idx int64 0 61 | prompt stringlengths 868 2.58k | answers listlengths 1 1 | gt stringlengths 1 53 |
|---|---|---|---|
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 |
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 find the coordinates of the point \\( P \\) on the line \\( y = -x + 6 \\) such that \\( P \\) is equidistant from the points \\( A(10, -10) \\) and \\( O(0, 0) \\).\n\n1. Let \\( P \\) be the point \\((x, y)\\). Since \\( P \\) lies on the line \\( y = -x + 6 \\), we can write \\... | (8,-2) |
27 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, let's denote the three consecutive integers as \\( n-1 \\), \\( n \\), and \\( n+1 \\).\n\n1. The product of these three integers is given by:\n \\[\n (n-1) \\cdot n \\cdot (n+1) = 120\n \\]\n\n2. The mean of these three integers is:\n \\[\n \\text{Mean} = \\frac{(n-1) + n + (n+1)}... | 6 |
18 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to set up a system of equations based on the given conditions. Let's denote the two-digit number as \\(10a + b\\), where \\(a\\) and \\(b\\) are the digits of the number. Here, \\(a\\) is the tens digit and \\(b\\) is the units digit.\n\nThe problem gives us two conditions:\n1. The s... | 143 |
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 find the quadratic polynomial \\(P(x) = ax^2 + bx + c\\) that satisfies the given inequalities and the condition \\(P(11) = 181\\).\n\nGiven the inequalities:\n\\[x^2 - 2x + 2 \\le P(x) \\le 2x^2 - 4x + 3\\]\nwe can infer that \\(P(x)\\) must be a linear combination of the polynom... | 406 |
42 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to understand the geometric properties of the square and the regular heptagon.\n\n1. **Square Properties**: Each internal angle of a square is \\(90^\\circ\\).\n2. **Regular Heptagon Properties**: Each internal angle of a regular heptagon is \\(\\frac{(7-2) \\times 180^\\circ}{7} = \... | \frac{270}7\text{ degrees} |
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 the problem, we need to evaluate the expression \\(441 + 2(21)(19) + 361\\). Let's break it down:\n\n1. First, calculate \\(2(21)(19)\\).\n2. Then, add the result to 441 and 361.\n\nLet's do this step-by-step using Python to ensure accuracy.\n```python\n# Step-by-step calculation\nterm1 = 441\nterm2 = 2 *... | 1600 |
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 solve the problem, we need to determine the $x$-intercept ($a$), the $y$-intercept ($b$), and the slope ($m$) of the line given by the equation $\\frac{x}{4} + \\frac{y}{12} = 1$. Then, we will compute the value of $a + b + m$.\n\n1. **Find the $x$-intercept ($a$):**\n The $x$-intercept is the value of $x$ wh... | 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-... | [
"To solve this problem, we need to determine the probability of obtaining face \\( F \\) on an unfair six-sided die and then use this information to find the probability of obtaining a sum of 7 when two such dice are rolled. We will then use the given probability to find the fraction \\( \\frac{m}{n} \\) and determ... | 29 |
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 these steps:\n\n1. First, we need to multiply the complex number $(1+2i)$ by 6.\n2. Then, we need to subtract $3i$ from the result obtained in step 1.\n\nLet's break it down and solve it step-by-step using Python and SymPy.\n\n```python\nimport sympy as sp\n\n# ... | 6+9i |
7 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To determine the degree of the polynomial \\( p(x) \\), we need to analyze the expression:\n\n\\[ p(x) = \\frac{(x - b)(x - c)}{(a - b)(a - c)} + \\frac{(x - a)(x - c)}{(b - a)(b - c)} + \\frac{(x - a)(x - b)}{(c - a)(c - b)}. \\]\n\nFirst, let's simplify the expression. We can observe that the polynomial \\( p(x)... | 0 |
50 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the value of \\( b \\) such that the line passing through the points \\((-5, 4)\\) and \\((-1, 3)\\) has a direction vector of the form \\(\\begin{pmatrix} -7 \\\\ b \\end{pmatrix}\\).\n\n1. **Find the direction vector of the line:**\n\n The direction vector of the lin... | \frac{7}{4} |
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 this problem, we can use a system of equations. Let's define:\n- \\( d \\) as the number of dimes.\n- \\( n \\) as the number of nickels.\n\nFrom the problem, we know the following:\n1. Hillary has a total of 11 coins, so \\( d + n = 11 \\).\n2. The total value of the coins is 75 cents. Since dimes are wo... | 7 |
