prompt list | ground_truth string |
|---|---|
[
{
"content": "Given integers \\( k \\) and \\( n \\) such that \\( 2 \\leq k \\leq n \\leq 2k - 1 \\), consider an \\( n \\times n \\) grid. Rectangles of size \\( 1 \\times k \\) or \\( k \\times 1 \\) are placed on the grid, each covering exactly \\( k \\) squares, with no two rectangles overlapping. Determin... | 2n - k |
[
{
"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... | 2.57 A |
[
{
"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 m |
[
{
"content": "Evaluate \\(7^{8^9} \\mod 100\\).",
"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... | [Na+] = 0.0537M
[NO3-] = 0.114M
[Pb2+] = 0.0303M
[I-] = 0.000484M |
[
{
"content": "Solve the equation \\(2^{-3x^3+5x^2-x}=\\frac{x^2+1}{x}\\) for all possible values of \\(x\\).",
"role": "user"
}
] | 1 |
[
{
"content": "In the RSA encryption system with modulus $m = 536813567 = 8191 \\cdot 65537$ and public exponent $e = 3602561$, determine the maximum value of a plaintext block (encoded as a number in the \"26-system\") that can be encrypted without exceeding the modulus $m$. Assume blocks of length 5 are used f... | 11881375 |
[
{
"content": "Determine the characteristics of the partial differential equation \\( \\frac{\\partial\\rho}{\\partial t} + 4 \\frac{\\partial\\rho}{\\partial x}= 0 \\).",
"role": "user"
}
] | x - 4t = C |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 9141 |
[
{
"content": "Evaluate the double integral \\( \\iint\\limits_{\\mathbb{R}^2} e^{-(x^2+y^2)} \\, dx \\, dy \\).",
"role": "user"
}
] | \pi |
[
{
"content": "Determine the domain of definition for the function \\( f(x) = \\int_{0}^{\\infty} \\frac{t^x}{1+t^4 \\sin^2 t} \\, dt \\).",
"role": "user"
}
] | (-1, 1) |
[
{
"content": "Determine whether an entire function with uncountably many zeros must be identically zero.",
"role": "user"
}
] | Yes |
[
{
"content": "Compute\n\\[\\tan \\left( 2 \\arctan \\frac{1}{5} - \\frac{\\pi}{4} \\right).\\]",
"role": "user"
}
] | -\frac{7}{17} |
[
{
"content": "Let $F$ be an algebraically closed field. Determine the dimension of the algebraic set $X = V(y - x^2, y^2 - y + z^2)$.",
"role": "user"
}
] | 1 |
[
{
"content": "Let \\((a_k)_{k\\geq 1}\\) be a positive sequence such that the series \\(\\sum_{k\\geq 1} a_k\\) converges to \\(L\\). Using the Cauchy-Schwarz inequality, for any \\(n\\geq 1\\), we have:\n\\[ f(n):=\\sum_{k=1}^n \\frac{1}{a_k}\\geq \\frac{n^2}{\\sum_{k=1}^n a_k}> \\frac{n^2}{L}. \\]\nDetermine ... | 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... | 12.01% |
[
{
"content": "Let \\( f: [0,1]\\to \\mathbb{R} \\) be a function of class \\( C^{1} \\) with \\( f(0)=0 \\). Determine which of the following inequalities is true:\n\n1. \\( \\displaystyle \\int_{0}^{1}\\bigl[f(t)\\bigr]^{2} dt \\leq \\dfrac{1}{4}\\int_{0}^{1}\\bigl[f'(t)\\bigr]^{2} dt \\)\n\n2. \\( \\displayst... | 3 |
[
{
"content": "Find \\( \\delta \\) such that \\( |f(x) - l| < \\varepsilon \\) for all \\( x \\) satisfying \\( 0 < |x - a| < \\delta \\), given \\( f(x) = \\frac{x}{1 + \\sin^2(x)} \\), \\( a = 0 \\), and \\( l = 0 \\).",
"role": "user"
}
] | \varepsilon |
[
{
"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... | 5.27e-22 |
[
{
"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... | 7.40% |
[
{
"content": "Determine the value of \\( \\gamma(0) \\) if \\( \\gamma(t) = e^{2\\pi it \\sin(1/t)} \\) for \\( t \\in (0, 2\\pi] \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate the limit: \\[ \\lim_{n\\to\\infty}\\frac{n!\\cdot e^n}{n^n} \\]",
"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... | 6.416 x 10^-7 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... | 8.86 years |
