prompt list | ground_truth string |
|---|---|
[
{
"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... | 8.10e-35 m |
[
{
"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... | $2,143 |
[
{
"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... | 131.66 mL |
[
{
"content": "Does there exist a finitely presented group that does not contain a nonabelian free group and is not residually amenable?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the number of homomorphisms from the alternating group $A_4$ to the cyclic group $\\mathbb{Z}_{12}$.",
"role": "user"
}
] | 3 |
[
{
"content": "Let $A$, $B$, and $C$ be three different real $3 \\times 3$ matrices with the following properties:\n\n- $A$ has the complex eigenvalue $\\lambda=3-5i$.\n- $B$ has eigenvalues $\\lambda=0$, $\\lambda=5$, $\\lambda=-5$.\n- $C = M M^T$ for some real $3 \\times 2$ matrix $M$.\n\nWhich of the matrices... | D |
[
{
"content": "Find the greatest common divisor (gcd) of $3 + \\sqrt{13}$ and $2 + 5\\sqrt{13}$ in the ring $\\mathbb{Z}[(1+\\sqrt{13})/2]$. Express your answer in the form $a + b\\sqrt{13}$, where $a$ and $b$ are integers.",
"role": "user"
}
] | 1 |
[
{
"content": "A point \\( P \\) moves along the parabola \\( y^2 = 2x \\). Points \\( B \\) and \\( C \\) lie on the y-axis. A circle defined by \\((x-1)^2 + y^2 = 1\\) is inscribed in \\(\\triangle PBC\\). Determine the minimum area of \\(\\triangle PBC\\).",
"role": "user"
}
] | 8 |
[
{
"content": "Consider the sequence of numbers: \\(2^1 - 1, 2^2 - 1, 2^3 - 1, \\ldots, 2^{100} - 1\\). Alice repeatedly selects two numbers \\(a\\) and \\(b\\) from the sequence, erases them, and writes \\(\\frac{ab - 1}{a + b + 2}\\) on the board. She continues this process until only one number remains. If th... | 100 |
[
{
"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... | 155 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... | 0.022L |
[
{
"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... | 8.49e4 m |
[
{
"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... | 449.82 |
[
{
"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... | 7.74 g |
[
{
"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... | 9.23e-8 N |
[
{
"content": "Determine the appropriate way to write the following inequality involving norms and mappings:\n\nGiven a continuous and bounded function $x : \\mathbf{R} \\to \\mathbf{C}$, where $x(t) \\neq 0$ only if $a \\leq t \\leq b$ and $|x(t)| \\leq 1$ for all $t$, consider the mappings:\n\\[ Aux_{\\tau} : ... | 1 |
[
{
"content": "Consider a closed irreducible algebraic set $V \\subseteq \\mathbb{A}^n_k$ over a closed field $k$. Construct the topological space $X$ consisting of all closed irreducible subsets of $V$, equipped with the Zariski topology. Define a continuous map $f: V \\to X$ by $f(x) = \\overline{\\{x\\}}$. Le... | True |
[
{
"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... | 9.53% |
[
{
"content": "Given real numbers $x, y, z$ such that $x^2 + y^2 + z^2 = 1$, determine the maximum value of the expression $\\sqrt{6}xy + 4yz$. Then, calculate the value of $2A^2 - 4$, where $A$ is the maximum value found.",
"role": "user"
}
] | 7 |
[
{
"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... | 8400 kg |
[
{
"content": "Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists a constant $C > 0$ such that for every rotation $R \\in SO(3)$, the area of every connected component of the set $\\{x \\in S^2 : u(x) > u(Rx)\\}$ is at least $C$. Is $u$ necessarily a constant function?",
"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... | 82.74 mins |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty} \\sum_{r=0}^n \\frac{\\binom{n}{r}}{(n^r)(r+3)} \\]",
"role": "user"
}
] | e - 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... | 2.25 g |
[
{
"content": "Determine whether the limit \\( \\lim_{(x, y) \\to (0, 0)} \\frac{\\ln(1+2x^2+4y^2)}{\\arctan(x^2+2y^2)} \\) exists. If it exists, find its value; otherwise, state that it is path-dependent.",
"role": "user"
}
] | 2 |
[
{
