prompt list | ground_truth string |
|---|---|
[
{
"content": "Determine whether the limit \\( \\lim_{x \\to a^-} (x-a)\\left\\lfloor \\frac{1}{x-a}\\right\\rfloor \\) exists. If it exists, find its value.",
"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... | 928.8 kPa |
[
{
"content": "Find the sum of the series \\( \\sum_{n=0}^\\infty \\frac{(x+1)^{n+2}}{(n+2)!} \\).",
"role": "user"
}
] | e^{x+1} - x - 2 |
[
{
"content": "Determine whether there exists a triple of distinct numbers \\(a, b, c\\) such that \\((a-b)^5 + (b-c)^5 + (c-a)^5 = 0\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Kelly's art teacher asked him to draw a rectangle with a length of $6$ inches and a width of $10$ inches. After Kelly does so, his teacher changes her mind and asks him to draw a proportional rectangle that has a length of $9$ inches. How many inches wide is the new rectangle?",
"role": "use... | 15 |
[
{
"content": "Determine whether the inequality \\( \\binom{2 \\cdot 2^{1000000}}{2^{1000000}} > 2^{2 \\cdot 2^{1000000}} \\) is true or false.",
"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... | 0.5DWL |
[
{
"content": "Calculate the variance of the random variable \\( \\frac{1}{T}\\int_0^T W_t \\, dt \\) for a standard one-dimensional Brownian motion \\( W(t) \\).",
"role": "user"
}
] | \dfrac{T}{3} |
[
{
"content": "In a parallelogram $ABCD$, points $E$ and $F$ are located on segments $AD$ and $CD$, respectively, such that \\( \\frac{|AE|}{|ED|} = \\frac{|DF|}{|FC|} = \\frac{1}{2} \\). Determine the ratio in which the line segment $\\overline{EF}$ divides the diagonal $\\overline{BD}$. Use the theorem about t... | \dfrac{7}{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... | $519.3 |
[
{
"content": "Consider the set \\( A = \\{a_1, a_2, \\ldots, a_{2016}\\} \\). Calculate the difference between the number of subsets of \\( A \\) with a number of elements divisible by 4 and the number of subsets of \\( A \\) with a number of elements divisible by 2 but not by 4.",
"role": "user"
}
] | 2^{1008} |
[
{
"content": "In triangle $\\triangle ABC$, point $D$ is the midpoint of side $BC$ such that $DA = DB = DC$. The angle bisectors $BE$ and $CF$ intersect $AC$ and $AB$, respectively. Perpendiculars $EG$ and $FH$ are drawn from $E$ and $F$ to side $BC$. Determine the measure of angle $HAG$.",
"role": "user"
... | 45^\circ |
[
{
"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... | 0.81e23 |
[
{
"content": "Determine the type of singularity and the order of the pole at $z_0 = 0$ for the function $f(z) = \\frac{1}{z - \\sin(z)}$ without using the Laurent series expansion.",
"role": "user"
}
] | 3 |
[
{
"content": "Evaluate the double integral \\( \\int_0^a \\int_0^a \\frac{dx \\, dy}{(x^2+y^2+a^2)^{\\frac{3}{2}}} \\).",
"role": "user"
}
] | \dfrac{\pi}{6a} |
[
{
"content": "Let \\( X_1 \\) and \\( X_2 \\) be independent and identically distributed random variables from the exponential distribution with parameter \\( \\lambda \\). Find the probability density function of the random variable \\( Y = \\frac{X_1}{X_1 + X_2} \\).",
"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... | 0.691m/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... | 6.44% |
[
{
"content": "Evaluate the definite integral: $$\\int _0^1 \\arctan(x^2-x+1)\\,dx$$",
"role": "user"
}
] | \ln 2 |
[
{
"content": "Determine the number of complex numbers $z \\in \\mathbb{C}$ such that $(2z + i\\overline{z})^3 = 27i$ and $\\text{Re}(z) \\geq \\text{Im}(z)$. Provide your answer as a single integer.",
"role": "user"
}
] | 2 |
[
{
"content": "For a random variable $X$ with probability density functions $f_0(x)$ and $f_1(x)$ defined as follows:\n\n\\[ f_0(x) = \\begin{cases} 1, & \\text{if } 0 \\leq x \\leq 1 \\\\ 0, & \\text{otherwise} \\end{cases} \\]\n\n\\[ f_1(x) = \\begin{cases} \\frac{1}{2}, & \\text{if } 0 \\leq x \\leq 2 \\\\ 0,... | 0.525 |
[
{
"content": "Calculate the degree of the field extension \\([\\mathbb{Z}_3(\\alpha) : \\mathbb{Z}_3]\\), where \\(\\alpha\\) is a root of the polynomial \\(x^3 + x + 2 \\in \\mathbb{Z}_3[x]\\).",
"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... | 1 second |