56 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to understand the pattern of the sum of the first $N$ positive odd numbers. The $n$-th positive odd number can be represented as $2n - 1$. The sum of the first $N$ positive odd numbers is given by:\n\n$$\n1 + 3 + 5 + \\cdots + (2N - 1) = N^2\n$$\n\nThis is a well-known result from th... | 11 |
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-... | [
"The given sequence is a geometric sequence with the first term \\( a = \\frac{1}{4} \\) and common ratio \\( r = \\frac{1}{2} \\).\n\nThe sum of the first \\( n \\) terms of a geometric sequence is given by the formula:\n\\[ S_n = a \\frac{1 - r^n}{1 - r} \\]\n\nGiven that the sum of the first \\( n \\) terms is \... | 8 |
59 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the area of the trapezoid, we can use the formula for the area of a trapezoid:\n\n\\[ \\text{Area} = \\frac{1}{2} \\times (b_1 + b_2) \\times h \\]\n\nwhere \\( b_1 \\) and \\( b_2 \\) are the lengths of the two parallel sides (bases), and \\( h \\) is the height of the trapezoid.\n\nFrom the given diagram... | 21 |
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 the equation $\\sqrt{3x-5}=2$, we need to follow these steps:\n\n1. Square both sides of the equation to eliminate the square root.\n2. Solve the resulting linear equation for $x$.\n3. Verify the solution by substituting back into the original equation to ensure it satisfies the original equation.\n\nLet'... | 3 |
58 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to use the properties of the interior angles of a convex polygon. The sum of the interior angles of a polygon with \\( n \\) sides is given by the formula:\n\n\\[ \\text{Sum of interior angles} = 180(n-2) \\]\n\nIf the sum of all but one of the interior angles is \\( 1070^\\circ \\),... | 8 |
5 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the value of \\( b \\) when \\( a = 4 \\) and the point \\((a, b)\\) lies on the line with the equation \\(3x + 2y = 12\\), we can substitute \\( a = 4 \\) into the equation and solve for \\( b \\).\n\nThe equation of the line is:\n\\[3a + 2b = 12\\]\n\nSubstituting \\( a = 4 \\) into the equation, we get:... | 0 |
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 |
39 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the given expression \\((3x-2)(4x+1)-(3x-2)4x+1\\) when \\(x=4\\), we will follow these steps:\n\n1. Substitute \\(x = 4\\) into the expression.\n2. Simplify the expression step by step.\n\nLet's start by substituting \\(x = 4\\) into the expression:\n\n\\[\n(3 \\cdot 4 - 2)(4 \\cdot 4 + 1) - (3 \\cdot 4 ... | 11 |
61 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to find the value of \\( k \\) that satisfies the equation \\((3^k)^6 = 3^6\\).\n\nFirst, we can simplify the left-hand side of the equation using the properties of exponents. Specifically, \\((a^m)^n = a^{mn}\\). So, \\((3^k)^6 = 3^{6k}\\).\n\nNow the equation becomes \\(3^{6k} = 3^6... | 1 |
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 find the unicycle's speed in inches per second, we need to determine the distance traveled by the unicycle in one second. Here are the steps to solve the problem:\n\n1. Calculate the circumference of the wheel. The circumference \\(C\\) of a circle is given by the formula \\(C = 2 \\pi r\\), where \\(r\\) is th... | 12\pi |
43 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find all two-digit numbers of the form $B6$ that are squares of positive integers. The number $B6$ can be written as $10B + 6$. We need to check which values of $B$ (from 0 to 9) make $10B + 6$ a perfect square.\n\nWe can do this by iterating through all possible values of $B$ and... | 2 |
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 find the minimum value of the function \\( f(x) = |x - p| + |x - 15| + |x - p - 15| \\) for \\( p \\leq x \\leq 15 \\), we need to analyze the behavior of the function within this interval.\n\n1. **Understand the Function:**\n - The function \\( f(x) \\) is composed of three absolute value terms.\n - The te... | 15 |