"content": "Suppose $a$ and $b$ are co-prime integers, and $p$ is a prime number of the form $4k+1$. Determine whether $3a^2 + b^2$ can have a prime divisor equal to $p$.",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the probability that a sequence \\((x_n)_{n \\ge 1}\\), defined by \\(x_1 = x_2 = 1\\) and \\(x_{n+2} = ax_{n+1} - bx_n\\), where \\(a\\) and \\(b\\) are distinct numbers chosen randomly from the set \\(\\{1, 2, \\ldots, 10\\}\\), converges.",
"role": "user"
}
] | \dfrac{1}{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... | 0.106M |
[
{
"content": "Calculate the limit: $$\\lim_{n \\to \\infty} \\sqrt[n]{|\\sin n|}$$",
"role": "user"
}
] | 1 |
[
{
"content": "Consider a game played with a fair coin where you start at 0. Each heads flip moves you forward one unit (+1), and each tails flip moves you backward one unit (-1). The game ends when you reach either -32 or +63. Calculate the expected number of coin flips required for the game to end.",
"role... | 2016 |
[
{
"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... | 0.04004 m |
[
{
"content": "Evaluate the following expression for $b_n$ and determine its value when $n=1$:\n\n$$b_n = 30 \\left[\\frac{\\sin(\\pi - n\\pi )}{\\pi - n\\pi } - \\frac{\\sin(\\pi + n\\pi )}{\\pi + n\\pi}\\right] - 200\\left(\\frac{\\sin (n\\pi )}{(n \\pi)^2}\\right) + 200\\left(\\frac{1}{n \\pi}\\right)$$\n\nSp... | 1 |
[
{
"content": "In a regular tetrahedron $P-ABC$ with all edges of length $1$, points $L$, $M$, and $N$ are the midpoints of edges $PA$, $PB$, and $PC$, respectively. Determine the area of the cross-section of the circumscribed sphere of the tetrahedron created by the plane $LMN$. Express your answer in simplest ... | \dfrac{\pi}{3} |
[
{
"content": "Show that the extension \\(\\mathbb{Q}(\\zeta_n)/\\mathbb{Q}(\\zeta_n+\\zeta_n^{-1})\\) has degree 2, where \\(\\zeta_n\\) is a primitive \\(n\\)-th root of unity with \\(n > 2\\).",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the limit: \\( \\lim_{n\\to\\infty}n\\cdot \\sin(2\\pi e(n!)) \\)",
"role": "user"
}
] | 2\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... | 285 K |
[
{
"content": "Let $f_n, f, g \\colon \\mathbb{R} \\to \\mathbb{R}$ be continuous functions. Assume that $f_n \\to f$ uniformly. Does it follow that $g(f_n) \\to g(f)$ uniformly? Justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Suppose $h$ maps the unit disk to its closure and $h(0) = 0$. If $|h(z)| \\leq 2$ for all $z$ in the unit disk, determine the maximum possible value of $|h(z)|$.",
"role": "user"
}
] | 2 |
[
{
"content": "Let $\\mathbb{R}$ denote the real line as a topological field. Determine whether a proper subfield $F$ of $\\mathbb{R}$ must be disconnected.",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( a, b \\in \\mathbb{R} \\) with \\( a < b \\). Show that the following limit holds:\n\\[ \\lim_{n \\to \\infty} \\int_a^b \\frac{((b-a)e^{2x})}{(b-a)n^2e^{2x}+b^2} \\, dx = 0. \\]",
"role": "user"
}
] | 0 |
[
{
"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... | M1, M2, M3, and M4 |
[
{
"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... | 0.4 N/C |
[
{
"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... | $500 per machine hour and $500 per labor hour |
[
{
"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... | 66.04 V |
[
{
"content": "Find all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that for all \\( x, y \\in \\mathbb{R} \\), \\[ x\\left[f(x+y)-f(x-y)\\right]=4yf(x). \\]",
"role": "user"
}
] | f(x) = ax^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... | Eastern Division ROI = 20.8%
Western Division ROI = 22.5% |
[
{
"content": "Suppose that \\(f: \\mathbb{R} \\rightarrow \\mathbb{R}\\) is a continuous function which is differentiable on all of \\(\\mathbb{R}_{0}\\) except at \\(0\\), and suppose that \\(\\lim_{x \\rightarrow 0}f'(x)\\) exists and is finite. Can we conclude that \\(f\\) is differentiable at \\(0\\)?",
... | Yes |
[
{
"content": "Solve the equation \\((\\cos x + 2 \\cos 6x)^{2} = 9 + \\sin^{2} 3x\\).",
"role": "user"
}
] | 2\pi n |
[
{
"content": "Let $V$ be a $\\mathbb{C}$-vector space, and let $f_1,\\dots,f_n \\in S^d(V^*)$ be homogeneous polynomials of degree $d$ such that $V(f_1,\\dots, f_n)=\\{0\\}$. Determine whether there must exist a positive integer $k\\geq d$ such that for all $v \\in S^k(V^*)$, there exist $g_1,\\dots, g_n \\in S... | Yes |
[
{
"content": "Compute the limit: $$\\lim_{n\\to\\infty} \\frac{1}{\\log(n)}\\sum _{k=1}^n \\frac{\\cos (\\sin (2 \\pi \\log (k)))}{k}$$",
