prompt list | ground_truth string |
|---|---|
[
{
"content": "Solve the equation: \\(2^{x/2} + 3^{x/2} = 13^{x/4}\\) for \\(x\\).",
"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... | $625 |
[
{
"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... | 6.29 mT |
[
{
"content": "Evaluate the improper integral: $$ \\int_{-\\infty}^{\\infty} \\left(\\frac{\\arctan(x)}{x}\\right)^2 \\, dx $$.",
"role": "user"
}
] | 2\pi \ln 2 |
[
{
"content": "Find the volume of the intersection of the two cylinders defined by the inequalities: \\(0 \\le x^2 + z^2 \\le 1\\) and \\(0 \\le y^2 + z^2 \\le 1\\).",
"role": "user"
}
] | \dfrac{16}{3} |
[
{
"content": "Evaluate the limit: \\[ L = \\lim_{x \\to 0} \\left(1 + \\arctan\\left(\\frac{x}{2}\\right)\\right)^{\\frac{2}{x}} \\]",
"role": "user"
}
] | e |
[
{
"content": "Given the recursive sequence \\( u_0 = 2 \\) and \\( u_{n+1} = \\frac{2}{3}u_n - n - \\frac{8}{3} \\), and another sequence defined as \\( v_n = u_n + x \\cdot n - 1 \\), find the value of \\( x \\) such that \\( v_n \\) forms a geometric sequence.",
"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... | $437.21 |
[
{
"content": "Evaluate the behavior of \\( \\lim_{x\\to 1} x^{\\frac{1}{x-a}} \\) as \\( a \\) approaches 0.",
"role": "user"
}
] | 1 |
[
{
"content": "Find the limit of \\(\\left(\\sqrt{\\frac{n}{(n+1)^2(n+2)}}t+\\frac{n}{n+1} \\right)^n\\) as \\(n\\) approaches infinity.",
"role": "user"
}
] | e^{t - 1} |
[
{
"content": "Let \\( f(x) = x^2 + px + q \\). It is known that the inequality \\( |f(x)| > \\frac{1}{2} \\) has no solutions on the interval \\([1, 3]\\). Find \\( \\underbrace{f(f(\\ldots f}_{2017}\\left(\\frac{3+\\sqrt{7}}{2}\\right)) \\ldots) \\). Round your answer to two decimal places if necessary.",
... | 0.18 |
[
{
"content": "What is the largest value among $\\operatorname{lcm}[12,2],$ $\\operatorname{lcm}[12,4],$ $\\operatorname{lcm}[12,6],$ $\\operatorname{lcm}[12,8],$ $\\operatorname{lcm}[12,10],$ and $\\operatorname{lcm}[12,12]?$ Express your answer as an integer.",
"role": "user"
}
] | 60 |
[
{
"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... | normative |
[
{
"content": "Let $S$ be the ring $\\mathbb{F}_2[x]/((x^2+x+1)(x^3+x+1))$, where $\\mathbb{F}_2$ is the finite field of two elements. If $I$ is the ideal generated by $\\overline{x}$, where $\\overline{x}$ is the residue of $x$, determine the cardinality of the ring $S/I$. \\( \\boxed{} \\)",
"role": "user"... | 1 |
[
{
"content": "Calculate the limit of the sequence as \\( n \\to \\infty \\):\n\n\\[\n\\lim _{n \\rightarrow \\infty} \\frac{n^{2}-\\sqrt{3 n^{5}-7}}{\\left(n^{2}-n \\cos n+1\\right) \\sqrt{n}}\n\\]",
"role": "user"
}
] | -\sqrt{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... | 78.07 g |
[
{
"content": "Let $a$ and $b$ be natural numbers such that $a \\neq b$. If the quadratic equations $(a-1)x^2-(a^2+2)x+a^2+2a=0$ and $(b-1)x^2-(b^2+2)x+b^2+2b=0$ have a common root, find the value of $ab$. \\( \\boxed{} \\)",
"role": "user"
}
] | 8 |
[
{
"content": "Find all even natural numbers \\( n \\) such that the sum of the reciprocals of all different divisors of \\( n \\) is equal to \\( \\frac{1620}{1003} \\).",
"role": "user"
}
] | 2006 |
[
{
"content": "Find the minimum value of \\( P = \\dfrac{1}{1 + x^2} + \\dfrac{1}{1 + y^2} + 2xy \\) subject to the constraint \\( xy \\geq 1 \\).",
"role": "user"
}
] | 3 |
[
{
"content": "Evaluate the sum \\( \\sum_{k=1}^{n} (2k-1) {n \\choose k} \\) using calculus, specifically by utilizing the binomial expansion of \\((1-x)^n\\).",
"role": "user"
}
] | (n - 1)2^n + 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... | 12.75% |
[
{
"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... | 2.88 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... | 0.48 |
[
{
"content": "Determine the number of Hausdorff topologies on a set with 5 elements.",
"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... | 1.55 * 10^8 L |
[
{
"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... | 12.08 m/s^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... | 8.93% |
[
{
