idx int64 0 61 | prompt stringlengths 868 2.58k | answers sequencelengths 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 |
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