[
{
"content": "In any metric space, does the diameter of a set remain the same for both open and closed balls?",
"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... | 1.2 ft |
[
{
"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... | Capsule |
[
{
"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... | 15489.76 |
[
{
"content": "Compare the numbers $e^\\pi$ and $\\pi^e$. Given that the function $f(x) = \\frac{\\ln(x)}{x}$ has a maximum at $(e, \\frac{1}{e})$, determine which of the two numbers is larger.",
"role": "user"
}
] | e^\pi |
[
{
"content": "Given the polynomial $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are distinct non-zero integers, and $p(a) = a^3$ and $p(b) = b^3$, find the value of $p(13)$. \n\nExpress your answer as a single integer.",
"role": "user"
}
] | 1927 |
[
{
"content": "Evaluate the double integral \\(\\int_{v=0}^{\\infty}\\int_{u=0}^{\\infty}\\frac{\\sqrt{uv}}{(u+v)^2}e^{-\\frac{(u+v)}{2}}dudv\\) using a change of variables.",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "In triangle $ABC$ with circumcircle $\\Gamma$, let $D$ be the midpoint of the minor arc $BC$. Points $E$ and $F$ lie on $\\Gamma$ such that $DE \\perp AC$ and $DF \\perp AB$. Lines $BE$ and $DF$ intersect at $G$, and lines $CF$ and $DE$ intersect at $H$. Given $AB = 8$, $AC = 10$, and $\\angle BAC... | 2\sqrt{3} |
[
{
"content": "Find the surface area of the solid obtained by revolving the curve $8y^2 = x^2 - x^4$ about the $x$-axis, where $x$ ranges from $-1$ to $1$. Use the formula for the surface area of revolution about the $x$-axis: $$S = 2\\pi \\int y \\, dS.$$",
"role": "user"
}
] | \dfrac{\pi}{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... | 96.4; C3H3F3 |
[
{
"content": "Determine the number of integer solutions for the equation \\(x_1 + x_2 - x_3 + x_4 - x_5 = 3\\) subject to the following constraints:\n\n\\[ \\begin{aligned} \n&0 < x_1 \\le 6, \\\\\n&-8 \\le x_2 < -2, \\\\\n&x_3 \\le 1, \\\\\n&3 < x_4, \\\\\n&2 \\le x_5 \\le 8 \n\\end{aligned} \\]",
"role": ... | 2016 |
[
{
"content": "Triangle ABC is an equilateral triangle and O is the center of its inscribed circle. If the area of the circle is $4\\pi$ sq cm, what is the area, in square centimeters, of triangle ABC? Express your answer in simplest radical form.\n\nNote: The area of a circle with radius $r$ is $\\pi r^2.$",
... | 12\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... | 0.68% |
[
{
"content": "Determine the number of group homomorphisms from the symmetric group $S_3$ to the alternating group $A_3$. Note that $S_3$ has order 6 and $A_3$ has order 3. Consider the properties of group homomorphisms, such as mapping the identity to the identity, and apply these to conclude the possible mappi... | 1 |
[
{
"content": "Compute the limit: \\[ \\lim_{n \\rightarrow\\infty}\\ln\\left(\\frac{n-1}{n+1}\\right)^n \\]",
"role": "user"
}
] | -2 |
[
{
"content": "In the acute-angled triangle \\(ABC\\), altitudes \\(BP\\) and \\(CQ\\) are drawn, and point \\(T\\) is the intersection point of the altitudes of \\(\\Delta PAQ\\). Given that \\(\\angle CTB = 90^\\circ\\), find the measure of \\(\\angle BAC\\).",
"role": "user"
}
] | 45^\circ |
[
{
"content": "Consider the region $\\Omega :x^2+y^2<4$ in $\\Bbb R^2$ with boundary $\\partial \\Omega$. Let $u(x,y)$ be the solution of the Dirichlet problem \\[\\frac{\\partial^2u}{\\partial x^2}+\\frac{\\partial^2u}{\\partial y^2}=0, \\hspace{1cm} (x,y)\\in \\Omega\\] with the boundary condition $u(x,y)=1+2x... | 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... | 16788.84 |