28 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To convert a decimal number to its base six equivalent, we need to repeatedly divide the number by 6 and keep track of the remainders. Here's the step-by-step process:\n\n1. Divide the decimal number by 6.\n2. Record the remainder.\n3. Update the decimal number to the quotient from the division.\n4. Repeat steps 1... | 4343_6 |
49 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to follow these steps:\n\n1. Identify the first term and the common ratio of the geometric series.\n2. Use the formula for the sum of an infinite geometric series to find the sum.\n3. Determine the condition for the sum to be a perfect square.\n4. Find the smallest possible value of ... | 4 |
55 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to use the properties of trapezoids and triangles. Here's the step-by-step reasoning:\n\n1. **Understand the Geometry:**\n - \\(AD \\parallel BC\\), so \\(ABCD\\) is a trapezoid.\n - \\(AC \\perp CD\\), so \\(\\angle ACD = 90^\\circ\\).\n - \\(AC\\) bisects \\(\\angle BAD\\), s... | 28 |
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 this problem, we need to find a nonzero digit \\( N \\) such that the number \\( 21420N \\) is divisible by 6. A number is divisible by 6 if and only if it is divisible by both 2 and 3.\n\n1. **Divisibility by 2**: The last digit \\( N \\) must be even. So \\( N \\) can be one of \\( \\{0, 2, 4, 6, 8\\} \... | 6 |
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. Expand the inner expression \\( x(1+x) \\).\n2. Add the result to \\( 2x \\).\n3. Multiply the entire result by \\( x \\).\n4. Subtract \\( 3(x^2 - x + 2) \\) from the previous result.\n\nLet's break it do... | x^3+3x-6 |
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 solve the problem, we need to understand the behavior of the function \\( F(z) = \\frac{z + i}{z - i} \\) and how it transforms the sequence \\( z_n \\). We will start by calculating the first few terms of the sequence to see if there is a pattern.\n\nGiven:\n\\[ z_0 = \\frac{1}{137} + i \\]\n\nWe need to find ... | 1+274i |
3 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the number of ways to distribute 6 cars into 3 lanes (left-turn, straight-going, and right-turn) such that each lane has at least one car. This is a combinatorial problem that can be approached using the principle of inclusion-exclusion.\n\nHere's the step-by-step reason... | 540 |
41 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the values of $\\lambda$ for which there exists an $x \\in [0,1]$ such that $f(x) \\neq x$ but $f(f(x)) = x$. Let's break down the problem into manageable steps:\n\n1. **Define the function $f(x)$:**\n \\[f(x) = \\lambda x(1 - x).\\]\n\n2. **Find the condition for $f(f(x)) ... | (3,4] |
31 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the probability that a randomly selected point $(x, y)$ within the rectangle defined by $0 \\le x \\le 8$ and $0 \\le y \\le 4$ satisfies the inequality $x + y \\le 4$.\n\n1. **Define the Total Area**: The total area of the rectangle is $8 \\times 4 = 32$.\n2. **Define t... | \frac{1}{4} |
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 find the remainder when $(5x + 9)^{611} + (x + 5)^{11} + (x - 1)^{11} + 3x^2 + 1$ is divided by $x + 2$, we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial $f(x)$ by $x - c$ is $f(c)$. In this case, we need to find the value of the polynomial when $... | 12 |
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 infinite sum:\n\n\\[\n\\sum_{n = 1}^\\infty \\frac{F_{n + 1}}{F_n F_{n + 2}},\n\\]\n\nwhere \\( F_n \\) denotes the \\( n \\)-th Fibonacci number. \n\nFirst, let's recall the definition of the Fibonacci sequence:\n- \\( F_0 = 0 \\)\n- \\( F_1 = 1 \\)\n- \\( F_{n+2} = F... | 2 |
49 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to find the value of \\(x\\) that satisfies the equation \\(\\arccos x + \\arccos 2x + \\arccos 3x = \\pi\\). We will then determine the cubic polynomial that \\(x\\) satisfies and find the smallest possible value of \\(|a| + |b| + |c| + |d|\\).\n\nFirst, let's use the property of the... | 27 |
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 solve this problem, we need to use the binomial probability formula. Here's the step-by-step reasoning:\n\n1. **Understand the Problem:**\n - We have $n$ fair 6-sided dice.\n - We want exactly two of them to show a number other than 1.\n - The probability of a single die showing a number other than 1 is $... | 4 |
