prompt list | ground_truth string |
|---|---|
[
{
"content": "Let $a,$ $b,$ $c,$ $p,$ $q,$ and $r$ be real numbers such that\n\\[\\begin{vmatrix} p & b & c \\\\ a & q & c \\\\ a & b & r \\end{vmatrix} = 0.\\]Assuming that $a \\neq p,$ $b \\neq q,$ and $c \\neq r,$ find the value of $\\frac{p}{p - a} + \\frac{q}{q - b} + \\frac{r}{r - c}.$",
"role": "user... | 2 |
[
{
"content": "Find the polynomial \\( P \\in \\mathbb{R}[x] \\) such that there exists a positive integer \\( n \\) for which the following holds for all \\( x \\in \\mathbb{Q} \\): \\[ P\\left(x+\\frac{1}{n}\\right) + P\\left(x-\\frac{1}{n}\\right) = 2P(x). \\]",
"role": "user"
}
] | P(x) = ax + b |
[
{
"content": "Evaluate the improper integral: \\[ \\int_0^\\infty \\frac{1}{1+x^2 \\sin^2 x} \\, dx. \\]",
"role": "user"
}
] | \infty |
[
{
"content": "Determine whether the following statement is true or false: If the sequence $\\left\\{ \\sqrt[n]{c_n} \\right\\}$ converges in $\\mathbb{R} \\cup \\{ \\pm \\infty \\}$, then the sequence $\\left\\{ \\frac{c_{n+1}}{c_n} \\right\\}$ also converges in $\\mathbb{R} \\cup \\{ \\pm \\infty \\}$ for any ... | B |
[
{
"content": "Let $A, B, C$ be subspaces of a vector space $V$. Determine if it is always true that $V = (A \\cap (B+C)) \\cap (C \\cap (B + A)) = A \\cap C$. Provide a proof or counterexample.",
"role": "user"
}
] | A \cap C |
[
{
"content": "Determine the order of magnitude of the number of involutions in the symmetric group $S_{256}$.",
"role": "user"
}
] | 10^{259} |
[
{
"content": "Calculate the limit: $$ \\lim_{n\\rightarrow\\infty}\\left(\\sum_{k=1}^{n} \\binom{n}{k} k! k \\frac{1}{n^k}\\right) $$",
"role": "user"
}
] | \infty |
[
{
"content": "In BINGO, a $5\\times5$ card is filled by marking the middle square as WILD and placing 24 other numbers in the remaining 24 squares.\n\n\nSpecifically a card is made by placing 5 numbers from the set $1-15$ in the first column, 5 numbers from $16-30$ in the second column, 4 numbers $31-45$ in the... | 50,\!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... | 8684J |
[
{
"content": "Suppose $0 \\neq X_n \\in \\mathbb{M}_{k(n)}(\\mathbb{C})$. If $\\lim_{n \\to \\infty} \\text{tr}(X_n^*X_n) = 0$, can we conclude that $\\lim_{n \\to \\infty} \\|X_n^*X_n\\| = 0$? Here, $\\text{tr}$ denotes the standard trace on complex matrices, and $\\|\\cdot\\|$ denotes the matrix norm.",
"... | 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... | -1.3959e-8 C/m |
[
{
"content": "Is it possible to find two functions $f(x)$ and $g(x)$ such that \\( \\lim_{x\\to 0} f(x) = \\lim_{x\\to 0} g(x) = 0 \\) and \\( \\lim_{x\\to 0} f(x)^{g(x)} = a \\) for a given $a > 0$?",
"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... | $1,845.38 |
[
{
"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... | 0.24 m/sec |
[
{
"content": "What is the greatest possible value of $x$ for the equation $$\\left(\\frac{4x-16}{3x-4}\\right)^2+\\left(\\frac{4x-16}{3x-4}\\right)=12?$$",
"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... | 400 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... | 1.475e-5 T.m^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 vials |
[
{
"content": "Evaluate the double integral \\( \\iint_D \\frac{xy}{\\sqrt{1-y^2}} \\, dx \\, dy \\), where \\( D \\) is the region in the first quadrant bounded by the circle \\( x^2 + y^2 = 1 \\).",
"role": "user"
}
] | \dfrac{1}{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... | 0.22 m/s |
[
{
"content": "Evaluate the limit \\( \\lim_{x \\rightarrow \\infty} \\frac{xF(x)}{f(x)} \\), where \\( f(x) = e^{x^2} \\) and \\( F \\) is an antiderivative of \\( f \\).",
"role": "user"
}
] | \dfrac{1}{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... | 357.70 s |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to 0} \\frac{(\\cos x)^{1/2} - (\\cos x)^{1/3}}{(\\sin x)^2}$$",