[
{
"content": "Determine the number of elements of order 30 in the symmetric group \\( S_{10} \\).",
"role": "user"
}
] | 120960 |
[
{
"content": "Find all functions \\(f:\\mathbb{Z}\\to\\mathbb{N}\\) such that, for every \\(x,y\\in\\mathbb{Z}\\), the following condition holds:\n\\[\\max(f(x+y),f(x-y))\\mid \\min(xf(y)-yf(x), xy)\\]",
"role": "user"
}
] | f(x) = 1 |
[
{
"content": "Compute the value of the integral \\( \\frac{1}{2\\pi i}\\int_\\mathcal{C} |1+z+z^2|^2 \\, dz \\), where \\( \\mathcal{C} \\) is the unit circle in the complex plane centered at the origin and taken in the anticlockwise direction.",
"role": "user"
}
] | 2 |
[
{
"content": "Compute the integral \\( \\int_{-\\infty}^{+\\infty} x^2 e^{-x^2/2} \\, dx \\).",
"role": "user"
}
] | \sqrt{2\pi} |
[
{
"content": "An ellipsoid in $\\mathbb{R}^3$ is defined by the equation $2x^2 + 2y^2 + z^2 = 338$. Find the radius of a sphere centered on the $z$-axis that touches the ellipsoid at points $(x, y, 10)$, where the intersection of the sphere and the plane $z = 10$ forms a circle with radius $\\sqrt{119}$.",
... | 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... | 9500 g |
[
{
"content": "Let \\( f : \\mathbb{R}^n \\to \\mathbb{R} \\) be a continuous function such that there exist points \\( x, y \\in \\mathbb{R}^n \\) with \\( f(x)f(y) < 0 \\). Determine if it is true that the Hausdorff dimension of the zero set of \\( f \\) is at least \\( n-1 \\).",
"role": "user"
}
] | True |
[
{
"content": "Evaluate the line integral \\( \\int_C (y+e^\\sqrt{x}) \\, dx + (xe^{y^2}) \\, dy \\) where \\( C \\) is the boundary of the triangle with vertices \\((0,0), (4,2), (0,2)\\), oriented counterclockwise.",
"role": "user"
}
] | e^4 - 5 |
[
{
"content": "Let $f\\colon \\mathbb{R}\\to\\mathbb{R}$ be a function. Assume that for all sequences $(x_n)_{n\\in\\mathbb{N}}\\subset\\mathbb{R}$ with $x_n\\to x$ in $\\mathbb{R}$, there exists a subsequence $(x_{n_k})_k$ such that $f(x_{n_k})\\to f(x)$ as $k\\to \\infty$. Does it follow that $f$ is sequential... | 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... | 15.60% |
[
{
"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... | 2.33e+03 N^{-1} |
[
{
"content": "Evaluate the limit:\n\\[\\lim_{x\\to0}\\left[\\left(\\frac{1}{n}\\right)^{1/\\sin^2x}+\\left(\\frac{2}{n}\\right)^{1/\\sin^2x}+\\cdots+\\left(\\frac{n}{n}\\right)^{1/\\sin^2x}\\right]^{\\sin^2x}\\]",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the integral \\( \\oint_{C}\\frac{\\sqrt{1-z^2}}{1+z^2}dz \\) using the upper semi-circle contour and the residue theorem.",
"role": "user"
}
] | \pi \sqrt{2} |
[
{
"content": "Consider the polynomial \\( P(X) = X^2 - 2 \\) over the field \\( \\mathbb{Q} \\). The splitting field of \\( P \\) is \\( L = \\mathbb{Q}(\\sqrt{2}) \\). Determine the number of distinct isomorphisms from \\( L \\) to itself that are not the identity map.",
"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... | 10.3 kJ |
[
{
"content": "Given matrices \\(A = \\left[ \\begin{array}{rrr} 2 & -2 & -4 \\\\ -1 & 3 & 4 \\\\ 1 & -2 & -3 \\end{array} \\right]\\) and \\(B = \\left[ \\begin{array}{rrr} -4 & -3 & -3 \\\\ 1 & 0 & 1 \\\\ 4 & 4 & 3 \\end{array} \\right]\\), find \\(|X|\\) where \\(X = A + A^2B^2 + A^3 + A^4B^4 + \\dots + A^{10... | 0 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty}\\frac{(n^2)!}{(2n)!} \\]",
"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... | 8,716,580 |
[
{
"content": "Compute the integral \\( \\int_\\gamma |z|^2 \\, \\mathrm{d}z \\) where \\( \\gamma \\) is the contour starting at 0, going vertically to \\( i \\), then going horizontally to \\( i + 1 \\), then going vertically to 1, and finally going horizontally to 0.",