[
{
"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... | 4kg particle: -5.86 m/s
6kg particle: -0.424 m/s |
[
{
"content": "Find the equation of a plane that is parallel to the plane \\(z = 2x + 3y\\) and tangent to the surface defined by the function \\(f(x, y) = x^2 + xy\\).",
"role": "user"
}
] | z = 2x + 3y + 3 |
[
{
"content": "Given that $x$ and $y$ are positive real numbers satisfying the equation \\( \\sqrt{x^2 + \\sqrt[3]{x^4 y^2}} + \\sqrt{y^2 + \\sqrt[3]{x^2 y^4}} = 512 \\), find the value of $x^{2/3} + y^{2/3}$.",
"role": "user"
}
] | 64 |
[
{
"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... | 200.0 rev |
[
{
"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... | 3.86e-2 |
[
{
"content": "Find the least positive integer $x$ that satisfies $x+4609 \\equiv 2104 \\pmod{12}$.",
"role": "user"
}
] | 3 |
[
{
"content": "Compute $35^{1723} - 16^{1723}$ modulo 6.",
"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... | 15.32 m/s |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to0}\\frac{1}{x^4}\\left(\\frac{6}{6+x^2}-\\frac{x}{\\sinh(x)}\\right). \\]",
"role": "user"
}
] | \dfrac{1}{120} |
[
{
"content": "Determine the radius of convergence of the Taylor series for the function \\( \\frac{2\\tan x}{1+4x^2} \\) centered at \\( x=0 \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Given the following data:\n\\[\\sum\\limits_{i=1}^{50} x_i = 1250, \\quad \\sum\\limits_{i=1}^{50} x_i^2 = 45000, \\quad \\sum\\limits_{i=1}^{50} y_i = 3100, \\quad \\sum\\limits_{i=1}^{50} y_i^2 = 250000\\]\nand knowing that there is a strong positive correlation between $x$ and $y$, determine if... | Yes |
[
{
"content": "Determine the number of real solutions \\((x, y, z, w)\\) to the system of equations:\n\\[ x + y = zw \\]\n\\[ z + w = xy \\]",
"role": "user"
}
] | \infty |
[
{
"content": "Calculate the covariance between the random vectors X and Y, where $X \\sim N_n(\\mu, O)$ and $Y|X \\sim N_n(X, \\Sigma)$. Provide your answer in terms of the given parameters.",
"role": "user"
}
] | 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... | -2.87e13 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... | 3.7e-12 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... | $471,935.48 |
[
{
"content": "Find the inverse Z-transform of \\( \\frac{z}{z+1} \\).",
"role": "user"
}
] | (-1)^n u[n] |
[
{
"content": "Determine the number of real roots of the equation \\(3^{x^{22}} + 28^{(1-x)^2} = 2020\\) without using derivatives.",
"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... | -$8,600 |
[
{
"content": "In the group $A_5$, let $x$ be an element of order 2 and $y$ be an element of order 3. If the subgroup generated by $x$ and $y$ is $A_5$, can the element $xy$ have an order less than 5?",
"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... | 3000 |
[
{
"content": "Determine whether there exist uncountable sets $A \\subset \\mathbb{R}$ such that the Hausdorff dimension $\\dim_H(A) = 0$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the sum of all positive integers $n$ such that $1.2n-4.4<5.2$.",
"role": "user"
}
] | 28 |
[
{
"content": "Calculate the number of non-commutative closed binary operations on a set $B$ where $|B|=5$.",
"role": "user"
}
] | 5^{25} - 5^{15} |
[
{
"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... | 1257s |
[
{
"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... | 4.276% |
[
{
"content": "Find the limit \\( \\lim_{n\\to+\\infty}\\left(n\\left(1+\\frac{1}{n}\\right)^n-ne\\right) \\) using the squeeze theorem, without using differentiation, integrals, or series.",
"role": "user"
}
] | -\dfrac{e}{2} |
[
{
"content": "Consider the recurrence relation defined as: $$x_0 = n, \\quad x_{i + 1} = x_i - \\log(x_i), \\quad \\forall i \\geq 0.$$ Let $N_n$ denote the smallest integer such that $x_{N_n} < 10$. Prove or disprove that $N_n \\in o(n)$. ",
"role": "user"
}
] | N_n \in o(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... | 198 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... | 468.75 J |
[
{