"content": "Evaluate the expression \\(\\sum_{k=0}^{\\infty}\\left[\\frac{n+2^{k}}{2^{k+1}}\\right]\\), where \\(n\\) is a natural number.",
"role": "user"
}
] | n |
[
{
"content": "Let \\( A \\) be a bounded subset of \\( \\mathbb{C}^d \\). If the closure \\( \\overline{A} \\) is convex, is \\( A \\) necessarily convex? 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... | 5.66 seconds |
[
{
"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... | 3364.31 seconds or 65.07 minutes |
[
{
"content": "Describe the topological space obtained by forming the adjunction space $X \\sqcup_{f} Y$, where $X = D^2$ is the closed unit disk in the plane, $A = S^1$ is the boundary of $D^2$, $Y$ is the singleton set $\\{1\\}$, and $f: S^1 \\to Y$ is the constant map defined by $f(x) = 1$. What well-known to... | S^2 |
[
{
"content": "Determine the equation of the curve that intersects all level curves of the function \\( g(x,y) = \\ln(x) + \\cos(y) \\) orthogonally and passes through the point (2,0).",
"role": "user"
}
] | y = 0 |
[
{
"content": "Consider the extended real line \\( \\bar{\\mathbb{R}} \\) and let \\( 0 < q < \\infty \\). Suppose \\( x_n \\) is a sequence in \\( \\bar{\\mathbb{R}} \\) with \\( x_n \\geq 0 \\) for all \\( n \\), and \\( x_n \\to +\\infty \\). Determine the limit \\( \\lim_{n \\to +\\infty} (x_n)^{-1/q} \\).",... | 0 |
[
{
"content": "Evaluate the sum: \\[ 2\\cos\\frac{2\\pi}{11} + 2\\cos\\frac{4\\pi}{11} + 2\\cos\\frac{6\\pi}{11} + 2\\cos\\frac{8\\pi}{11} + 2\\cos\\frac{10\\pi}{11} \\]",
"role": "user"
}
] | -1 |
[
{
"content": "Evaluate the integral: $$\\int_{0}^{\\pi}\\left\\{\\int_{y^2}^{\\pi}\\frac{y\\sin x}{x} dx\\right\\}dy$$",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate the value of the integral \\( \\int_{-\\infty}^{\\infty} e^{-2t} \\delta'(t) \\, dt \\), where \\( \\delta(t) \\) is the Dirac delta function.",
"role": "user"
}
] | 2 |
[
{
"content": "A $4\\times 4\\times 4$ open cubical box contains 64 identical small cubes that exactly fill the box. How many of these small cubes touch the bottom or one of the four lateral sides of the box?",
"role": "user"
}
] | 52 |
[
{
"content": "Solve the equation \\( x^{\\frac{\\log2}{\\log3}} = x^{\\frac{1}{2}} + 1 \\) algebraically.",
"role": "user"
}
] | 9 |
[
{
"content": "On Friday, a snowboard originally priced at $\\$100$ was discounted $50\\%$. On Monday, that sale price was reduced by $30\\%$. In dollars, what is the price of the snowboard after the Monday reduction?",
"role": "user"
}
] | 35 |
[
{
"content": "Determine the possible eigenvalues of a $3\\times3$ matrix $P$ given that $P^3 = P$ and $P^2 \\neq P$. Provide your answer as a set of numbers.",
"role": "user"
}
] | \{-1, 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... | 9.90 g/L |
[
{
"content": "Determine the cardinality of the set of continuous functions \\( f : [0,1] \\rightarrow \\mathbb{R}^2 \\) such that \\( f(0) = a \\) and \\( f(1) = b \\), where \\( a \\) and \\( b \\) are distinct points in \\( \\mathbb{R}^2 \\). Express your answer in terms of beth numbers.",
"role": "user"
... | \beth_1 |
[
{
"content": "Let $p_1, \\dots, p_r$ be $r$ distinct odd primes. Define the group \\( G = \\prod_{i=1}^r \\mathrm{SL}_3(\\mathbb{F}_{p_i}) \\). Determine the maximum possible rank of a subgroup of \\( G \\), denoted as \\( \\max \\{d(H) : H \\leq G\\} \\), where \\( d(K) \\) is the smallest cardinality of a gen... | 2r |
[
{
"content": "What is the minimum number of students that must be lined up in such a way that there is either a sequence of 10 students in increasing order of height or a sequence of 12 students in decreasing order of height, given that no two students have the same height?",
"role": "user"
}
] | 100 |
[
{
"content": "Let $K$ and $L$ be two natural numbers such that $K$ has $L$ divisors and $L$ has $\\frac{K}{2}$ divisors. Determine the number of divisors of $K + 2L$. \\( \\text{Express your answer as a single number.} \\)",
"role": "user"
}
] | 4 |
[
{
"content": "Is it possible to extend a smooth function defined on a closed subset of Euclidean space to a smooth function on the entire space? Provide a yes or no answer.",
