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... | -4.23e6 J |
[
{
"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... | 72 m/s |
[
{
"content": "Find the smallest positive integer $N$ such that $N = 3a^2 - ab^2 - 2b - 4$ for positive integers $a$ and $b$. Provide your answer as a single integer.",
"role": "user"
}
] | 2 |
[
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers. Find the minimum value of\n\\[\\frac{\\sqrt{x^2 + y^2} + \\sqrt{x^2 + z^2} + \\sqrt{y^2 + z^2}}{x + y + z}.\\]",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "Calculate the limit of the following expression as \\( n \\) approaches infinity: \\[ \\lim _{n \\to \\infty} {\\ln^{2}\\left(n\\right) \\over n^{ 1/\\ln\\left(\\,{\\ln\\left(\\,{n}\\,\\right)}\\,\\right)}} \\]",
"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... | 3.8 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... | $78,125 |
[
{
"content": "Evaluate the integral \\( \\int_{[0,1]} f \\, d\\mu \\) using the Lebesgue integral, where \\( d\\mu \\) is the Borel measure and \\( f \\) is defined as follows: \\[ f(x) = \\begin{cases} x & \\text{if } x \\in C, \\\\ 0 & \\text{if } x \\in [0,1] \\setminus C, \\end{cases} \\] with \\( C \\) bei... | 0 |
[
{
"content": "Determine the values of $n$ for which the number $10^{2^n} + 1$ is composite, where $n$ is a positive integer.",
"role": "user"
}
] | n \geq 2 |
[
{
"content": "Evaluate the double integral \\( \\int_{0}^{1}\\int_{0}^{1}\\left\\lfloor\\frac{x-y}{x+y}\\right\\rfloor\\,dx\\,dy \\), where \\( \\left\\lfloor z \\right\\rfloor \\) denotes the floor function.",
"role": "user"
}
] | -\dfrac{1}{2} |
[
{
"content": "Let $X$ and $Y$ be random subsets of $\\{1, 2, \\dots, k\\}$, each picked uniformly at random from all $2^k$ subsets, and independent of each other. Determine the variance of the random variable $|X \\cup Y|$. \\( \\text{Express your answer in terms of } k. \\)",
"role": "user"
}
] | \dfrac{3k}{16} |
[
{
"content": "Calculate the volume of the solid generated by revolving the region under the first arc of the cycloid, given by the parametric equations \\(x(t) = a(t - \\sin t)\\) and \\(y(t) = a(1 - \\cos t)\\), about the \\(x\\)-axis. Use the formula \\(V = \\pi \\int [f(y)]^2 \\, dy\\) to find the volume.",
... | 5\pi^2 a^3 |
[
{
"content": "Given a positive integer S, determine the maximum value of N such that the sum of all elements in a 3D array A of dimensions N x N x N, where each element A[i][j][k] is defined as ik(i+j+k) for 1 <= i, j, k <= N, does not exceed S. Constraints: 3 <= S <= 10^{16}. For example, if S = 720, the maxim... | 3 |
[
{
"content": "A line parallel to $3x-7y = 65$ passes through the point $(7,4)$ and $(0,K)$. What is the value of K?",
"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... | weak tie |
[
{
"content": "Evaluate the contour integral \\(\\int\\limits_{L}z\\cos(2\\pi z \\bar{z})\\,dz\\) where \\(L\\) is the semi-circle defined by \\(|z| = 1\\) for \\(0 \\le \\arg(z) \\le \\pi\\).",
"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... | Cr_{2}O_{7}^{2-}+8H^{+}+6e^{-}
ightarrow Cr_{2}O_{3}+4H_{2}O |
[
{
"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... | 1.5e12 J |
[
{
"content": "Find the splitting field of the polynomial \\(x^6 + x^3 + 1\\) over \\(\\mathbb{Q}\\) and determine the degree of this field extension.",