"role": "user"
}
] | 1 - i |
[
{
"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... | 2.97e5 Pa |
[
{
"content": "Let point \\( O \\) be a point inside triangle \\( ABC \\) that satisfies the equation\n\\[\n\\overrightarrow{OA} + 2 \\overrightarrow{OB} + 3 \\overrightarrow{OC} = 3 \\overrightarrow{AB} + 2 \\overrightarrow{BC} + \\overrightarrow{CA}.\n\\]\n\nFind the value of \\(\\frac{S_{\\triangle AOB} + 2 S... | \dfrac{11}{6} |
[
{
"content": "Let $ABCD$ be a convex quadrilateral, and let $M$ and $N$ be the midpoints of $\\overline{AC}$ and $\\overline{BD},$ respectively. Then there exists a constant $k$ so that\n\\[AB^2 + BC^2 + CD^2 + DA^2 = AC^2 + BD^2 + k \\cdot MN^2.\\]Find $k.$\n\n[asy]\nunitsize(0.8 cm);\n\npair A, B, C, D, M, N... | 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... | 1.25 |
[
{
"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... | 9.08 g of K_2Cr_2O_7 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.858n |
[
{
"content": "Compute the integral \\( \\int_{\\sigma} \\omega \\) of the differential form \\( \\omega = x_3 \\, dx_1 \\wedge dx_2 \\) over the surface parametrized by \\( \\sigma(\\theta,\\phi) = (\\sin\\phi\\cos\\theta,\\sin\\phi\\sin\\theta,\\cos\\phi) \\) for \\( (\\theta, \\phi) \\in [0,2\\pi]\\times[-\\p... | 0 |
[
{
"content": "Calculate the area of the largest rectangle that can be inscribed in an ellipse using integrals. Express the maximum area in terms of the ellipse's semi-major axis \\(a\\) and semi-minor axis \\(b\\).",
"role": "user"
}
] | 2ab |
[
{
"content": "Let \\( n \\geq 3 \\). Determine whether there exist \\( n \\) points not lying on a single line, such that the pairwise distances between them are irrational, and the areas of all triangles with vertices at these points are rational.",
"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... | $93,258,217 |
[
{
"content": "Determine the cardinality of the set \\( \\{ f \\in \\mathbb{N}^{\\mathbb{N}} \\mid f \\le h \\} \\), where \\( h: \\mathbb{N} \\to \\mathbb{N} \\) is defined by \\( h(n) = n + 1 \\) and the partial order relation \\( f \\le g \\) is defined as \\( \\forall n \\in \\mathbb{N}, f(n) \\le g(n) \\)."... | 2^{\aleph_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, -1>, <-2, 0>, <-1, sqrt(3)/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... | 29.1 N |
[
{
"content": "Solve for $x$: $$81^{2x} = 27^{3x - 4}.$$",
"role": "user"
}
] | x = 12 |
[
{
"content": "In triangle \\( \\triangle ABC \\), the sides are \\( AB = 30 \\), \\( BC = 40 \\), and \\( CA = 50 \\). Squares \\( A_1A_2BC \\), \\( B_1B_2AC \\), and \\( C_1C_2AB \\) are constructed externally on the sides of \\( \\triangle ABC \\). The lines \\( A_1A_2 \\), \\( B_1B_2 \\), and \\( C_1C_2 \\) ... | 124 |
[
{
"content": "Find the limit superior of \\( \\sqrt[k]{\\lvert a_k \\rvert} \\) for the sequence \\( a_k \\) defined by \\( a_k = \\dfrac{(-1)^n}{\\ln(n)} \\) when \\( k = 3n - 1 \\) and \\( a_k = 0 \\) otherwise.",
"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... | 411 lines per millimeter |
[
{
"content": "Let $a_n$ be the integer obtained by writing all the integers from $1$ to $n$ from left to right. For example, $a_3 = 123$ and $a_{11} = 1234567891011$. Compute the remainder when $a_{44}$ is divided by $45$.",
"role": "user"
}
] | 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... | 4.7 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... | Revenue: $4,320,000
Variable cost of goods sold:($1,440,000)
Fixed cost of goods sold: ($360,000)