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.\n\n1. First, let's analyze the numerator \\(|x^2 - 81|\\). The absolute value function \\(|x^2 - 81|\\) is always non-negative, and it is zero when \... | (0,9) \cup (9,36) |
61 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To convert the fraction $\\frac{57}{160}$ to a terminating decimal, we can perform the division $57 \\div 160$.\n\nLet's do this step by step using Python to ensure accuracy.\n```python\n# Perform the division to convert the fraction to a decimal\nfraction = 57 / 160\nprint(fraction)\n```\n```output\n0.35625\n```\... | .35625 |
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 |
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 determine the exchange rate between lire and dollars based on the given information, and then use that exchange rate to find out how many dollars are equivalent to 1,000,000 lire.\n\n1. First, we need to find the exchange rate. We know that 1200 lire is equivalent to $1.50. So, th... | 1250 |
15 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the values of \\(N\\) that satisfy the following conditions:\n1. \\(N \\equiv 5 \\pmod{8}\\)\n2. \\(N \\equiv 3 \\pmod{6}\\)\n\nWe can start by iterating through possible values of \\(N\\) and checking which values satisfy both conditions. Since \\(N < 50\\), we can limit our... | 66 |
13 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the probability that the product of the numbers on two fair 6-sided dice is a multiple of 5. \n\nHere's the step-by-step reasoning:\n\n1. **Identify the total number of outcomes**: When two dice are thrown, there are \\(6 \\times 6 = 36\\) possible outcomes.\n\n2. **Iden... | \frac{11}{36} |
18 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the smallest distance between the origin and a point on the graph of \\( y = \\frac{1}{\\sqrt{2}}(x^2 - 3) \\), we need to minimize the distance function \\( D = \\sqrt{x^2 + y^2} \\). Since \\( y = \\frac{1}{\\sqrt{2}}(x^2 - 3) \\), we can substitute this into the distance function to get:\n\n\\[ D = \\sq... | 12 |
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 the problem, we need to follow the given operation definition:\n\n\\[\n\\frac{m}{n} @ \\frac{p}{q} = (m)(p)\\left(\\frac{q}{n}\\right)\n\\]\n\nGiven the fractions $\\frac{7}{30}$ and $\\frac{10}{21}$, we can identify $m = 7$, $n = 30$, $p = 10$, and $q = 21$. Substituting these values into the operation d... | 49 |
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 the greatest integer less than \\((\\sqrt{7} + \\sqrt{5})^6\\). We can use the binomial theorem and properties of conjugates to help us with this calculation.\n\nFirst, let's consider the expression \\((\\sqrt{7} + \\sqrt{5})^6 + (\\sqrt{7} - \\sqrt{5})^6\\). By the binomial t... | 13535 |
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 determine the values of \\( x \\) that are not in the domain of the function\n\n\\[ f(x) = \\frac{1}{|x^2 + 3x - 4| + |x^2 + 9x + 20|}, \\]\n\nwe need to identify when the denominator is zero. The function will be undefined when the denominator is zero because division by zero is not allowed.\n\nThe denominator... | -4 |
42 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the equation \\((3)5^2 - 4(5 - a)^2 \\div 3 = 63\\), we need to follow these steps:\n\n1. Simplify the equation.\n2. Solve for \\(a\\).\n3. Sum the values of \\(a\\) that satisfy the equation.\n\nLet's start by simplifying the equation:\n\n\\[\n(3)5^2 - \\frac{4(5 - a)^2}{3} = 63\n\\]\n\nThis simplifies 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 solve this problem, we'll use the given equations and find the values of \\(x\\), \\(y\\), and \\(z\\). Here are the steps we'll follow:\n\n1. Multiply all three equations together: \\((xy)(xz)(yz) = 24 \\cdot 48 \\cdot 72\\).\n2. This gives us \\((xyz)^2 = 24 \\cdot 48 \\cdot 72\\).\n3. Solve for \\(xyz\\) by ... | 22 |
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 this problem, we need to count the number of ways to form a subcommittee of 5 members from a Senate committee of 8 Republicans and 6 Democrats, ensuring that the subcommittee has at least one member from each party.\n\nWe can break this down into manageable steps:\n\n1. Calculate the total number of ways ... | 1940 |
