prompt list | ground_truth string |
|---|---|
[
{
"content": "Calculate the value of the definite integral \\( \\int_{0}^{1} \\frac{e^{-x^{2}}-1}{x} \\, dx \\) with an accuracy up to \\(10^{-3}\\).",
"role": "user"
}
] | -0.398 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 188 g/mol |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.216 g |
[
{
"content": "How many homomorphisms exist from the dihedral group $D_5$ onto the group $\\mathbb{Z}_2 \\oplus \\mathbb{Z}_2$?",
"role": "user"
}
] | 0 |
[
{
"content": "Given a cubic polynomial $g(x)$ such that $g(1)=3$, $g'(1)=-2$, and $g''(1)=6$, evaluate the integral \\( \\int_{0}^{2} g(x) \\, dx. \\)",
"role": "user"
}
] | 8 |
[
{
"content": "An exam paper contains 4 multiple-choice questions, each with three options: (A), (B), and (C). For any group of 3 students taking the exam, there is always at least one question where their answers are all different. Each student answers every question. What is the maximum number of students who ... | 9 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.45 nm |
[
{
"content": "Find the integer value of \\( K \\) for the expression \\[ K = \\left\\lfloor\\sum_{r=1}^{80}\\int_0^1x^{\\sqrt r-1}dx\\right\\rfloor \\] after evaluating the integral and simplifying the sum.",
"role": "user"
}
] | 16 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $300,000 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $15.16 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 6.10% |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | C > D > B > F > E > A |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 433.61 m/s |
[
{
"content": "Find the arc length of the function \\( y = (x-x^2)^{1/2} + \\sin^{-1}(x^{1/2}) \\) over the interval \\([0, 1]\\).",
"role": "user"
}
] | 2 |
[
{
"content": "Provide an example of a well-known set with a cardinality greater than or equal to $\\aleph_2$. What is the set?",
"role": "user"
}
] | \mathcal{P}(\mathbb{R}) |
[
{
"content": "Evaluate the infinite sum \\( \\sum_{m=1}^{\\infty}\\frac{(-1)^{m+1}}{2m-1} \\) by considering the Fourier series of the function \\( f(x) = 1 + x \\) on the interval \\(-1 < x < 1\\). First, find the Fourier series representation of \\( f(x) \\), and then evaluate it at \\( x = \\frac{1}{2} \\) t... | \dfrac{\pi}{4} |
[
{
"content": "Find the minimum value of \\( f \\) such that for all non-zero real numbers \\( x, y, z, w \\), the inequality \\[ \\frac{6 x y + 5 y z + 6 z w}{x^{2} + y^{2} + z^{2} + w^{2}} \\leq f \\] holds.",
"role": "user"
}
] | \dfrac{9}{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 196.0 km/min |
[
{
"content": "Compute the limit \\( \\lim_{(x, y, z) \\to (0,0,0)} \\frac{2x^2 y \\cos(z)}{x^2 + y^2} \\) if it exists.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the range of values for \\( p \\) such that the series \\( \\sum_{n=2}^{\\infty} \\frac{1}{n (\\ln n)^p} \\) converges.",
"role": "user"
}
] | p > 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $10,475.5 |
[
{
"content": "Determine the smallest integer $n$ such that the alternating group $A_n$ contains an element of order $4$. Provide a general approach to constructing such an element in $A_n$.",
"role": "user"
}
] | 6 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 6.803 m/s |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 47.3% |
[
{
"content": "Given that $f'(x) = g(x)$ and $g'(x) = f(x)$ for all real $x$, with initial conditions $f(0) = 1$ and $f'(0) = 1$, find the value of $f^2(5) + g^2(5)$. Provide your answer as a single number.",
"role": "user"
}
] | 2e^{10} |
[
{
"content": "Determine the number of continuous functions on the interval $[0,1]$ that satisfy the equation $f(x) = \\frac{1}{\\sqrt{2}}f\\left(\\frac{x}{2}\\right)$ and are bounded on this interval.",
"role": "user"
}
] | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 10.55 N |
[
{
"content": "Find all non-empty sets $A$ of real numbers such that for any real numbers $a$ and $b$, if $a + b \\in A$, then $a \\cdot b \\in A$. Such a set $A$ is called complete.",
"role": "user"
}
] | \mathbb{R} |
[
{
