prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the integral \\( \\int\\limits_6^{16}\\left(\\frac{1}{\\sqrt{x^3+7x^2+8x-16}}\\right)\\,\\mathrm{d}x \\) given that it equals \\( \\frac{\\pi }{k} \\), where \\( k \\) is a constant. Simplify the integrand by factoring the denominator and then use a suitable substitution to find the const... | 6\sqrt{5} |
[
{
"content": "Find the minimal polynomial of an $n \\times n$ matrix $A$ with $a$ on the diagonal and $1$ on the diagonal above the actual diagonal, given that the characteristic polynomial of $A$ is $(x-a)^n$. Express the minimal polynomial in the form $(x-a)^k$, where $k \\in \\{0, 1, \\ldots, n\\}$.",
"r... | (x - a)^n |
[
{
"content": "Given the function \\( g(x) \\) such that \\( g(x+1) + g(x-1) = g(x) \\) for all \\( x \\in \\mathbb{R} \\), determine the value of \\( p \\) for which \\( g(x+p) = g(x) \\).",
"role": "user"
}
] | 6 |
[
{
"content": "Determine the radius of convergence of the function \\( f(z) = \\frac{z^2}{1 + \\cosh(z)} \\) at \\( z_0 = 0 \\), using the identity \\( \\cosh(z) = \\frac{e^z + e^{-z}}{2} \\).",
"role": "user"
}
] | \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... | 72 N/C |
[
{
"content": "Find the general solution of the differential equation $y = xy' - {y'}^2$.",
"role": "user"
}
] | y = Cx - C^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... | 16,820 |
[
{
"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 cm |
[
{
"content": "If matrices $A$ and $B$ are similar to matrices $C$ and $D$, respectively, is $A + C$ similar to $B + D$? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "Let $X$ and $Y$ be two homeomorphic topological spaces. Let $d(X)$ denote the minimal cardinality of a subset $A \\subseteq X$ such that $\\overline{A} = X$, i.e., $A$ is dense in $X$. Determine whether $d(X) = d(Y)$ when $X$ and $Y$ are homeomorphic.",
"role": "user"
}
] | d(X) = d(Y) |
[
{
"content": "Determine the number of ordered pairs \\((F_1, F_2)\\) such that \\(F_1 \\not\\subseteq F_2'\\), where \\(F_1, F_2 \\subseteq \\{a, b, c, d\\}\\) and \\(F_2' = F_2 \\cup \\{c \\mid d \\in F_2\\}\\).",
"role": "user"
}
] | 157 |
[
{
"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... | v1=0.4 m/s, v2=2.4 m/s |
[
{
"content": "Find the determinant of the matrix:\n\\[\n\\begin{bmatrix}\n283 & 5 & \\pi & 347.86\\times10^{15^{83}} \\\\\n3136 & 56 & 5 & \\cos(2.7402) \\\\\n6776 & 121 & 11 & 5 \\\\\n2464 & 44 & 4 & 2\n\\end{bmatrix}\n\\]\nwithout using a calculator. Hint: The answer is 6.",
"role": "user"
}
] | 6 |
[
{
"content": "Consider the polynomial \\(f(x) = x^3 - 2 \\in \\mathbb{Q}[x]\\). The roots of \\(f(x)\\) are \\(2^{\\frac{1}{3}}, 2^{\\frac{1}{3}}\\omega, 2^{\\frac{1}{3}}{\\omega}^2\\). Given that \\(\\mathbb{Q}(2^{\\frac{1}{3}},\\omega)\\) is the minimal splitting field of \\(f(x)\\) over \\(\\mathbb{Q}\\), de... | 2 |
[
{
"content": "The arithmetic progressions $\\{2, 5, 8, 11, \\ldots\\}$ and $\\{3, 10, 17, 24, \\ldots \\}$ have some common values. What is the largest value less than 500 that they have in common?",
"role": "user"
}
] | 479 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 20.24% |
[
{
"content": "Evaluate the integral \\( \\int_1^\\infty \\frac{dx}{x^2\\sqrt{x^2-1}} \\).",
"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... | 2.82e-19 C |
[
{
"content": "Suppose $P$ is a poset and let $G$ be $P$-generic over $V$. If $M$ is a model of ZFC such that $V \\subset M \\subset V[G]$, does there exist a poset $Q \\subset P$ such that $M = V[H]$ for some $H$ which is $Q$-generic over $V$?",