47 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find out how much of the pie Louis and Jack eat altogether, we need to add the fractions $\\frac{1}{5}$ and $\\frac{2}{3}$. To add these fractions, we need a common denominator. The least common multiple of 5 and 3 is 15. So, we convert $\\frac{1}{5}$ to $\\frac{3}{15}$ and $\\frac{2}{3}$ to $\\frac{10}{15}$. A... | \frac{13}{15} |
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 integer roots of the polynomial equation \\(x^4 + 5x^3 + 9x^2 - x - 14 = 0\\), we can use the Rational Root Theorem. The Rational Root Theorem states that any rational solution, expressed in its lowest terms, must be such that the numerator is a factor of the constant term (-14) and the denominator is ... | -2,1 |
28 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to determine for how many real values of \\(x\\) the expression \\(\\sqrt{120 - \\sqrt{x}}\\) is an integer.\n\n1. Let \\( \\sqrt{120 - \\sqrt{x}} = k \\), where \\( k \\) is an integer.\n2. Then, \\( 120 - \\sqrt{x} = k^2 \\).\n3. Rearrange to solve for \\(\\sqrt{x}\\): \\( \\sqrt{x}... | 11 |
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 to find the midp... | (-1,6) |
59 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to expand the expression \\((\\sqrt{2} + \\sqrt{3})^3\\) and express it in the form \\(a\\sqrt{2} + b\\sqrt{3} + c\\sqrt{6}\\). We can use the binomial theorem for this expansion.\n\nThe binomial theorem states that:\n\\[\n(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\n\\]\nSubstituting \\(x ... | 20 |
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 determine the total number of outfits that can be made consisting of one shirt, one pair of pants, and one hat, we can use the fundamental principle of counting. According to this principle, if there are \\(m\\) ways to do one thing, and \\(n\\) ways to do another, then there are \\(m \\times n\\) ways to do bo... | 240 |
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 area of the shaded triangle, we can use the formula for the area of a triangle, which is given by:\n\n\\[ \\text{Area} = \\frac{1}{2} \\times \\text{base} \\times \\text{height} \\]\n\nFrom the diagram, we can see that the base of the triangle is 10 cm and the height is 3 cm. So, we can substitute thes... | 15\mbox{ cm}^2 |
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 the problem $99^2 + 99 + 1$ in our heads, we can break it down into simpler steps. Here's the step-by-step reasoning:\n\n1. First, calculate $99^2$.\n2. Then, add 99 to the result.\n3. Finally, add 1 to the result.\n\nLet's do this step by step.\n\n1. Calculate $99^2$:\n \\[\n 99^2 = (100 - 1)^2 = 100... | 9901 |
26 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve $513^2 - 487^2$, we can use the difference of squares formula, which states that $a^2 - b^2 = (a - b)(a + b)$. Here, $a = 513$ and $b = 487$. So we have:\n\n\\[513^2 - 487^2 = (513 - 487)(513 + 487)\\]\n\nNow let's calculate the values inside the parentheses:\n\n\\[513 - 487 = 26\\]\n\\[513 + 487 = 1000\\... | 26000 |
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 substitute \\(a = 5\\) and \\(b = 1\\) into the expression. Let's calculate it step by step.\n\nThe expression for \\(5 \\star 1\\) is:\n\\[5 \\star 1 = 9(5) + 2(1) - (5)(1) + 5\\]\n\nLet's break it down:\n1... | 47 |
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 determine the equation of the hyperbola given the foci and the difference in distances from any point on the hyperbola to the foci. The standard form of a hyperbola centered at \\((h, k)\\) is:\n\n\\[\n\\frac{(x - h)^2}{a^2} - \\frac{(y - k)^2}{b^2} = 1\n\\]\n\nGiven:\n- Foci \\(F... | 16 |
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 use trigonometric identities and properties of angles. Given that $\\sin \\angle RPQ = \\frac{7}{25}$, we can use the Pythagorean identity to find $\\cos \\angle RPQ$. The Pythagorean identity states that $\\sin^2 \\theta + \\cos^2 \\theta = 1$. Therefore, we can solve for $\\cos ... | -\frac{24}{25} |
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 equation \\(\\sqrt{x + \\sqrt{3x + 6}} + \\sqrt{x - \\sqrt{3x + 6}} = 6\\), we will follow these steps:\n\n1. Let \\(a = \\sqrt{x + \\sqrt{3x + 6}}\\) and \\(b = \\sqrt{x - \\sqrt{3x + 6}}\\). Then the equation becomes \\(a + b = 6\\).\n2. Square both sides of the equation to eliminate the square root... | 10 |