"content": "Find the minimum value of \\( \\frac{(x^2+4)(y^2+4)(z^2+4)}{ (x+y+z)^2} \\), where \\(x, y, z\\) are positive real numbers.",
"role": "user"
}
] | 12 |
[
{
"content": "Given two bounded closed regions $A, B \\subset \\mathbb{R}^n$, let $A \\bigtriangleup B$ denote the symmetric difference of $A$ and $B$, defined as $(A - B) \\cup (B - A)$. If $m(A \\bigtriangleup B) = 0$, where $m(S)$ is the Lebesgue measure of $S$, does it necessarily follow that $A = B$? In ot... | 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... | 18.076 g |
[
{
"content": "Is the cardinality of the sequence $I_r$ equal to $\\varkappa$?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the intersection of the plane $z = 4 + x + y$ with the cylinder $x^2 + y^2 = 4$. Then, calculate the surface integral \\( \\int\\int_{S}(x^2z+y^2z)\\,dS \\), where \\( S \\) is the surface defined by this intersection.",
"role": "user"
}
] | 32\sqrt{3}\pi |
[
{
"content": "For any function $f$ satisfying $|f(x)| \\le Cx^3$ for some constant $C$, determine if it is necessarily true that $f'(0) = 0$. Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral \\( \\int_C (2xy) \\, dx + (xy^2) \\, dy \\) using Green's Theorem, where \\( C \\) is the closed, piecewise curve formed by traveling in straight lines between the points \\((-2,1), (-2,-3), (1,-1), (1,5)\\), and back to \\((-2,1)\\), in that order.",
"role": "user"
}
... | 61 |
[
{
"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 General Journal Debit Credit
3-Dec Accounts receivable 461,100
Sales revenue 461,100
Cost of goods sold 327,900
Merchandise inventory 327,900
8-Dec Sales returns and allowance 25,300
Accounts receivable 25,300
Merchandise inventory 16,000
Cost of goods sold 16,000
13-Dec Cash 435,800
Accounts receivable... |
[
{
"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... | d = 5a |
[
{
"content": "Let $(X_n)_{n\\ge 1}$ be a sequence of dependent nonnegative random variables, where each $X_n$ has a density with respect to the Lebesgue measure on $[0,n]$ and satisfies $\\mathbb{E}(X_n) < \\infty$. Suppose $X_n$ converges weakly to a random variable $X$, which has a density with respect to the... | No |
[
{
"content": "Given a matrix $A$ with $m$ rows and $n$ columns, where $m \\leq n$, each row of $A$ has a norm of $1$, and any two rows of $A$ are orthogonal. Prove or disprove that the sum of the squares of the minors of order $m$ of matrix $A$ is equal to $1$.",
"role": "user"
}
] | 1 |
[
{
"content": "Consider an arbitrary non-zero ring $R$ and an $R$-scheme $X$. Is it possible for the ring of global sections $H^0(X, \\mathcal{O}_X)$ to be a zero ring?",
"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... | 4.4 m/s |
[
{
"content": "Let $A \\in M_n(\\mathbb{C})$ and $\\lambda$ be an eigenvalue of $A$. Determine whether there exists a sequence of diagonalizable matrices $D_n$ and a sequence $\\{\\lambda_n\\}$ of complex numbers such that each $\\lambda_n$ is an eigenvalue of $D_n$, $D_n \\to A$, and $\\lambda_n \\to \\lambda$.... | 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... | Inverse relationship |
[
{
"content": "Let $X \\subset \\mathbb{R}$ and let $f,g : X\\rightarrow X$ be continuous functions such that $f(X)\\cap g(X) = \\emptyset$ and $f(X)\\cup g(X) = X$. Which one of the following sets cannot be equal to $X$?\\nA. $[0, 1]$\\nB. $(0, 1)$\\nC. $[0, 1)$\\nD. $\\mathbb{R}$",
"role": "user"
}
] | A |
[
{
"content": "Given that \\(p = 4k + 3\\) is a prime number and \\(r\\) is a primitive root modulo \\(p\\), find the order of \\(-r\\) modulo \\(p\\) if \\((-r)^{2k+1} \\equiv 1 \\pmod{p}\\).",
"role": "user"
}
] | 2k + 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... | 6.75e-7 J |
[
{
"content": "The dartboard below has a radius of 6 inches. Each of the concentric circles has a radius two inches less than the next larger circle. If nine darts land randomly on the target, how many darts would we expect to land in a non-shaded region?\n\n[asy]import graph;\nfill(Circle((0,0),15),gray(0.7));\... | 6 |
[
{