"role": "user"
}
] | J_0(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... | $3.39 |
[
{
"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... | 5.62e3 m |
[
{
"content": "Determine the value of $k$ for which 100 arcs can be placed on a circle such that each arc intersects exactly $k$ other arcs.",
"role": "user"
}
] | 98 |
[
{
"content": "Determine whether the statement is true: If $N$ is a multiple of $100$, then $N!$ ends with $\\left(\\frac{N}{4}-1 \\right)$ zeroes.",
"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... | 3.8 m |
[
{
"content": "Compute the degree of the morphism \\( f: X \\to \\mathbb{A}_{\\mathbb{C}}^1 \\), where \\( X = \\{(t,t^2,t^3) : t \\in \\mathbb{C}\\} \\) and \\( f(t,t^2,t^3) = t \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Does there exist a sequence of real numbers whose set of subsequential limits is the interval $[0,1]$? Consider the sequence given by $1, \\frac{1}{2}, \\frac{2}{2}, \\frac{3}{2}, \\frac{1}{3}, \\frac{2}{3}, \\frac{3}{3}, \\frac{4}{3}, \\ldots, \\frac{1}{n}, \\frac{2}{n}, \\frac{3}{n}, \\ldots, \\... | [0,1] |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0^+} (\\sin x)^\\frac{1}{\\ln x} \\]",
"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... | 904110 |
[
{
"content": "Can a multivariate Gaussian distribution with a non-diagonal covariance matrix be transformed into a distribution with independent variables by changing the basis to the eigenvectors of its covariance matrix?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral \\( \\int_0^1 \\frac{x^a-1}{\\ln x}\\,\\mathrm{d}x \\) using Feynman's integration under the integral sign technique.",
"role": "user"
}
] | \ln(a + 1) |
[
{
"content": "Let $V$ be a $K$-vector space and $E/K$ be a field extension. If $\\dim_K V$ is infinite, does it follow that $\\dim_E(V\\otimes_K E)$ is also infinite? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Compute the closed-form expression for the integral \\( R^* = \\int\\int \\max_i\\{\\theta_i+\\epsilon_i-1,0\\}dF(\\theta_i)dG(\\epsilon_i) \\), where \\( \\theta_i \\) is uniformly distributed over \\([1,2]\\) and \\( \\epsilon_i \\) is uniformly distributed over \\([-1/2,1/2]\\), with both rando... | \dfrac{25}{48} |
[
{
"content": "Calculate the radius of convergence for the power series \\(x^2\\left(1+\\sum_{n=1}^{\\infty}\\frac{(-1)^n}{(2n)!}2^{2n-1}x^{2n}\\right)\\), which represents \\(x^2\\cos(x)^2\\).",
"role": "user"
}
] | \infty |
[
{
"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... | 0.079 mg |
[
{
"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... | 1.76e7 K |
[
{
"content": "Calculate the expectation \\( E\\left[\\left(\\sum_{i=1}^{\\tau} X_{i}\\right)^2 \\mid \\tau\\right] \\), where \\( X_{i} \\) are independent random variables taking values \\( 1 \\) and \\( -1 \\) with equal probability \\( \\frac{1}{2} \\), and \\( \\tau \\) is a stopping time independent of \\(... | \tau |
[
{
"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... | 7.56*10^-5 A |
[
{
"content": "Is it always true that the derivative of a differentiable and increasing function is Lebesgue integrable?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the sum of the series \\( \\sum_{k=0}^\\infty \\frac{(2x)^{2k}}{(k!)^2} \\).",
"role": "user"
}
] | I_0(4x) |
[
{
"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... | $7,500 |
[
{
"content": "Given that $3x + y = 10$ and $x + 3y = 14$, find $10x^2 + 12xy + 10y^2$.",
"role": "user"
}
] | 296 |
[
{
"content": "Determine the order of a $p$-Sylow subgroup of the group $GL_{50}(\\mathbb{F}_p)$, where $p$ is a prime number. Choose the correct answer from the following options:\n\n- $p^{50}$\n- $p^{125}$\n- $p^{1250}$\n- $p^{1225}$",
"role": "user"
}
] | D |
[
{
"content": "Let $A_1, A_2, A_3, A_4$ be events such that for every $i, j, k = 1, 2, 3, 4$, $P(A_i) = \\frac{1}{2}$, $P(A_i \\cap A_j) = \\frac{1}{3}$ for $i \\ne j$, $P(A_i \\cap A_j \\cap A_k) = \\frac{1}{4}$ for $i \\ne j, j \\ne k, k \\ne i$, and $P(A_1 \\cap A_2 \\cap A_3 \\cap A_4) = \\frac{1}{5}$. Find ... | \dfrac{1}{4} |
[
{