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 convert the bicycle's speed from feet per minute to inches per second, we need to follow these steps:\n\n1. Convert feet to inches. Since there are 12 inches in a foot, we multiply the speed in feet per minute by 12.\n2. Convert minutes to seconds. Since there are 60 seconds in a minute, we divide the result by... | 4 |
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 calculate the total cost for both Susan and Pam after applying their respective discounts, and then find the difference between the two amounts.\n\n1. Calculate the cost for Susan:\n - Full price of one ticket: \\$20\n - Discount for Susan: 25%\n - Number of tickets Susan bu... | 10 |
33 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the number of ways to assign 3 people to 5 different offices, with each person getting exactly one office. This is a permutation problem where we are selecting 3 offices out of 5 and arranging 3 people in those offices.\n\nThe number of ways to assign 3 people to 5 diffe... | 60 |
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 the equation \\( |x+5| - |3x-6| = 0 \\), we need to consider the different cases for the absolute values. The absolute value function changes its behavior at the points where the expression inside the absolute value is zero. So, we need to consider the points \\( x = -5 \\) and \\( x = 2 \\) (since \\( x+... | \frac{11}{2} |
55 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we can set up a system of linear equations. Let's denote the cost of a pencil by \\( p \\) and the cost of a jumbo eraser by \\( e \\). The given information can be translated into the following equations:\n\n1. \\( 3p + e = 1.24 \\)\n2. \\( 5p + e = 1.82 \\)\n\nWe can solve these equations ... | 29 |
19 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to determine the matrix \\(\\mathbf{A}\\) and then use it to find \\(\\mathbf{A} \\begin{pmatrix} -13 \\\\ 3 \\\\ 4 \\end{pmatrix}\\).\n\nGiven:\n\\[\n\\mathbf{A} \\begin{pmatrix} 3 \\\\ 1 \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} 2 \\\\ 4 \\\\ -3 \\end{pmatrix}\n\\]\n\\[\n\\mathbf{A}... | \begin{pmatrix} -2 \\ -14 \\ -7 \end{pmatrix} |
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 convert the fraction $\\frac{21}{2^2 \\cdot 5^7}$ to a terminating decimal, we need to follow these steps:\n\n1. Simplify the denominator to a power of 10.\n2. Multiply the numerator and the denominator by a suitable number to achieve this.\n3. Perform the division to get the decimal representation.\n\nThe deno... | .0000672 |
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 determine the type of graph represented by the equation \\(\\left(\\frac{x}{2} - 3\\right)^2 + y^2 = 10\\), we can follow these steps:\n\n1. Simplify the equation to a standard form.\n2. Identify the type of conic section based on the simplified form.\n\nLet's start by simplifying the given equation. The equati... | \text{ellipse} |
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 simplify the expression \\(\\frac{1}{5} \\cdot \\frac{8}{7} \\div \\frac{12}{20}\\), we can follow these steps:\n\n1. First, rewrite the division as multiplication by the reciprocal. So \\(\\frac{1}{5} \\cdot \\frac{8}{7} \\div \\frac{12}{20}\\) becomes \\(\\frac{1}{5} \\cdot \\frac{8}{7} \\cdot \\frac{20}{12}\... | \frac{8}{21} |
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 determine for which prime numbers \\( p \\) the congruence \\( 8x \\equiv 1 \\pmod{p} \\) has no solutions. This congruence has a solution if and only if 8 has an inverse modulo \\( p \\). The inverse of 8 modulo \\( p \\) exists if and only if \\(\\gcd(8, p) = 1\\). Since \\( p \\... | 2 |
61 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To determine the domain of the function \\( f(x) = \\frac{2-x}{\\log(2-\\log(x-2))} \\), we need to ensure that the expression inside the logarithm is positive and that the denominator is not zero. Let's break it down step by step.\n\n1. The argument of the outer logarithm must be positive:\n \\[\n 2 - \\log(x... | (2,12) \cup (12,102) |
