compute \\( f(1) = 2^1 = 2 \\).\n3. Compute \\( f(f(1)) = f(2) = 2^2 = 4 \\).\n4. Compute \\( f(f(f(1))) = f(4) = 2... | 256 |
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 this problem, we need to calculate the probability that after the process, the contents of the two bags are the same. Let's break down the steps:\n\n1. **Initial State**: Both Alice and Bob have a bag with balls of colors blue, green, orange, red, and violet.\n2. **Alice's Move**: Alice randomly selects o... | \frac{1}{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 Eng... | 16 |
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 convert the repeating decimals into fractions and then perform the arithmetic operations.\n\n1. Convert \\(0.\\overline{7}\\) to a fraction.\n2. Convert \\(0.\\overline{4}\\) to a fraction.\n3. Convert \\(0.\\overline{2}\\) to a fraction.\n4. Perform the arithmetic operations with ... | \frac 59 |
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.\n\nThe formula for the number o... | 9 |
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 convert the fraction $\\frac{9}{2}$ to a decimal, we perform the division of 9 by 2. The result is 4.5. Therefore, the decimal representation of $\\frac{9}{2}$ is:\n\n\\[\n\\boxed{4.5}\n\\] To verify the correctness of the answer, we can perform the division of 9 by 2 using Python. Let's do that.\n```python\n# ... | 4.5 |
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 need to perform the division of 57 by 160. We can do this using Python to ensure accuracy. Let's perform the calculation.\n```python\n# Perform the division to convert the fraction to a decimal\ndecimal_value = 57 / 160\nprint(decimal_value)\n`... | .35625 |
15 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem \\(5 \\star 1\\) using the given operation \\(a \\star b = 9a + 2b - ab + 5\\), we can follow these steps:\n\n1. Identify the values of \\(a\\) and \\(b\\). Here, \\(a = 5\\) and \\(b = 1\\).\n2. Substitute these values into the expression \\(9a + 2b - ab + 5\\).\n3. Simplify the expression to... | 47 |
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