"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... | 14.72e5 N |
[
{
"content": "Determine the number of homomorphisms from the dihedral group $D_5$ of order 10 to the Klein four-group $V_4$. Provide your answer as a single integer.",
"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... | The dart will hit the balloon at t = 0.85 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... | 56 m/s |
[
{
"content": "Let $G$ be a finitely generated subgroup of $\\text{PGL}(2,\\mathbb{C})$ such that each element $g \\in G$ has finite order. Is $G$ necessarily finite? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.10 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... | 0.05 kg |
[
{
"content": "Find the limit: \\[ \\lim_{n \\to \\infty}\\frac{ \\ln n^{\\ln n}}{n!} \\]",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $1,290.24 |
[
{
"content": "Determine the value of \\( c \\in \\mathbb{R} \\) for which the limit \\( \\lim_{(x,y) \\to (0,0)} f(x,y) \\) exists, where the function \\( f(x,y) \\) is defined as follows:\n\\[\nf(x,y)=\n\\begin{cases}\n\\frac{1-\\cos(2xy)}{x^2y\\sin(\\pi y)} & \\text{ if } x\\neq 0, y\\neq 0 \\\\\nc & \\text{ ... | \dfrac{2}{\pi} |
[
{
"content": "A safe has a lock with 3 wheels, each of which can be set in 8 different positions. Due to a defect, the safe will open if any two of the three wheels are in the correct position. What is the minimum number of combinations that must be tried to ensure the safe can be opened, assuming the correct c... | 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... | 196 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... | 1255 mL |
[
{
"content": "Find the limit: \\( \\lim_{n \\to \\infty}\\frac{a^n}{n^{n/2}} \\)",
"role": "user"
}
] | 0 |
[
{
"content": "Given the joint probability distribution \\( \\mathbb{P}(X=k, Y=m) = \\frac{1}{36} \\left( \\frac{5}{6} \\right)^{k-2} \\) for random variables \\(X\\) and \\(Y\\) with \\(1 \\leq m < k\\), calculate the product of the expected values \\(\\mathbb{E}[X]\\) and \\(\\mathbb{E}[Y]\\).",
"role": "u... | 72 |
[
{
"content": "What is the dimension of the real numbers \\(\\mathbb{R}\\) as a vector space over itself?",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\rightarrow\\infty} \\sum^{n}_{k=0}\\left|\\frac{2\\pi\\cos(k\\pi(3-\\sqrt{5}))}{n}\\right| \\).",
"role": "user"
}
] | 4 |
[
{
"content": "Given the ellipse \\(C: \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\) and a line \\(l\\) that intersects the ellipse \\(C\\) at points \\(P\\) and \\(Q\\) without being perpendicular to the \\(x\\)-axis, point \\(S\\) is the reflection of point \\(P\\) about the \\(x\\)-axis. Find the maximum are... | \dfrac{1}{2}ab |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{1} \\int_{0}^{\\sqrt{1-x^2}} \\sqrt{x^2 + y^2} \\,dy\\,dx \\) using a change of variables.",
"role": "user"
}
] | \dfrac{\pi}{6} |
[
{
"content": "Calculate the integral of the pullback of the form \\( \\omega = \\frac{xdy-ydx}{x^2+y^2} \\) over the interval \\([0, 2\\pi]\\), where \\( g(t) = (e^t \\sin(17t), e^{t^2} \\cos(17t)) \\).",
"role": "user"
}
] | 34\pi |
[
{
"content": "Simplify the expression \\( \\frac{1}{\\log_2(100!)} + \\frac{1}{\\log_3(100!)} + \\frac{1}{\\log_4(100!)} + \\ldots + \\frac{1}{\\log_{100}(100!)} \\) without using logarithms.",
"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... | 19.6 N/m |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\int_{0}^{1} \\left[x^n + (1-x)^n \\right]^{1/n} \\, \\mathrm{d}x \\]",
"role": "user"
}
] | \dfrac{3}{4} |