"role": "user"
}
] | 6 |
[
{
"content": "The sum of fifty positive numbers is 1. What is the maximum value of the sum of their inverses?",
"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... | 104m/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... | $x = \frac{1}{\omega}\sin(\omega t)$, $z= 0$, and $y = \frac{1}{2\omega(\omega-\sigma)}\sin((\omega-\sigma)t) - \frac{1}{2\omega(\omega+\sigma)}\sin((\omega+\sigma)t)$ |
[
{
"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... | -0.807 m/s |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\rightarrow \\infty}\\frac{1}{n}\\int^{\\pi}_{0}\\lfloor n \\sin x \\rfloor \\, dx \\]",
"role": "user"
}
] | 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... | 88.89% |
[
{
"content": "In a game of craps, if the first roll is neither a natural nor craps, the player can make an additional bet equal to their original one, betting that they will make their point before a seven appears. The payoff odds are as follows: 2:1 for a point of four or ten, 3:2 for a point of five or nine, ... | 0 |
[
{
"content": "Is it possible to find a symmetric matrix $S$ that is not positive semidefinite, such that there exist two different indices $i_1 \\neq i_2$ for which the submatrices $A_{-i_1}$ and $A_{-i_2}$ are positive semidefinite?",
"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... | 25.3 m/s |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to \\infty} \\frac{3}{n}\\left(1+\\sqrt{\\frac{n}{n+3}}+\\sqrt{\\frac{n}{n+6}}+\\cdots+\\sqrt{\\frac{n}{4n-3}}\\right) \\]",
"role": "user"
}
] | 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... | 1842500 |
[
{
"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... | 15.24% |
[
{
"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... | 9.9 |
[
{
"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... | 20.15 years |
[
{
"content": "Determine whether the following inequality holds for all continuous functions $f$ and $g$ on $[0,1]$ in the sup-norm metric:\n\n$$\\sup_{x\\in[0,1]}\\int_{0}^{x}|f(t)-g(t)|\\ dt \\leq \\sup_{x\\in[0,1]}|f(x)-g(x)|$$",
"role": "user"
}
] | Yes |
[
{
"content": "Is the set \\( N_{b}=\\{ f \\in {\\rm L}^{2}[0,1]:f(x)=0, \\text{ for almost every } x \\in [b,1]\\} \\) a closed subspace of \\( {\\rm L}^{2}[0,1] \\) for \\( 0 \\leq b \\leq 1 \\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the remainder when \\(45^{17^{17}}\\) is divided by 204.",
"role": "user"
}
] | 45 |
[
{
"content": "Determine whether the following condition implies the other: Given two conditions, (1) For all unit vectors \\(\\mathbf{x}\\) and \\(\\mathbf{y}\\), if \\(\\left|\\left|\\mathbf{x}+\\mathbf{y}\\right|\\right| = 2\\), then \\(\\mathbf{x} = \\mathbf{y}\\). (2) For all vectors \\(\\mathbf{x}\\) and \... | Yes |
[
{
"content": "Solve the equation for real and positive parameters $a$, $b$, and $c$:\n\\[ \\sqrt{a+bx}+\\sqrt{b+cx}+\\sqrt{c+ax}=\\sqrt{b-ax}+\\sqrt{c-bx}+\\sqrt{a-cx} \\]",
"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... | October: $41,296
November: $67,546
December: $60,492 |
[
{
"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... | 6.5, 4, 5, 24.44 |
[
{
"content": "Determine the domain of convergence for the power series \\( \\sum_{n\\ge1}{(\\cos{\\frac{1}{n}})^{\\frac{n^2+2}{n+2}}x^n} \\).",
"role": "user"
}
] | (-1, 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... | 3430J |