"role": "user"
}
] | -\dfrac{1}{12} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.94 N |
[
{
"content": "How many prime numbers between 30 and 65 have a prime remainder when divided by 10?",
"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... | Consumption, investment, and net exports would decrease. However, net exports might increase due to lower import purchases. |
[
{
"content": "Determine the values of $p$ for which the series \\( \\sum_{n=1}^{\\infty} \\frac{n^p}{\\ln n} \\) converges.",
"role": "user"
}
] | (-\infty, -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... | Depreciation expense = $748,220
Accumulated depreciation - Equipment = $440,600
Accumulated depreciation - Buildings = $209,000
Accumulated depreciation - Infrastructure = $98,620 |
[
{
"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 = \int_0^m \left( 2\pi r^3\sigma \right)\sqrt{1+\left(\frac{2hr}{a^2}\right)^2} |
[
{
"content": "Determine the shortest possible length of an algorithm that generates a harlequin pattern on a Rubik's cube. The pattern should use at most 6 colors, with no more than 2 facelets of the same color, and no adjacent facelets of the same color.",
"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... | 115.719 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... | $1-\frac{2GM}{\hat r}$ |
[
{
"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... | 8/3 ft/s |
[
{
"content": "Evaluate the contour integral \\( \\int_{C} \\frac{z}{i} \\, dz \\), where \\( C \\) is the part of the circle \\( z = e^{i\\theta} \\) for \\( 0 \\leq \\theta \\leq \\frac{\\pi}{2} \\).",
"role": "user"
}
] | i |
[
{
"content": "Is the ideal \\((1+x^2, x^3)\\) in \\(\\mathbb{Q}[x]\\) a principal ideal? Justify 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... | J_{i,t+1} = \begin{cases} 0, & \text{if } x_{i,t} = 1\\ J_{i,t} + 1, & \text{otherwise} \end{cases} |
[
{
"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... | inelastic |
[
{
"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... | 1.09e-8 |
[
{
"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... | Activity level 84,600 104,700 124,800
Direct material ($4) 338,400 418,800 499,200
Direct labor ($7) 592,200 732,900 873,600
Overhead ($11) 930,600 1,151,700 1,372,800
Fixed costs 721,000 721,000 721,000
Total Costs 2,582,200 3,024,400 3,466,600 |
[
{
"content": "Determine the degree of the field extension \\([\\mathbb{Q}(2^{1/5}, e^{2\\pi i/5}) : \\mathbb{Q}]\\).",
"role": "user"
}
] | 20 |
[
{
"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... | 9.1 |
[
{
"content": "Let $f$ be a continuous self-map of a topological space $X$ such that $f \\circ f = f$. Is $X$ homotopic to its image $f(X)$?",
"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... | 0.007m |
[
{
"content": "Determine the number of solutions to the equation \\(e^{-z} = 3z-2\\) in the annulus \\(\\{z \\in \\Bbb C : 1 < |z| < 2\\}\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the largest value of $s$ such that the point $x_0 = (0,0)$ and direction $\\xi_0 = (0,1)$ do not belong to the $H^s$ wavefront set of the 2D step function $f(x,y)$, which is 1 for $y > 0$ and 0 for $y \\leq 0$. Consider a cutoff function $\\phi$ that is not zero at $x_0$ and a conic neig... | \dfrac{1}{2} |
[
{
"content": "Find all functions \\( f : \\mathbb{R} \\to \\mathbb{R} \\) such that \\( f(xy) = f'(x)f(y) \\) for all \\( x, y \\in \\mathbb{R} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "What is the sum of all the positive two-digit integers divisible by both the sum and product of their digits?",
"role": "user"
}
] | 72 |
[
{
"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... | 66,666.67 ly/yr |