[
{
"content": "If \\(f:[0,1] \\rightarrow \\mathbb{R}\\) is a strictly increasing discontinuous function, can \\(f([0,1])\\) be a subset of \\(\\mathbb{Q}\\)?",
"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... | Mutations |
[
{
"content": "Let \\( F \\) be a field and \\( x, y, z \\) be indeterminates. Consider the field extension \\( L = F(xz, yz, xy) \\subseteq K = F(x, y, z) \\). Show that the degree of the extension \\([K:L]\\) is finite and calculate its value.",
"role": "user"
}
] | 2 |
[
{
"content": "In an acute triangle \\(ABC\\), the altitudes from vertices \\(A\\) and \\(B\\) intersect at point \\(H\\), and \\(\\angle AHB = 120^\\circ\\). The angle bisectors from vertices \\(B\\) and \\(C\\) intersect at point \\(K\\), and \\(\\angle BKC = 130^\\circ\\). Determine the measure of \\(\\angle ... | 40 |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\rightarrow\\infty}\\frac{\\sqrt{n}}{e^n} \\) using the squeeze theorem and Stirling's approximation.",
"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... | 1102.7 nm |
[
{
"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... | 0.0896 M |
[
{
"content": "Find a polynomial $q(x)$ of degree at most 2 such that $\\phi^4 = q(\\phi)$, given that $\\phi : V \\rightarrow V$ has a minimal polynomial $x^3 + 2x + 1$. What is $q(x)$?",
"role": "user"
}
] | -2x^2 - x |
[
{
"content": "In a normed linear space $F$ with a norm that is not strictly convex, and assuming $F$ contains more than just the zero element, determine if there always exist distinct vectors $f_1$ and $f_2$ in $F$ such that $\\lVert f_1 \\rVert = \\lVert f_2 \\rVert = 1$ and $\\lVert f_1 + f_2 \\rVert = 2$. Pr... | 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... | Commander Company
Balance sheet on December 31 2007
Current Assets:
Cash $45,000
Accounts receivable20,000
Inventory 156,000
Total current assets $ 221,000
Long-Term Assets:
Office Equipment40,000
Total assets $261,000
LIABILITIES
Current Liabilities:
Accounts payable$10,000
STOCKHOLDERS' EQUITY
Common stock$200,0... |
[
{
"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... | 9*10^-2 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... | $18,600 |
[
{
"content": "Consider the space $X$ which is $\\\\mathbb{R}\\\\$ endowed with the right topology generated by \\\\Gamma=\\\\{(a,\\\\infty):a\\\\in \\\\mathbb{R}\\\\}\\\\. Determine which of the following statements is false:\n\nI. $X$ is $\\\\sigma$-compact (it is the union of countably many compact subsets)\n... | II |
[
{
"content": "Let $\\{X(t), t \\ge 0\\}$ be a process defined by $X(t) = N(t) - \\lambda t$, where $N$ is a Poisson process with rate $\\lambda > 0$. Compute $\\mathbb{E}[X(t_1)X(t_2)]$ for $0 < t_1 \\le t_2$. \\(\\lambda\\) is a positive constant.",
"role": "user"
}
] | \lambda t_1 |
[
{
"content": "Find the value of \\(x\\) that minimizes the function \\(f(x) = \\int_{0}^{1} |t-x|^3 dt\\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Find all points on the portion of the plane \\(x+y+z=5\\) in the first octant where the function \\(f(x,y,z)=xy^2z^2\\) attains its maximum value.",
"role": "user"
}
] | (1, 2, 2) |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to\\frac{\\pi}{2}}\\frac{(1-\\tan x)(1-\\sin x)}{(1+\\tan x)(\\pi-2x)^2} \\]",
"role": "user"
}
] | -\dfrac{1}{8} |
[
{
"content": "Find the maximum value of \\(7x + 12y\\) subject to the constraint \\(2x^2 + 6xy + 9y^2 - 2x - 6y \\leq 24\\) without using Lagrange multipliers.",
"role": "user"
}
] | 29 |
[
{