[
{
"content": "Find the residue of the function \\( f(z) = \\tan{z} \\) at any of its poles.",
"role": "user"
}
] | -1 |
[
{
"content": "Find the limit of the sequence as \\( n \\) approaches infinity: $$\\lim_{ n \\to \\infty }\\sin ^2(\\pi\\sqrt{n^2+n})$$",
"role": "user"
}
] | 1 |
[
{
"content": "Determine whether the following proposition is true or false: If $a_n \\neq 0$ for all natural numbers $n$, $\\sum_{n=1}^{+\\infty} a_n$ converges, and $\\lim_{n\\to+\\infty} \\frac{a_n}{b_n} = 1$, then $\\sum_{n=1}^{+\\infty} b_n$ converges.",
"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... | $450,660.9222 |
[
{
"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... | 0.290 |
[
{
"content": "In the diagram, $P$ is a point on the semicircle with diameter $AB$. The point $L$ is the foot of the perpendicular from $P$ onto $AB$, and $K$ is the midpoint of $PB$. The tangents to the semicircle at $A$ and $P$ meet at point $Q$. Given that $PL$ intersects $QB$ at point $M$, and $KL$ intersect... | 12 |
[
{
"content": "Suppose the series \\( \\sum c_n z^n \\) has a radius of convergence \\( R \\). Determine the radius of convergence of the series \\( \\sum n^p c_n z^n \\).",
"role": "user"
}
] | R |
[
{
"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... | 77.53 |
[
{
"content": "Find the number of ordered pairs \\((p, q)\\) such that \\(p\\) and \\(q\\) are coprime positive integers, \\(\\frac{p}{q} \\in (0,1)\\), and \\(pq = 20!\\).",
"role": "user"
}
] | 128 |
[
{
"content": "Find the smallest integer greater than 1 that, when divided by \\(\\frac{5}{7}, \\frac{7}{9}, \\frac{9}{11}, \\frac{11}{13}\\) respectively, yields mixed numbers with fractional parts \\(\\frac{2}{5}, \\frac{2}{7}, \\frac{2}{9}, \\frac{2}{11}\\) respectively.",
"role": "user"
}
] | 3466 |
[
{
"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... | $7.28 per carton |
[
{
"content": "Let $U = \\cap_{n \\in \\mathbb{N}} U_n$ be a non-empty $G_{\\delta}$-set, where each $U_n$ is open in a normal topological space $X$. If $\\phi: X \\rightarrow Y$ is a homeomorphism, is $\\phi(U)$ a $G_{\\delta}$-subset of $Y$? 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... | 1026 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... | 69.6025 percent by volume |
[
{
"content": "Find the distance between the point \\( p = (1,0,0,1) \\) and the subspace \\( L = (\\operatorname{span}{(1,0,2,1),(2,1,2,3),(0,1,-2,1)})^\\perp \\) in \\( \\mathbb{R}^4 \\).",
"role": "user"
}
] | \dfrac{\sqrt{14}}{3} |
[
{
"content": "Given the function \\( f(z) = \\frac{2}{z^2 - 1} \\), find the sum of the coefficients \\( a_{-3} \\) and \\( a_{2} \\) in its Laurent series expansion around \\( z = 2 \\). The official answer is \\( \\frac{26}{27} \\). Verify this result.",
"role": "user"
}
] | \dfrac{26}{27} |
[
{
"content": "Determine the number of real roots of the equation \\(x^{13} - e^{-x} + x - \\sin{x} = 0\\). Choose one of the following options:\n\n- No real root\n- More than two real roots\n- Exactly two real roots\n- Exactly one real root",
"role": "user"
}
] | D |
[
{
"content": "Determine if the function \\( g: \\mathbb{R}^2 \\rightarrow \\mathbb{R} \\), defined by\n\n\\[ g(x,y) = \\frac{|x| y}{|x| + |y|} + 3x - 2y, \\]\n\nhas a linear approximation at the point \\((1,1)\\).",