"content": "Given a parabola with equation \\( y^2 = 4ax \\), the distances of two points \\( P \\) and \\( Q \\) from the focus are 4 and 9, respectively. Find the distance from the focus to the point of intersection of the tangents at \\( P \\) and \\( Q \\).",
"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... | 20 W |
[
{
"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... | 103 degrees |
[
{
"content": "Determine the limiting distribution of the sequence:\n\\[ \\frac{1}{\\sqrt{n}} \\sum_{i=1}^n\\left(X_{2 i}-X_{2 i-1}\\right) \\]",
"role": "user"
}
] | \mathcal{N}(0, 2\sigma^2) |
[
{
"content": "Polycarp wrote a multiplication problem of two three-digit numbers on the board and mistakenly wrote 0 instead of the multiplication sign, resulting in a seven-digit number. This seven-digit number is an integer multiple of the product of the two numbers. Determine the exact factor by which the se... | 73 |
[
{
"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... | 4.466 |
[
{
"content": "If $\\exp(tz_1) = \\exp(tz_2)$ for all $t \\in \\mathbb{R}$, can we conclude that $|z_1 - z_2|$ is arbitrarily small?",
"role": "user"
}
] | 0 |
[
{
"content": "Does Theorem 1 from the paper by Gidas, Ni, and Nirenberg ensure that the radial solution $u(r)$ of a second-order elliptic equation is bounded for $0 \\leq r \\leq R$?",
"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... | $212,000.00 |
[
{
"content": "Evaluate the integral \\( \\iiint_V xe^{-(x^2+y^2+z^2)^2}\\,\\mathrm{d} V \\) using spherical coordinates, where \\( V \\) is the solid that lies between \\( y=0 \\) and the hemisphere \\( x^2+y^2+z^2=4 \\) in the right half-space \\( (y>0) \\).",
"role": "user"
}
] | 0 |
[
{
"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... | Adopting a sedentary lifestyle, increased sugar intake, eating simple sugars instead of complex carbs including starch and high fibre, insufficient sleep, and skipping meals can decrease metabolism. |
[
{
"content": "What is $2121_3 - 212_3?$ Express your answer in base 3.",
"role": "user"
}
] | 1202_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... | 84 m |
[
{
"content": "The polynomial $P(x) = x^3 + ax^2 + bx +c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $y$-intercept of the graph of $y=\nP(x)$ is 2, what is $b$?",
"role": "user"
}
] | -11 |
[
{
"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... | 739.2 mm Hg |
[
{
"content": "Find the derivative of the function \\( f(z) = z^c \\), where \\( c \\in \\mathbb{C} \\setminus \\{0\\} \\).",
"role": "user"
}
] | c z^{c - 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... | ### DateParticularsDebit ($)Credit ($)
Mar. 1, 2014Investemnt-Buncombe company50,000
Interest receivable250
Cash 50,250
Mar. 16, 2014Investemnt-French broad84,000
Interest receivable175
Cash 84,175
Aug. 1, 2014Cash 1,500
Interest receivable-Buncombe company 250
Interest revenue 1,250
Aug. 31, 2014Cash ... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\rightarrow \\infty } \\int_0^1 \\sqrt{ \\frac 1 x + n^2 x^{2n} } \\ \\mathrm dx \\]",
"role": "user"
}
] | 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... | 10.79% |
[
{
"content": "Determine the number of real roots of the polynomial \\( P_n(x) = 1 + 2x + 3x^2 + 4x^3 + \\, \\ldots \\, + (n+1)x^n \\) when \\( n \\) is an odd number.",
"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... | 18.3% |
[
{
"content": "Determine the number of distinct integer solutions to the equation \\(a+b+c+d+e+f+g=18\\) such that no two variables differ by more than 1.",
"role": "user"
}
] | 35 |
[
{
"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... | 0.109 |
[
{
"content": "In an acute triangle $ABC$, the segment $CD$ is an altitude and $H$ is the orthocenter. If the circumcenter of the triangle lies on the line containing the bisector of the angle $DHB$, find all possible values of $\\angle CAB$. \\( \\text{Provide your answer in degrees.} \\)",
"role": "user"
... | 60 |
[
{
"content": "Let $S$ be a group and $G$ be a finite group that is $S$-free, meaning no quotient group of any subgroup of $G$ is isomorphic to $S$. If $N \\lhd S$, is $G$ an $N$-free group? Answer yes or no and justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Calculate \\( 7^{50} \\cdot 4^{102} \\mod 110 \\).",
"role": "user"
}
] | 104 |
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