"content": "Determine whether the binomial coefficient \\( \\binom{2n}{n} \\) is divisible by \\( 2n-1 \\) for all positive integers \\( n \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Find a closed-form analytic expression for the function \\( f(x) \\) given by:\n\\[\nf(x) = \\int_{0}^{2\\pi } \\left[ H\\left(x + \\left| \\cos\\left( \\phi + \\beta\\right) \\right| \\right) - H\\left(x - \\left| \\cos\\left(\\phi + \\beta\\right) \\right| \\right) \\right] d\\phi,\n\\]\nwhere \... | 4 \arccos(|x|) |
[
{
"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... | $3,965 |
[
{
"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... | 243 kg |
[
{
"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... | 1.67% |
[
{
"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... | Jan: $92,000, Feb: $89,000, Mar: $101,000 |
[
{
"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... | 313.2 A |
[
{
"content": "Find the limit of the sequence \\( a_n = \\frac{n!(0.5)^{n-1}}{\\sqrt{n}\\{[(n-1)/2]!\\}^2} \\) as \\( n \\to \\infty \\).",
"role": "user"
}
] | \sqrt{\dfrac{2}{\pi}} |
[
{
"content": "Find a polynomial $f(x)$ with real coefficients that satisfies the equation $$(x+1)f(x-1) = (x-2)f(x)$$ given that $f(-1) = 0$ and $f(1) = 0$. What is the polynomial $f(x)$?",
"role": "user"
}
] | x^3 - x |
[
{
"content": "Determine the limits of the sequences defined by:\n\n\\[ x_n = \\sum_{k=0}^{2n} 2^{-k} \\cos\\left(\\sqrt{\\frac{k}{n}}\\right) \\]\n\nand\n\n\\[ y_n = \\sum_{k=0}^{2n} 2^{-\\frac{nk}{n+k}} \\]",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\frac{e^{-x} \\cos(x)}{\\sin{\\frac{1}{x}}} \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the norm of the linear functional \\( \\ell^{2} \\ni \\{x_n\\}^{\\infty}_{n=0} \\rightarrow \\sum^\\infty_{n=0}\\frac{x_n}{\\sqrt{2^n}}\\in \\mathbb{K} \\).",
"role": "user"
}
] | \sqrt{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... | H2S(aq) + 2KOH(aq) -> K2S(aq) + 2H2O(l) |
[
{
"content": "Given that \\( r, s, t \\) are integers, and the set \\( \\{a \\mid a = 2^r + 2^s + 2^t, 0 \\leq t < s < r\\} \\) forms a sequence \\(\\{a_n\\} \\) from smallest to largest as \\(7, 11, 13, 14, \\cdots\\), find \\( a_{36} \\).",
"role": "user"
}
] | 131 |
[
{
"content": "Simplify the infinite series: $$\\sum_{i=0}^\\infty \\frac{4^i}{(i!)^2}$$",
"role": "user"
}
] | I_0(4) |
[
{
"content": "Determine the number of solutions to the congruence \\(x^3 \\equiv 1 \\pmod{162}\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Determine whether the following statement is true or false: For a square matrix $X$, is it true that $\\|X\\|_2^2 = \\|X^TX\\|_2$, where $\\|\\cdot\\|_2$ denotes the operator norm? Justify your answer.",
"role": "user"
}
] | True |
[
{
"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... | 8.49e+9 Joules |
[
{
"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... | 39.1% |
[
{
"content": "Given \\((x, y) \\in \\mathbb{R}\\), if \\(x^2 + y^2 + xy = 4\\) and \\(x^4 + y^4 + (xy)^2 = 8\\), calculate the value of \\(x^6 + (xy)^3 + y^6\\).",
"role": "user"
}
] | 19 |
[
{
"content": "Determine which is larger: $e^{3\\pi}$ or $3^{e\\pi}$.",
"role": "user"
}
] | e^{3\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... | 10183.2 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... | The velocity of M1 is 4.4 trillion and the velocity of M2 is 7.3 trillion. |
[
{
"content": "Determine the number of integers \\( n \\) such that the ideal \\( \\langle n, x \\rangle \\) in \\( \\mathbb{Z}[x] \\) is a principal ideal.",
"role": "user"
}
] | 3 |
[
{