"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... | 7.07 cm |
[
{
"content": "Find the area of the intersection between the sphere \\(x^2 + y^2 + z^2 = 1\\) and the cylinder \\(x^2 + y^2 - y = 0\\).",
"role": "user"
}
] | 2\pi - 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... | 0.610 m |
[
{
"content": "Let A be a diagonalizable square matrix with real eigenvalues greater than zero. Is it necessarily true that there exists a matrix S such that \\( A = S^2 \\)? Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the optimal step size \\(\\nabla t\\) for the Forward-Backward Splitting algorithm, which minimizes the sum of two convex functions \\(f(x)\\) and \\(g(x)\\). The algorithm iteration is given by:\n\\[x_{t+1}=\\arg\\min_u\\left\\{\\frac{1}{2}\\|u-(w_t-\\nabla t \\partial f(w_t))\\|^2+\\na... | \frac{1}{L} |
[
{
"content": "Determine the condition on the parameter $s \\in \\mathbb{R}$ such that the series \\( \\sum_{\\xi\\in\\mathbb{Z}^n}\\frac{1}{(1+|\\xi|^2)^{s/2}} \\) converges, where \\(|\\xi|\\) is the Euclidean norm of \\(\\xi\\).",
"role": "user"
}
] | s > n |
[
{
"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... | 0.909 s^-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... | 50560 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{\\cos x - \\sqrt{2 - e^{x^2}}}{\\ln(\\cos x) + \\frac{1}{2} x \\sin x} \\cdot \\frac{(x+2)^{2017}}{(x-2)^{2015}} \\]",
"role": "user"
}
] | 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... | 4.77 m |
[
{
"content": "Find the limit of the sequence \\( \\lim_{n\\to\\infty}\\frac{\\prod_{k=1}^n(2k-1)}{(2n)^n} \\).",
"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... | $220,696.10 |
[
{
"content": "Evaluate the integral \\( \\int_{|C|=2} \\frac{dz}{z^2 + 2z + 2} \\) using the Cauchy-Goursat theorem, where the contour \\( C \\) is a circle of radius 2 centered at the origin. Express your answer in terms of \\( \\pi \\).",
"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... | I_2 + 10NO_3^- + 8H^+ -> 10NO_2 + 2IO_3^- + 4H_2O |
[
{
"content": "Let $f(x)$ be a continuous function satisfying the differential equation \\[ f(x) = (1+x^2)\\left(1+\\int_0^x \\frac{f^2(t) \\, dt}{1+t^2}\\right) \\]. Find the value of $f(1)$. \\( \\boxed{} \\)",
"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... | Revenue - Expenses + Gains - Losses |
[
{
"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... | 9.17 years |
[
{
"content": "Determine the value of $r$ in the logistic map's bifurcation diagram where the transition from a 2-cycle to an unstable state occurs, specifically at $r=1+\\sqrt{6}$. What is this $r$ value?",
"role": "user"
}
] | 1 + \sqrt{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... | 3.21 cm |
[
{
"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... | $12,923.07 |
[
{
"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... | 24% |
[
{
"content": "Calculate the sum \\(\\sum_{k=0}^{n}\\binom{n}{k}\\left(\\binom{m+k+1}{m-n}+(-1)^{k+1}\\binom{m+2(n-k)+1}{m}\\right)\\) for integers \\(m\\) and \\(n\\) where \\(m \\ge n \\ge 0\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let $S^{1} \\subset \\mathbb{R}^2$ be the unit circle, and let $\\pi_1 : S^1 \\to \\mathbb{R}$ be the natural projection onto the first coordinate. Determine whether the preimage $\\pi_{1}^{-1} ( [0,\\frac{1}{2}])$ is compact.",
"role": "user"
}