"role": "user"
}
] | Yes |
[
{
"content": "If $\\log_6 (4x)=2$, find $\\log_x 27$. Express your answer in simplest fractional form.",
"role": "user"
}
] | \frac32 |
[
{
"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... | $63,076.92 |
[
{
"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... | 0.864 grams |
[
{
"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... | 114,009,662.69 |
[
{
"content": "Find the minimum positive root of the equation \\(\\sin{3x}=a\\), where \\(a>0\\), given that there exist two roots with a difference of \\(\\frac{\\pi}{4}\\).",
"role": "user"
}
] | \dfrac{\pi}{24} |
[
{
"content": "Find the function \\( f(x) \\) that satisfies the functional equation: \\[ f(x) + x f(-x) = x + 1. \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Let \\( A \\in GL_{3} (\\mathbb{C}) \\) and \\( n \\in \\mathbb{N} \\). Determine whether there always exists a matrix \\( B \\in Mat_{3, 3}(\\mathbb{C}) \\) such that \\( B^n = A \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Find the value of \\( f^{-1}(3) \\) for the function \\( f(x) = x^5 + x^3 + x \\).",
"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... | sqrt(3gL) |
[
{
"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... | 10.04 amu |
[
{
"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... | 1.45e-4 H |
[
{
"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... | CHO_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... | 7205 |
[
{
"content": "Compute \\[\\frac{(N-1)!(N)}{(N+1)!}.\\]",
"role": "user"
}
] | \frac{1}{N+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... | 23.8 N |
[
{
"content": "Determine if the function \\( F(x,y)=e^{x+y} \\) is differentiable at the point \\((0,0)\\) using the definition of differentiability.",
"role": "user"
}
] | Yes |
[
{
"content": "If $A$ and $B$ are both lower triangular matrices in $M_{n \\times n}$, are they simultaneously triangularizable?",
"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... | S = \frac{w \sin (i - j)}{\cos j} |
[
{
"content": "Use Green's Theorem to evaluate the line integral \\( \\int_C y^3 \\, dx + x^3 \\, dy \\), where \\( C \\) is the ellipse \\( \\frac{x^2}{9} + \\frac{y^2}{25} = 1 \\).",
"role": "user"
}
] | -180\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... | 20837.87 |
[
{
"content": "Round $6287215$ to the nearest ten thousand.",
"role": "user"
}
] | 6290000 |
[
{
"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... | 9.46Pa |
[
{
"content": "Given a vector space $V$ of $3 \\times 3$ matrices and a matrix $A \\in V$ with $\\det(A) = 32$ and minimal polynomial $(x-4)(x-2)$, find the trace of the linear map $L_A: V \\to V$ defined by $L_A(B) = AB$ for $B \\in V$. What is the trace of $L_A$?",
"role": "user"
}
] | 30 |
[
{
"content": "Determine the number of zeros of the function \\( J(x) = 3^x - 1 - 2x^2 \\) given that \\( 1.09 < \\ln 3 < 1.1 \\).",
"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... | 1.36 s |
[
{
"content": "A school buys 60% of its light bulbs from supplier A. The light bulbs from supplier A have exponentially distributed lifetimes with a mean of 600 hours, and the lifetimes of all bulbs are independent of each other. Using a normal approximation, find the probability that 100 light bulbs from suppli... | 0.9522 |
[
{
"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... | 207000 |
[
{
"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 |
[
{
"content": "Evaluate the limit: $$\\lim_{a \\to +\\infty} \\int_a^{a+1} \\frac{x}{x+\\ln x} \\, \\text{d}x$$",
"role": "user"
}
] | 1 |
[
{
"content": "Let $(x_k)$ be a sequence of real numbers defined by $x_1 = 1$ and $x_{k+1} = x_k + \\sqrt{x_k}$ for $k > 0$. Determine the value of \\( \\lim_{k \\to \\infty} \\frac{x_k}{k^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... | For fundamental frequency, $F_1 = 49.71Hz$
For the second mode, $F_2 = 99.42Hz$
For the third mode, $F_3 = 149.1Hz$
For the forth mode, $F_4 = 198.8Hz$ |
[
{
"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... | 1.44e6 g |
[
{