"content": "Determine whether there exist values of $a$, $b$, $c$, and a positive integer $n$ such that the equation $a|x|^n + b|x|^{n-1} + c|x|^{n-2} = a|y|^n + b|y|^{n-1} + c|y|^{n-2}$ has solutions other than $|x| = |y|$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | {
"Date": "Mar 31",
"Description": "Machine rent expense",
"Debit": "$600",
"Credit": {
"Prepaid expense": "$600"
}
} |
[
{
"content": "Determine whether there exists a countable subset $C$ of polynomials with rational coefficients such that every continuous function on a closed interval $[a, b]$ can be uniformly approximated by polynomials from $C$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "A $5 \\times 8$ rectangle can be rolled to form two different cylinders with different maximum volumes. What is the ratio of the larger volume to the smaller volume? Express your answer as a common fraction.",
"role": "user"
}
] | \frac{8}{5} |
[
{
"content": "Solve for $x$: $3^{2x} = \\sqrt{27}$. Express your answer as a common fraction.",
"role": "user"
}
] | \frac{3}{4} |
[
{
"content": "Determine the intervals where the function \\(f(x) = (x+\\frac{1}{x})^{x}\\) is increasing and where it is decreasing.",
"role": "user"
}
] | (0, \infty) |
[
{
"content": "Find the limit of the sequence \\( \\frac{\\sqrt{12 + a_n} - \\sqrt{4a_n}}{a_n^2 - 2a_n - 8} \\) as \\( a_n \\to 4 \\) and \\( a_n > 4 \\).",
"role": "user"
}
] | -\dfrac{1}{16} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | In the long run, under perfect competition, firms make normal profit, meaning they don't make any economic profit. Thus the profit equals 0. The output for each firm is 12 units at a price of $280, where MC=AC. The firm's profit curve coincides with the x-axis because there is no economic profit. |
[
{
"content": "Find the maximum order of a permutation in the symmetric group $S_7$. Provide a specific permutation in $S_7$ that achieves this maximum order.",
"role": "user"
}
] | 12 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | osmosis |
[
{
"content": "Determine whether the integral \\( \\int_{-\\infty}^\\infty e^{-\\pi (a+z)^2} \\, da \\) converges for every \\( z \\in \\mathbb{C} \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the area of the portion of the sphere \\(x^{2} + y^{2} + z^{2} = a^{2}\\) that lies within the cylinder \\(x^{2} + y^{2} = ax\\) and above the \\(xy\\)-plane. Use the parametrization of the upper half of the sphere and evaluate the integral correctly to find the surface area.",
"role... | a^2(\pi - 2) |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 9810 Pa |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 0}(\\cos x)^{\\dfrac{1}{\\ln(\\sin^2(x))}}$$ without using L'Hospital's rule.",
"role": "user"
}
] | 1 |
[
{
"content": "All numbers from 1 to 155 are written on a blackboard, one time each. We randomly choose two numbers and delete them, replacing one of them with their product plus their sum. We repeat the process until there is only one number left. What is the average value of this final number?",
"role": "u... | 156! - 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.02 rad |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 547409.23 |
[
{
"content": "Find the remainder when \\(9^{9^{9}}\\) is divided by 100.",
"role": "user"
}
] | 89 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.0877 mol NaClO |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.7 kg |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 5.333 |
[
{
"content": "Determine whether there exists an interval $I \\subset (0.8856, +\\infty)$ such that the derivative $(\\Gamma^{-1})'(x) > 1$ for all $x$ in $I$. The function $\\Gamma^{-1}$ is the principal inverse of the gamma function, which is increasing and concave on $(0.8856, +\\infty)$. Provide a justificat... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | The cost of equity will reduce from 10.01% to 8.65% |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty} \\frac{\\sum_{r=0}^n\\binom{2n}{2r}3^r}{\\sum_{r=0}^{n-1}\\binom{2n}{2r+1}3^r} \\]",
"role": "user"
}
] | \sqrt{3} |
[
{
"content": "Determine whether there exists a commutative ring with a finite number of idempotents that is not equal to $2^n$ for some integer $n$. Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 4.52 m/s, 59.4 degrees above the horizontal |