[
{
"content": "Determine the greatest possible value of $a + b + c$ for which there exist at least two values of $n$ such that $C_n - B_n = A_n^2$. Here, for a positive integer $n$ and nonzero digits $a$, $b$, and $c$, $A_n$ is the $n$-digit integer with each digit equal to $a$, $B_n$ is the $n$-digit integer wi... | 18 |
[
{
"content": "Let $\\triangle ABC$ be an acute scalene triangle with circumcircle $\\omega$. The tangents to $\\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT = CT = 16$, $BC = 22$, and $TX^2 + TY^2 + XY^2 = 1143$. Find $XY^2... | 717 |
[
{
"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... | $1.48 trillion |
[
{
"content": "Determine a condition on the vector \\( v \\in \\mathbb{R}^n \\) such that for any function \\( f: \\mathbb{R}^n \\longrightarrow \\bar{\\mathbb{R}} \\), the conjugate satisfies \\( f^*(u+v) = f^*(u) \\) for all \\( u \\in \\mathbb{R}^n \\).",
"role": "user"
}
] | v = 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... | Magnesium Chloride and Potassium Carbonate |
[
{
"content": "Calculate the average value of the expression \\( S = x_1x_2 + x_2x_3 + \\cdots + x_{10}x_1 \\) over all permutations \\( x_1, x_2, \\ldots, x_{10} \\) of the numbers \\( 1, 2, 3, \\ldots, 10 \\).",
"role": "user"
}
] | \dfrac{880}{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... | 573279 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... | 1.5 m/s |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to+\\infty} n^{(4/3)} \\left( \\sqrt[3]{n^2 +1} + \\sqrt[3]{3- n^2} \\right) \\]",
"role": "user"
}
] | \dfrac{4}{3} |
[
{
"content": "Let \\([x]\\) denote the greatest integer less than or equal to the real number \\(x\\). Define\n\\[ A = \\left[\\frac{19}{20}\\right] + \\left[\\frac{19^2}{20}\\right] + \\cdots + \\left[\\frac{19^{2020}}{20}\\right]. \\]\n\nFind the remainder when \\(A\\) is divided by 11.",
"role": "user"
... | 2 |
[
{
"content": "Let $M$ be a compact connected orientable $3$-manifold without boundary. If $H_1(M; \\mathbb{Z}) = 0$, does it follow that $M$ has the same homology groups as a $3$-sphere? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Consider a pentagon with vertices labeled as \\( P_{i} \\) for \\( i = 1, \\ldots, 5 \\). Each vertex \\( P_{i} \\) is assigned an integer \\( x_{i} \\), such that the sum \\( s = \\sum x_{i} \\) is positive. You are allowed to perform the following operation: select a negative \\( x_{i} \\), repl... | 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... | $27,073.57 |
[
{
"content": "Determine the integer value of \\( x \\) such that \\( x^{677} \\equiv 3 \\pmod{2020} \\).",
"role": "user"
}
] | 543 |
[
{
"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... | DateAccountDebitCredit
DateLand$452,400
Building$210,600
Equipment$117,000
Cash$780,000 |
[
{
"content": "Determine whether the expression \\( \\frac{1}{1+W(x)} \\) approaches 0 as \\( x \\to +\\infty \\), given that \\( W(x) = \\log x - \\log\\log x + \\cdots \\) and \\( x \\in \\mathbb{R}^+ \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Solve for $x$ in the equation: \\[ 0 = \\frac{x^{36} - 1}{x - 1} - 10 \\]",
"role": "user"
}
] | 0.9025 |
[
{
"content": "Suppose that $g$ is a Lebesgue integrable function. Show that \\( \\int |g(x + \\epsilon) - g(x)|\\,dx \\to 0 \\) as \\( \\epsilon \\to 0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the maximum and minimum values of the expression \\(a^2\\sin^2\\theta + b^2\\csc^2\\theta\\) for real numbers \\(a\\) and \\(b\\).",
"role": "user"
}
] | 2|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... | 11.25 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... | 6601 years |
[
{