"content": "Let $n > 1$ be an integer, and let $K$ be a field such that $n$ does not divide the characteristic of $K$ and $K$ contains the $n$-th roots of unity. Consider $\\Phi \\in \\text{Hom}(K^{\\times}/K^{\\times n}, \\mu_n)$, where $\\mu_n \\subseteq K$ is the set of $n$-th roots of unity. For $x \\in K... | 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... | 3.14 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... | 19000 |
[
{
"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... | 4.30*10^10 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... | Balance sheet of Warren Clinic as of December 31, 2015:
Assets
Cash: $30,000
Accounts receivable, net: $60,000
Total current Assets: $90,000
Long-term investments: $100,000
Net property and equipment: $150,000
Other assets: $40,000
Total assets: $380,000
Liabilities and Equity
Account Payable: $20,000
Total current lia... |
[
{
"content": "Determine the number of groups of order 45 up to isomorphism.",
"role": "user"
}
] | 2 |
[
{
"content": "Compute the differential of $\\phi(t) = e^{A+tB}$ at $0$, where $A$ and $B$ are commuting square matrices of size $n\\times n$. Explain why $e^{A}o(hB)$ is $o(h)$ in the expression $\\phi(h) = e^{A}+he^{A}B+e^{A}o(hB)$. Provide a justification for this result.",
"role": "user"
}
] | e^A B |
[
{
"content": "Given the equations:\n\\[ a + b = 6 \\]\n\\[ ax + by = 10 \\]\n\\[ ax^2 + by^2 = 24 \\]\n\\[ ax^3 + by^3 = 62 \\]\nFind the value of \\( ax^4 + by^4 \\).",
"role": "user"
}
] | 162 |
[
{
"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... | AB+ |
[
{
"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... | $1,500 |
[
{
"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... | 0.667 s |
[
{
"content": "Find the residue of the function \\(\\frac{(z+2\\pi)^3\\cos(z)}{1 - \\sin(3z)}\\) at \\(z=\\frac{\\pi}{3}\\).",
"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... | 0.3017M |
[
{
"content": "Let $W$ be the set of all real sequences $\\{b_j\\}_{j=1}^\\infty$ that satisfy the recurrence relation $b_j = Db_{j-1} + Eb_{j-2} + Fb_{j-3}$ for all $j \\geq 4$, where $D,E,F$ are real constants. Determine if $W$ is a vector subspace of the vector space of all real sequences and find its dimensi... | 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... | $1.11 |
[
{
"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... | 60-70% |
[
{
"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... | 0.138 |
[
{
"content": "Find the remainder when \\(45^{42} + 61^{34}\\) is divided by 2205.",
"role": "user"
}
] | 2131 |
[
{
"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 single, 2 double |
[
{
"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... | $63.02 |
[
{
"content": "Given $n \\in \\mathbb{N}$, let the unit simplex $S$ be defined as $$S \\equiv \\left\\{\\mathbf{x} \\in \\mathbb{R}_+^n \\,\\middle|\\, \\sum_{i=1}^n x_i = 1\\right\\}.$$ For each $i \\in \\{1, \\ldots, n\\}$, consider non-negative continuous functions $f^i: S \\to \\mathbb{R}_+$ and non-negative... | 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... | 10.48 rad/s |
[
{
"content": "Determine the Euler-Lagrange equation for the functional \\( \\int_{0.5}^2 (y')^2(1 + y'^2) \\,dx \\) subject to the boundary conditions \\( y(0.5) = 20 \\) and \\( y(2) = 2 \\).",
"role": "user"
}
] | y'' = 0 |
[
{
"content": "Let $V$ be a finite-dimensional vector space over a field $K$ with $\\dim_K(V) \\geq 1$. Suppose $M$ is a non-empty proper subset of $V$, i.e., $\\emptyset \\neq M \\subsetneq V$. Can $M$ exist such that its annihilator $M^0 = V$? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0^-} \\left( \\frac{\\ln(1-x)}{3(1-x)^{1/3}+1}\\right)^{\\frac{x}{\\cos^2x} } \\] using L'Hopital's Rule.",
"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... | 5.3e6 m/s |
[
{
"content": "How many distinct Jordan normal forms exist for matrices with the characteristic polynomial \\((x-b)^6\\)? Consider the constraints imposed by the minimal polynomial and the geometric multiplicity of the eigenvalue \\(b\\).",
"role": "user"
}
] | 11 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty} \\int^\\infty_0 \\frac{1}{1+(\\frac{x}{a})^n} \\, dx$$",
"role": "user"
}
] | a |
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