[
{
"content": "Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that $f(5) = 3$ and\n\\[f(4xy) = 2y[f(x + y) + f(x - y)]\\]for all real numbers $x$ and $y.$ Find $f(2015).$",
"role": "user"
}
] | 1209 |
[
{
"content": "Find the remainder when $6x^4-14x^3-4x^2+2x-26$ is divided by $2x - 6.$",
"role": "user"
}
] | 52 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $95.24 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.9 s |
[
{
"content": "Determine whether the sequence \\( \\frac{1}{n}\\sum_{k=1}^n \\frac{k}{(k+1)\\log^2(k+1)} \\) converges to 0 as \\( n \\to \\infty \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Consider the function \\( f: A \\to \\mathbb{R} \\) where \\( A \\) is a subset of \\( \\mathbb{R}^2 \\). If \\( f^{-1}(C) \\) is closed in \\( A \\) for every closed set \\( C \\) in \\( \\mathbb{R} \\), does it imply that \\( f \\) is continuous?",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | (i) alkanes; (ii) alkyl halides; (iii) amines; (iv) alcohols; (v) acid. |
[
{
"content": "Let $p$ and $q$ be integers such that $q < 0 < p$. Prove that $M_q < G < M_p$ for positive real numbers $x_1, x_2, ..., x_n$ that are not all equal, where $G$ is the geometric mean.",
"role": "user"
}
] | M_q < G < M_p |
[
{
"content": "Compute the correlation coefficient \\( r(X_{(1)},X_{(3)}) \\) for the order statistics of three independent random variables \\( X_1, X_2, \\) and \\( X_3 \\), each following an \\( \\text{Exp}(1) \\) distribution.",
"role": "user"
}
] | \dfrac{2}{7} |
[
{
"content": "Find the limit of the function \\( p_n = \\frac{n}{(2n+1)^2} \\frac{1}{\\sum_{d=1}^n \\dfrac{1}{(2n+d)^2} } \\) as \\( n \\to \\infty \\).",
"role": "user"
}
] | \dfrac{3}{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $35,000 |
[
{
"content": "Solve for \\( x \\) in the equation \\( x^{x^{x^{.^{.^.}}}} = 2 \\).",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $71,656.03 |
[
{
"content": "Given positive real numbers $a, b, c$ such that $ab + bc + ca = 1$, find the maximum value of the expression:\n$$P = \\frac{a}{1+a^2} + \\frac{b}{1+b^2} - \\frac{1}{c^2+1}$$",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.86 N, repel each other |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 584 mm Hg |
[
{
"content": "Determine the angle of rotation for the rotation matrix $$ A=\\left[\\begin{array}{ccc} 0 & 0 & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\end{array}\\right] $$ given its diagonal form in a new basis $(x', y', z')$: $$ D=\\left[\\begin{array}{ccc} 1 & 0 & 0 \\\\ 0 & -\\left(\\frac{1+\\sqrt{3} i}{2}\\right)... | \dfrac{2\pi}{3} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Ni^{2+} |
[
{
"content": "Determine the limit of the sequence \\(y_n=\\left(\\frac{(2n)^{2n}}{(2n)!}\\right)^{\\frac{1}{2n}}\\) as \\(n\\) approaches infinity.",
"role": "user"
}
] | e |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $2,340 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 17.7m |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.1 L |
[
{
"content": "Calculate the probability that when 36 people are randomly split into 12 groups of 3, and then split again into 12 groups of 3, at least one group will have the same members as in the first grouping.",
"role": "user"
}
] | \dfrac{12}{595} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 5495.96 joules |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -1.5e-6 J |
[
{
"content": "Consider the power series function \\( f(z) = \\sum_{n\\geq0} nz^n = z + 2z^2 + 3z^3 + \\ldots \\) defined for \\(|z| < 1\\). Determine which of the following statements is correct about \\( f(z) \\):\n\n(a) \\( f(z) \\) is not injective, but attains every complex value at least once.\n(b) \\( f(z... | B |
[
{
"content": "Find the minimum value of the modulus of the complex number \\(m\\) such that the quadratic equation \\((4+3i) x^{2} + m x + (4-3i) = 0\\) has real roots.",
"role": "user"
}
] | 8 |
[
{