] | Yes |
[
{
"content": "Given the function \\( v = \\frac{x-y}{x^2+y^2} \\), find the analytic function \\( f(z) = u(x, y) + iv(x, y) \\). Choose the correct option:\n\n(a) \\( z+c \\) \n(b) \\( z^{-1}+c \\) \n(c) \\( \\frac{1-i}{z}+c \\) \n(d) \\( \\frac{1+i}{z}+c \\)",
"role": "user"
}
] | D |
[
{
"content": "Suppose $A \\in {\\mathbb C^{n \\times n}}$ is a matrix with $\\left\\| A \\right\\| \\le \\varepsilon$, and $v \\in {\\mathbb C^n}$ is a vector such that ${v^*}v = 1$. Is it true that $\\left\\| {{v^*}Av} \\right\\| \\le \\varepsilon$?",
"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... | 17.3 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... | 423 m/s in the +y direction |
[
{
"content": "Two regular triangular pyramids, \\( P-ABC \\) and \\( Q-ABC \\), share the same base \\( ABC \\) and are inscribed in the same sphere. The angle between a side face and the base of pyramid \\( P-ABC \\) is \\( 45^{\\circ} \\). Determine the tangent of the angle between a side face and the base of... | 4 |
[
{
"content": "Determine whether the growth of the function \\( f(n) = \\frac{(n+1)(n+2)\\ldots(n+\\sqrt{n})}{(n-1)(n-2)\\ldots(n-\\sqrt{n})} \\) follows a polynomial pattern. Specifically, do there exist constants \\( k \\) and \\( m \\) such that \\( f(n) < n^k \\) for all \\( n > m \\)?",
"role": "user"
... | Yes |
[
{
"content": "Find all integer values of \\( n \\) such that \\( \\sqrt{\\frac{4n-2}{n+5}} \\) is a rational number.",
"role": "user"
}
] | 13 |
[
{
"content": "Evaluate the limit \\( \\lim_{h \\to 0} \\frac{2\\sin\\left(\\frac{1}{2}\\ln\\left(1+\\frac{h}{x}\\right)\\right)}{h} \\) without using L'Hopital's rule, given that \\( \\lim_{x \\to 0} \\frac{\\ln(1+x)}{x} = 1 \\).",
"role": "user"
}
] | \dfrac{1}{x} |
[
{
"content": "Determine the minimum number of unit squares that must be removed from a \\(202 \\times 202\\) grid so that it can be tiled with T-tetrominoes.",
"role": "user"
}
] | 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... | 1.5 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... | 5Fe^(2+) (aq) + 8H^+ (aq) + MnO4^- (aq) -> 5Fe^(3+) (aq) + Mn^(2+) (aq) + 4H2O (l) |
[
{
"content": "Let $f:I\\to\\mathbb{R}$ be a nonnegative, bounded, integrable, and continuous function with $I\\subseteq\\mathbb{R}$ being an interval. If $\\int_I f \\, d\\mu=1$, does $\\mu(f>0)>0$ hold, where $\\mu$ is the Lebesgue measure?",
"role": "user"
}
] | Yes |
[
{
"content": "The graph of the rational function $\\frac{p(x)}{q(x)}$ is shown below, with a horizontal asymptote at $y = 0$. If $q(x)$ is quadratic, $p(2)=2$, and $q(-1) = 18$, find $p(x) + q(x).$\n[asy]\nsize(8cm);\nimport graph;\n\nLabel f; \nf.p=fontsize(6); \n\nreal f(real x) {return 2*(x-1)/(3*(x-2)*(x-1)... | 3x^2 - 7x + 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... | 128.75 pence |
[
{
"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... | 20500 |
[
{
"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... | 1.37 L |
[
{
"content": "Determine the coefficient of the term $x^{60}$ in the power series expansion of $C(x) = \\frac{1}{1-x} \\cdot \\frac{1}{1-x^2} \\cdot \\frac{1}{1-x^5}$. This coefficient represents the cardinality of the set $A = \\{(N, D, Q) \\mid 0.05N + 0.1D + 0.25Q = 3 \\text{ and } N, D, Q \\in \\mathbb{N}\\}... | 205 |
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