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 determine the range of the function \\( y = \\log_2 (\\sqrt{\\sin x}) \\) for \\( 0^\\circ < x < 180^\\circ \\), we need to analyze the behavior of the function within this interval.\n\n1. **Understand the Range of \\(\\sin x\\):**\n - For \\( 0^\\circ < x < 180^\\circ \\), the value of \\(\\sin x\\) ranges f... | (-\infty, 0] |
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} |
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} = \\phi\\), where \\(\\phi = \\frac{1 + \\sqrt{5}}{2}\\) is the golden ratio. We know that \\(\\phi\\) satisfies the equation \\(\\phi^2 = ... | -2 |
7 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the number of different possible rational roots of the polynomial \\(2x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 1 = 0\\), we can use the Rational Root Theorem. The Rational Root Theorem states that any rational root of the polynomial \\(a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0 = 0\\) must be of the form \... | 4 |
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 distance between two points \\((x_1, y_1)\\) and \\((x_2, y_2)\\) in a Cartesian plane, we can use the distance formula:\n\n\\[\nd = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\n\\]\n\nGiven the points \\((2, -6)\\) and \\((-4, 3)\\), we can substitute these values into the formula:\n\n\\[\nd = \\sqrt{((-4) ... | 3\sqrt{13} |
60 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to find the value of the function \\( f(n) \\) for \\( n = 1, 2, 3, \\ldots, 1985 \\) given the recurrence relation \\( f(n + 1) = (-1)^{n + 1} n - 2f(n) \\) and the initial condition \\( f(1) = f(1986) \\). We will then compute the sum \\( f(1) + f(2) + f(3) + \\dots + f(1985) \\).\n... | 331 |
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 consider the set of all triples \\((a, b, c)\\) of positive integers for which there exist triangles with side lengths \\(a\\), \\(b\\), and \\(c\\). This means that the side lengths must satisfy the triangle inequality: \\(a + b > c\\), \\(a + c > b\\), and \\(b + c > a\\).\n\nWe ... | \frac{17}{21} |
47 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the vertical asymptotes of the function \\( f(x) = \\frac{2x}{x^2 - 5x - 14} \\), we need to determine where the denominator is zero. The denominator is \\( x^2 - 5x - 14 \\). We can factor this quadratic expression to find the roots.\n\n\\[ x^2 - 5x - 14 = (x - 7)(x + 2) \\]\n\nSo, the vertical asymptotes... | 5 |
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 determine who earns the most after working an eight-hour day, we need to convert their hourly wages into a common currency, such as US dollars. Here are the steps:\n\n1. Convert Navin's hourly wage from Mauritian rupees to US dollars.\n2. Convert Luka's hourly wage from Croatian kuna to US dollars.\n3. Convert ... | \text{Navin} |
27 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to find the determinant of the matrix \\(\\begin{vmatrix} x_1 & y_1 & 1 \\\\ x_2 & y_2 & 1 \\\\ x_3 & y_3 & 1 \\end{vmatrix}\\) and then square it. The determinant of the matrix can be expressed as:\n\\[\n\\text{det} = \\begin{vmatrix} x_1 & y_1 & 1 \\\\ x_2 & y_2 & 1 \\\\ x_3 & y_3 ... | 144 |
40 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the expression \\(\\tan 100^\\circ + 4 \\sin 100^\\circ\\), we can use trigonometric identities and properties. Let's break it down:\n\n1. Recall that \\(\\tan 100^\\circ = \\tan (180^\\circ - 80^\\circ) = -\\tan 80^\\circ\\).\n2. We can use the identity \\(\\sin 100^\\circ = \\sin (180^\\circ - 80^\\circ... | -\sqrt{3} |
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 largest integer less than 2010 that satisfies the following system of congruences:\n\\[ x \\equiv 5 \\pmod{7} \\]\n\\[ x \\equiv 10 \\pmod{11} \\]\n\\[ x \\equiv 10 \\pmod{13} \\]\n\nFirst, we can combine the last two congruences since they both have the same remainder:\n... | 1440 |
9 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To find the integer value of \\( x \\) in the arithmetic sequence \\( 3^2, x, 3^4 \\), we need to recognize that in an arithmetic sequence, the difference between consecutive terms is constant. This means that the difference between \\( x \\) and \\( 3^2 \\) is the same as the difference between \\( 3^4 \\) and \\... | 45 |