"content": "Let $(X_1,X_2)$ be a bivariate normal random variable with mean vector $\\mu = \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix}$ and covariance matrix $\\Sigma = \\begin{pmatrix} 3 & -1\\\\ -1 & 3 \\end{pmatrix}$. Calculate the probability that $X_1 + X_2 < 2$ and $X_1 - X_2 > 0$. \\(\\text{Express your a... | 0.1116 |
[
{
"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... | 1.71e-7 T |
[
{
"content": "Determine the second homology group $H_2(\\mathbb{Q},\\mathbb{Z})$ where the action is trivial.",
"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... | 34.41 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... | 3 mL |
[
{
"content": "Find the expression for \\( E(Z^2 | Z \\leq 0) \\) where \\( Z \\sim N(0,1) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Find the minimal polynomial of \\( \\sqrt{3} + \\sqrt{5} \\) over \\( \\mathbb{Q} \\).",
"role": "user"
}
] | x^4 - 16x^2 + 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... | 0.30 |
[
{
"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... | 2e-3 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... | 1.33e-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... | $301,061.41 |
[
{
"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... | 2.82 years |
[
{
"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... | -89mW |
[
{
"content": "What is the sum of all of the odd divisors of $6300$?",
"role": "user"
}
] | 3224 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\left( 1 + \\tan 3x \\right) ^{\\dfrac{1}{\\sin x}} \\]",
"role": "user"
}
] | e^3 |
[
{
"content": "Let \\( f : \\mathbb{R} \\to \\mathbb{R} \\) be a function that is continuous at 0 with \\( f(0) = 1 \\). The function satisfies the relation \\( f(x) - f\\left(\\frac{x}{2}\\right) = \\frac{3x^2}{4} + x \\) for all \\( x \\). Find \\( f(3) \\).",
"role": "user"
}
] | 16 |
[
{
"content": "Given the joint probability density function \\( f(x,y) = 1120x^{3}y^{3} \\) for \\( 0 \\leq x, 0 \\leq y, \\) and \\( x+y \\leq 1 \\), calculate the correlation coefficient \\( \\text{Corr}(X,Y) \\).",
"role": "user"
}
] | -\dfrac{4}{5} |
[
{
"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... | 1287 s |
[
{
"content": "Evaluate the integral \\( \\int_{|z|=1} \\frac{dz}{(z-2i)(z+2i)} \\) using the substitution \\( z = e^{i\\theta} \\).",
"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... | -9060 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... | H_2(g) + Cl_2(g) o 2HCl |
[
{
"content": "Evaluate the limit: $$L=\\lim_{x \\to -0.5^{-}} \\left\\lfloor\\frac{1}{x} \\left\\lfloor \\frac{-1}{x} \\right\\rfloor\\right\\rfloor$$",
"role": "user"
}
] | -2 |
[
{
"content": "Let $q$ be a prime number such that $q \\equiv 1 \\pmod{12}$. Find the sum of all solutions to the congruence $x^3 \\equiv 8 \\pmod{q}$, and express your answer modulo $q$. What is the value of this sum?",
"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... | 1.027 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... | 8.21% |
[
{
"content": "Calculate the expectation \\( \\tilde{E}\\left(B_t - \\int_0^t B_s \\,ds\\right) \\) where \\( B \\) is a standard Brownian motion with \\( B_0 = 0 \\) under the probability measure \\( P \\), and \\( \\tilde{B} = B_t - \\int_0^t B_s ds \\) for \\( t \\in [0,T] \\), \\( T>0 \\). The expectation \\... | 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... | -$1593 |
[
{
"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... | 0.04 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... | $42,480 |
[
{
"content": "Determine the maximum number of distinct labels that can be assigned to the vertices of a graph such that each vertex is adjacent to at least one vertex with a different label.",
"role": "user"
}
] | 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... | $1.62 |
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