"content": "An isosceles triangle with equal sides of 5 inches and a base of 6 inches is inscribed in a circle. What is the radius, in inches, of the circle? Express your answer as a mixed number.",
"role": "user"
}
] | 3\frac18 |
[
{
"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 dot product of F and n is $F \cdot n = \frac{x^2 + y^2 + z^2}{a} = \frac{a^2}{a} = a$ |
[
{
"content": "Calculate the index \\([G:H]\\) for the group \\(G\\), the group of all non-singular \\(n \\times n\\) matrices over the finite field \\(GF(11)\\), and its subgroup \\(H\\), the set of matrices with determinant 1.",
"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... | 240 kW |
[
{
"content": "Given a real number $x$, let $\\lfloor x\\rfloor$ denote the greatest integer less than or equal to $x$. For a certain integer $k$, there are exactly 70 positive integers $n_1$, $n_2, \\ldots,$ $n_{70}$ such that\n\\[k = \\lfloor \\sqrt[3]{n_1} \\rfloor = \\lfloor \\sqrt[3]{n_2} \\rfloor =\\cdots=... | 553 |
[
{
"content": "Given that \\(b^2 - 4ac = 0\\) with \\(a \\neq 0\\) and \\(a, b, c \\in \\mathbb{R}\\), and \\(x, y\\) satisfy the system of equations:\n\\[ ax^2 + (b+3)x + c = 3y \\]\n\\[ ay^2 + (b+3)y + c = 3x \\]\nfind the value of \\(\\frac{x}{y}\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\frac{1}{10^x} \\frac{1-e^{it}}{1-e^{\\frac{ti}{10}x}}. \\]",
"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... | 3.53 m/s^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... | 13.5% |
[
{
"content": "Given the matrix $M=\\begin{pmatrix} 1 & 1 & 0 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 1 \\end{pmatrix}$, compute the value of $$\\frac{1}{e}\\sum_{i=1}^{3}\\sum_{j=1}^{3}b_{ij}$$ where $e^M=[b_{ij}]$ and $e^M=I+M+\\frac{1}{2!}M^2+\\cdots$. What is the result?",
"role": "user"
}
] | \dfrac{11}{2} |
[
{
"content": "Find the sum of the areas of all triangles with vertices at lattice points (0, 0), (a, 0), and (0, b), where a and b are positive integers, such that each triangle contains exactly 100 lattice points in its interior. Exclude lattice points on the sides of the triangles.",
"role": "user"
}
] | 3118 |
[
{
"content": "Given a dictionary \\( D^{m \\times n} \\) of \\( n \\) basis vectors of size \\( m \\) and an input vector \\( x^{m \\times 1} \\), find a vector \\( \\alpha^{n \\times 1} \\) that minimizes the following constrained optimization problem for some constant \\( L \\):\n\\[\n\\min_{\\alpha} ||x-D\\a... | \alpha |
[
{
"content": "Given the polynomial \\(1-x+x^2-x^3+\\ldots-x^{17}\\), express it in terms of \\(y = x + 1\\) as \\(a_0 + a_1y + a_2y^2 + \\ldots + a_{17}y^{17}\\). Determine the value of \\(a_{17}\\).",
"role": "user"
}
] | -1 |
[
{
"content": "Find the number of integers in the range of the function \\( g(x) = \\sin x + \\cos x \\).",
"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... | 0.15 g |
[
{
"content": "Find the coefficient of $x$ in the determinant of the matrix:\n\\[ \\begin{pmatrix} x & (1+\\sin x)^3 & \\cos x \\\\ 1 & \\ln(1+x) & 2 \\\\ x^2 & 1+x^2 & 0 \\end{pmatrix} \\]",
"role": "user"
}
] | -2 |
[
{
"content": "Determine if the limit \\( \\lim_{n\\to\\infty}\\left( n-\\sum_{k=1}^ne^{\\frac{k}{n^2}}\\right) \\) can be solved using Riemann sums.",
"role": "user"
}
] | -\dfrac{1}{2} |
[
{
"content": "Evaluate the limit of the following sum as \\( n \\) approaches infinity:\n\n\\[\n\\lim _{n \\rightarrow \\infty} \\sum_{k=0}^{n} \\frac{k^{2}+3k+1}{(k+2)!}\n\\]",
"role": "user"
}
] | 2 |
[
{
"content": "Determine whether there exists a metric space $(\\mathbb{R}, d)$ such that the sequence $\\left\\{\\frac{1}{n}\\right\\}$ converges to $3$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Assume $a_n > 0$ and $\\lim\\limits_{n \\to \\infty}\\dfrac{a_1+a_2+\\cdots+a_n}{n}=a<+\\infty$. Find the limit \\( \\lim\\limits_{n \\to \\infty}\\dfrac{a_1^p+a_2^p+\\cdots+a_n^p}{n^p} \\) where $p > 1$. \\( \\boxed{} \\)",
"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... | <img>/cimages/multimages/16/marvinjs-output_-_2021-08-16t151743.0044597717118440987278.png</img> |
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