"content": "Let $C$ be the curve defined as the locus of points of intersection of the lines $x=2+m$ and $my=4-m$. A circle $S$ is given by the equation $(x-2)^2+(y+1)^2=25$. The circle $S$ intersects the curve $C$ at four points $P, Q, R, S$. If $O$ is the center of the curve $C$, calculate $OP^2 + OQ^2 + OR... | 100 |
[
{
"content": "A series of natural numbers is divided into groups as follows: \\((1), (2,3,4), (5,6,7,8,9,10,11), (12,13,\\ldots,26)\\), and so on. The sum of numbers in the \\(n^{th}\\) group is given by the expression \\(a \\cdot 2^{2n-1} - (2n+b) \\cdot 2^{2n-1} + n + 1\\). Determine the value of \\(a + b\\).... | 8 |
[
{
"content": "Determine the radius of convergence of the series \\( \\sum_{n=1}^\\infty \\frac{2^n x^{n!}}{n} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate the surface area of the solid generated by rotating the parametric curve given by \\( x=\\cos\\left(\\frac{\\pi u}{2}\\right) \\) and \\( y=\\sin\\left(\\frac{\\pi u}{2}\\right) \\) for \\( 0\\le u\\le 1 \\) about the x-axis through \\( 360^\\circ \\).",
"role": "user"
}
] | 2\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... | 0.224 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... | {
"June": 14575,
"July": 17825,
"August": 18925,
"Total": 51325
} |
[
{
"content": "Let $M$ and $N$ be subgroups of a group $G$ such that $G = M \\times N$. Suppose $H$ is a subgroup of $G$ satisfying $HM = G = HN$ and $H \\cap M = H \\cap N = \\{1\\}$. Prove that $M \\cong N$.",
"role": "user"
}
] | M \cong 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... | $37.54 |
[
{
"content": "Let \\( a, b, c \\) be natural numbers such that the product \\( ab \\) is divisible by \\( 5c \\), the product \\( c \\) is divisible by \\( 13a \\), and the product \\( ca \\) is divisible by \\( 31b \\). Find the smallest possible value of the product \\( abc \\).",
"role": "user"
}
] | 20301125 |
[
{
"content": "Let $A$ be a complex $n \\times n$ matrix with $n$ distinct eigenvalues. Does there exist a vector $u \\in \\mathbb{C}^n$ such that the vectors $u, Au, A^2u, \\ldots, A^{n-1}u$ are linearly independent?",
"role": "user"
}
] | Yes |
[
{
"content": "Point \\( P \\) is a moving point on the ellipse \\(\\frac{x^{2}}{16}+\\frac{y^{2}}{9}=1\\) in the first quadrant. From point \\( P \\), two tangents are drawn to the circle \\( x^{2}+y^{2}=9 \\), touching the circle at points \\( A \\) and \\( B \\) respectively. The line \\( AB \\) intersects th... | \dfrac{27}{4} |
[
{
"content": "Suppose that $n$ is a given even number, and let $a_1, a_2, \\ldots, a_n$ be nonnegative real numbers such that \\(a_1 + a_2 + \\cdots + a_n = 1.\\) Determine the maximum possible value of the expression: \\[ \\sum_{1 \\leq i < j \\leq n} \\min\\left\\{ (i-j)^2, (n+i-j)^2 \\right\\} a_i a_j. \\]",... | \dfrac{n^2}{16} |
[
{
"content": "Find the minimum value of \\(\\left|1-(ab+bc+ca)\\right| + \\left|1-abc\\right|\\) given that \\(a, b, c \\in \\mathbb{R}\\) and \\(a+b+c=1\\).",
"role": "user"
}
] | \dfrac{44}{27} |
[
{
"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... | 54 |
[
{
"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... | $127,335.88, $84,889.74, $56,592.59, $37,728.02 |
[
{
"content": "Calculate the arc length of the curve given by the polar equation \\( r = 1 + \\cos\\theta \\) from \\( \\theta = 0 \\) to \\( \\theta = \\pi \\). Use the formula for arc length in polar coordinates: \\[ \\int_{0}^{\\pi} \\sqrt{r^2 + \\left(\\frac{dr}{d\\theta}\\right)^2} \\, d\\theta \\] and then... | 8 |
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