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 rewrite the quadratic expression \\(x^2 + 2.6x + 3.6\\) in the form \\((x + b)^2 + c\\). This process is known as completing the square.\n\n1. Start with the quadratic expression \\(x^2 + 2.6x + 3.6\\).\n2. To complete the square, we take the coefficient of the \\(x\\) term, which ... | 3.21 |
54 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To convert the fraction $\\frac{3}{20}$ to a decimal, we can perform the division of 3 by 20. Here's the step-by-step process:\n\n1. Set up the division: $3 \\div 20$.\n2. Since 3 is less than 20, we add a decimal point and a zero to 3, making it 30.\n3. Divide 30 by 20, which gives us 1 with a remainder of 10.\n4... | 0.15 |
42 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to use some geometric properties and trigonometric identities. Here's the step-by-step reasoning:\n\n1. **Understanding the Geometry:**\n - Since \\(AB = AC\\), triangle \\(ABC\\) is isosceles with \\(AM\\) as the altitude, which means \\(M\\) is the midpoint of \\(BC\\).\n - Let... | 11 \sqrt{5} + 11 |
16 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem $1 - 2 + 3 - 4 + 5 - \\dots + 99 - 100$, we can observe the pattern in the series. The series alternates between positive and negative terms. We can group the terms in pairs:\n\n\\[\n(1 - 2) + (3 - 4) + (5 - 6) + \\dots + (99 - 100)\n\\]\n\nEach pair $(1 - 2)$, $(3 - 4)$, $(5 - 6)$, etc., eval... | -50 |
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-... | [
"Given that $\\sin D = 0.7$ and $EF = 7$, we need to find the length of $DE$.\n\nIn the right triangle $DEF$, $\\sin D$ is defined as the ratio of the length of the opposite side to the hypotenuse. So, we have:\n\\[\n\\sin D = \\frac{EF}{DE}\n\\]\nGiven that $\\sin D = 0.7$ and $EF = 7$, we can substitute these val... | \sqrt{51} |
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 |
33 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve the problem, we need to understand the geometry of the pentagon and the star formed by extending its sides. The key is to recognize that the angles in the star are related to the angles of the pentagon.\n\n1. **Interior Angle of a Regular Pentagon:**\n The interior angle of a regular pentagon is given b... | 36^\circ |
31 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"To solve this problem, we need to convert the decimal number 629 to its base 7 and base 8 representations, and then compare the digits in these representations to find out how many digits are the same.\n\n1. Convert 629 from base 10 to base 7.\n2. Convert 629 from base 10 to base 8.\n3. Compare the digits of the b... | 3 |
46 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"Given that $\\omega^{1997} = 1$ and $\\omega \\neq 1,$ we need to evaluate the sum\n\\[\n\\frac{1}{1 + \\omega} + \\frac{1}{1 + \\omega^2} + \\dots + \\frac{1}{1 + \\omega^{1997}}.\n\\]\n\nFirst, let's recall that $\\omega$ is a primitive 1997th root of unity. This means that $\\omega^k$ for $k = 1, 2, \\dots, 199... | \frac{1997}{2} |
37 | <|im_start|>system
You are a mathematical reasoning assistant. For each problem, follow these steps strictly:
1. Solve the problem using step-by-step reasoning and output the final answer within \boxed{}.
2. Perform a self-evaluation:
- You may include reasoning to verify correctness.
- However, your final self-... | [
"A regular pentagon has 5 sides. When it is rotated, it will coincide with its original position when the rotation is a multiple of the angle that evenly divides the full circle (360 degrees) by the number of sides. \n\nTo find the minimum number of degrees the pentagon must be rotated until it coincides with its o... | 72 |
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 a set of distinct positive integers whose product is 84 and whose sum is minimized. We can start by factorizing 84 into its prime factors and then combining these factors in different ways to form the set of integers.\n\nThe prime factorization of 84 is:\n\\[ 84 = 2^2 \\times... | 14 |
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 simplify the given expression, we need to break it down step by step. The expression is:\n\n\\[\n\\sqrt{\\dfrac{\\dfrac{5}{\\sqrt{80}}+\\dfrac{\\sqrt{845}}{9}+\\sqrt{45}}{\\sqrt{5}}}\n\\]\n\nFirst, let's simplify the terms inside the fraction:\n\n1. Simplify \\(\\dfrac{5}{\\sqrt{80}}\\):\n\\[\n\\dfrac{5}{\\sqrt... | \frac{13}{6} |
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