"content": "Determine whether the following statement is true or false: If $M$ is a monoid and $S$, $R$, and $T$ are submonoids of $M$ with $S \\subseteq R \\cup T$, then either $S \\subseteq R$ or $S \\subseteq T$. Provide a justification for your answer.",
"role": "user"
}
] | False |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 8.44 |
[
{
"content": "Find the relation between $x$ and $y$ by eliminating the parameter $t$ from the equations:\n\\[ x = \\frac{3 + 3t^2}{t^2 - 1} \\quad \\text{and} \\quad y = \\frac{3t}{t^2 - 1} \\]",
"role": "user"
}
] | x^2 - 4y^2 = 9 |
[
{
"content": "Calculate the line integral \\( \\oint_\\Gamma x \\, dx + (x+y) \\, dy + (x+y+z) \\, dz \\) using Stokes' theorem, where the curve \\( \\Gamma \\) is parameterized by \\( x = \\sin{t}, y = \\cos{t}, z = \\sin{t} + \\cos{t} \\) for \\( t \\in [0, 2\\pi] \\).",
"role": "user"
}
] | -\pi |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 5.173e16 |
[
{
"content": "Given that $3 \\cdot 40 A$ is the time needed to complete a job, which is $m$ times the time needed by $B$ and $C$ together to complete the job, $B$ needs $n$ times the time needed by $A$ and $C$ together to complete the job, and $C$ needs $x$ times the time needed by $A$ and $B$ together to compl... | E |
[
{
"content": "Let $x=\\frac{\\sum\\limits_{n=1}^{44} \\cos n^\\circ}{\\sum\\limits_{n=1}^{44} \\sin n^\\circ}$. What is the greatest integer that does not exceed $100x$?",
"role": "user"
}
] | 241 |
[
{
"content": "Given that all roots of the equation \\(z^4+az^3+bz^2+cz+d=0\\) (where \\(a, b, c, d \\in \\mathbb{R}\\)) are of unit modulus, determine which of the following statements are correct:\n\n(A) \\(|a| \\leq 4\\)\n\n(B) \\(|b| \\leq 4\\)\n\n(C) \\(|b| \\leq 6\\)\n\n(D) \\(|c| \\leq 4\\)",
"role": ... | ACD |
[
{
"content": "Solve the equation \\(\\log_{1-2x}(6x^2-5x+1) - \\log_{1-3x}(4x^2-4x+1) = 2\\) for \\(x\\), given that \\(x\\) must satisfy the conditions \\(1-2x > 0\\), \\(6x^2-5x+1 > 0\\), \\(1-3x > 0\\), and \\(x \\neq \\frac{1}{2}\\).",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 13.97 mL |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 200 nm |
[
{
"content": "Let \\( a_{1}, a_{2}, a_{3}, \\ldots, a_{100} \\) be integers such that \\[ \\frac{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+\\cdots+a_{100}^{2}}{a_{1}+a_{2}+a_{3}+\\cdots+a_{100}}=100. \\] Determine the maximum possible value of \\( a_{1} \\).",
"role": "user"
}
] | 550 |
[
{
"content": "Given that for any prime factors $p_i$ and $q_i$, if $p_i$ divides $q_i$ ($p_i | q_i$), then $p_i$ does not divide any other $q_j$ where $j \\neq i$. Also, if $p_i$ does not divide $q_i$, then it is false that $p_i$ divides some $q_j$ where $j \\neq i$. Does this imply that $p_i$ never divides any... | Yes |
[
{
"content": "What is the name of the convex set defined by \\( \\mathcal{A} = \\left\\{\\sum_{j=1}^N a_j x_j : -1 \\le a_j \\le 1, \\, \\, \\forall j=1,...,N\\right\\} \\), where \\( x_1,\\dots,x_N \\) are points in \\( \\mathbb{R}^n \\)?",
"role": "user"
}
] | zonotope |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3CuSO_4(aq) + 2Na_3PO_4(aq) \to Cu_3(PO_4)_2(s) + 3Na_2SO_4(aq) |
[
{
"content": "Evaluate the triple integral \\(\\int\\int\\int_V (x^2 + y^2 + z^2)\\, \\mathrm{d}x\\, \\mathrm{d}y\\, \\mathrm{d}z\\) over the region \\(V\\) bounded by the planes \\(x = 0\\), \\(y = 0\\), \\(z = 0\\), and \\(x + y + z = b\\) where \\(b > 0\\).",
"role": "user"
}
] | \dfrac{b^5}{20} |
[
{
"content": "Let \\( n \\geq 3 \\) be an integer. \\( n \\) balls are arranged in a circle and labeled with numbers \\( 1, 2, \\ldots, n \\) in order. There are 10 colors, and each ball is colored with one of them such that two adjacent balls have different colors. The sum of the labeled numbers of balls of ea... | 20 |
[
{
"content": "Find the supremum of the function \\( f(x) = \\sin(x) + \\sin(\\varphi x) \\), where \\( \\varphi \\) is the golden ratio.",
"role": "user"
}
] | 2 |
[
{
"content": "The base of a pyramid is a square with each side measuring one unit. One of its lateral edges is also one unit long and coincides with the height of the pyramid. Determine the largest face angle of the pyramid.",
"role": "user"
}
] | 90^\circ |
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