Net income(sum): $2,520,000 |
[
{
"content": "Evaluate the integral \\( \\int_0^{2\\pi} \\log |1-w e^{i\\theta}| \\, d\\theta \\) where \\( w \\) is a complex number such that \\( |w| < 1 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral:\n\\[ I = \\int_{eV/2}^{\\infty} d\\epsilon_1 \\int_{-\\infty}^{-eV/2} d\\epsilon_2 \\int_{0}^{\\infty} d\\epsilon' \\int_{-\\infty}^{0} d\\epsilon \\, \\delta(\\epsilon_2 - \\epsilon_1 + \\epsilon - \\epsilon') \\]\nwhere \\( \\delta(x) \\) is the Dirac delta function.",
... | 0 |
[
{
"content": "Find a matrix \\( P \\) such that \\( PAP^{-1} = B \\), where \\( A \\) and \\( B \\) are given matrices.",
"role": "user"
}
] | P |
[
{
"content": "Given the function \\(f(x) = 2x^3 + 5x + 3\\), find the value of \\(x\\) such that \\(f^{-1}(x) = 1\\).",
"role": "user"
}
] | 10 |
[
{
"content": "Given finitely many distinct real numbers $x_1, x_2, \\ldots, x_n$, determine whether there exists a real number $y$ such that all of $y + x_1, y + x_2, \\ldots, y + x_n$ are irrational.",
"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... | 98.825 Mpa |
[
{
"content": "Determine which of the following expressions correctly represents \\( \\sin^2(x) \\cos(x) \\):\n\n1. \\( \\sin^2(x) \\cos(x) = \\sum x^2\\frac{(-4)^n (9^{n+1}-1)x^{2n}}{(2n+2)!4^{n+1}} \\)\n\n2. \\( \\sin^2(x) \\cos(x) = \\frac{1\\sum{(-1)^n x^{2n}(1-9)^n}}{4\\cdot(2n)!} \\)\n\nProvide a justifica... | 1 |
[
{
"content": "Let $f(x)$ be a real-valued function defined for all real numbers such that $|f(x) - f(y)| \\leq \\frac{1}{2} |x - y|$ for all $x$ and $y$. Determine the number of points where the graph of $y = f(x)$ intersects the line $y = x$. \\( \\text{Provide your answer as a single number.} \\)",
"role"... | 1 |
[
{
"content": "Let $K_1$ and $K_2$ be disjoint convex sets in $\\mathbb{C}$. Let $z_1 \\in \\partial K_1$ and $z_2 \\in \\partial K_2$ be the points that minimize the distance $\\text{dist}(K_1, K_2) = \\inf{|x-y|}$ where $x \\in K_1$ and $y \\in K_2$. Can the separating hyperplane be chosen to be orthogonal to ... | Yes |
[
{
"content": "Apples are being packed in a box, where each apple has an expected weight of 200 g and a variance of 20 g. Packing stops as soon as the total weight reaches 10 kg or more. Calculate the probability that the number of apples, $N$, in the box is less than or equal to 49, i.e., find $P(N \\leq 49)$. ... | 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... | 6.851e-5 M |
[
{
"content": "Find all natural numbers \\( n \\ge 3 \\) such that the numbers \\( 1, 2, \\ldots, n \\) can be arranged at the vertices of a regular \\( n \\)-sided polygon so that the sum of the numbers at each three consecutive vertices is even.",
"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... | 678260.87 |
[
{
"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... | 3km/h |
[
{
"content": "Determine the values of \\( a > 0 \\) for which the improper integral \\( \\int_{0}^{\\infty}\\frac{\\sin x}{x^{a}} \\, dx \\) converges.",
"role": "user"
}
] | (0, 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... | 21.8 kN |
[
{
"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 m/s^2 |
[
{
"content": "Given the curve \\( C \\):\n\\[\n(x - \\arcsin a)(x - \\arccos a) + (y - \\arcsin a)(y + \\arccos a) = 0\n\\]\nIf the line \\( x = \\frac{\\pi}{4} \\) intersects the curve creating a chord of length \\( d \\), what is the minimum value of \\( d \\) as \\( a \\) varies?\n\n(A) \\(\\frac{\\pi}{4}\\)... | C |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | F=(mv^2)/r or F=m omega^2r |
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