prompt list | ground_truth string |
|---|---|
[
{
"content": "Find the Laurent series and the residue of the function \\( f(z) = \\frac{(z-2)^3}{e^{5/z^3}} \\) at \\( z = 0 \\). Use the formula for the coefficient function \\( c(n) \\) in the Laurent series: \\[ c(n) = \\frac{1+(-1)^{n}}{2}\\cdot(a(n)-b(n))+b(n) \\] where \\( a(2n) \\) generates coefficients... | 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... | 2.46e-3 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... | 1.65 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... | Sr{eq}^{2+}, Ba{eq}^{2+}, K{eq}^+, Rb{eq}^+ |
[
{
"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... | Cash flows from operating activities
Net Profit (Loss) $28,829
Adjustments for:
Depreciation $30,945
Amortization $4,090
Working capital changes:
(Increase) / Decrease in trade and other receivables $12,929
(Increase) / Decrease in inventories $20,322
(Increase) / Decrease in prepaids $(856)
Increase / (Decrease) in trade payables $4,092
Increase / (Decrease) in trade accruals $2,017
Net cash generated (used) from operating activities $102,368
Cash flows from investing activities
Purchase of property, plant and equipment $(37,568)
Sale (Acquisition) of portfolio investments $(52,926)
Net cash generated (used) in investing activities $(90,494)
Cash flows from financing activities
Dividends $(5,825)
Payment of short-term borrowings $(2,166)
Payment of long-term borrowings $(6,967)
Net cash generated (used) in financing activities $(14,958)
Net increase in cash and cash equivalents $(3,084)
Cash and cash equivalents at beginning of period $9,181
Cash and cash equivalents at end of period $6,097 |
[
{
"content": "Calculate the expectation \\( E[W_s e^{W_s - W_t}] \\) for a Brownian process \\( W_s \\) where \\( 0 \\leq s < t \\) and \\( W_0 = 0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the following statement is true or false: If a point is a source for a one-dimensional (smooth) map $f(x)$, meaning $x$ satisfies $|f'(x)| > 1$, then this point is sensitive to initial conditions. This means that for every $\\epsilon > 0$ and $d>0$, there exist $\\bar{x}$ and $k$... | 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... | $25.23 |
[
{
"content": "Consider an $n \\times n$ nilpotent complex matrix $A$. Determine the minimum integer $d$ such that $A^d = 0$ for any such matrix $A$.",
"role": "user"
}
] | n |
[
{
"content": "Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$.",
"role": "user"
}
] | 19 |
[
{
"content": "Solve the equation \\(x^{3} - \\lfloor x \\rfloor = 3\\), where \\(\\lfloor x \\rfloor\\) represents the greatest integer less than or equal to \\(x\\).",
"role": "user"
}
] | \sqrt[3]{4} |
[
{
"content": "Let \\( a, b, c, d \\) be odd numbers such that \\( 0 < a < b < c < d \\), and \\( ad = bc \\), \\( a+d = 2^k \\), \\( b+c = 2^m \\), where \\( k \\) and \\( m \\) are integers. Determine the value of \\( a \\).",
"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... | 5.32 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... | 6.33% |
[
{
"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... | 14.05 L |
[
{
"content": "Find the area of an equilateral triangle $\\triangle ABC$ if there is a point $I$ inside the triangle such that $IA = 8$, $IB = 10$, and $IC = 6$. Express your answer in simplest form.",
"role": "user"
}
] | 36 + 25\sqrt{3} |
[
{
"content": "Let $N$ be the number of ordered pairs of nonempty sets $\\mathcal{A}$ and $\\mathcal{B}$ such that:\n\n- $\\mathcal{A} \\cup \\mathcal{B} = \\{1,2,3,4,5,6,7,8,9,10,11,12\\}$,\n- $\\mathcal{A} \\cap \\mathcal{B} = \\emptyset$,\n- The number of elements in $\\mathcal{A}$ is not an element of $\\mat... | 772 |
[
{
"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... | -2.4% |
[
{
"content": "Can a group of order 42 be a subgroup of SL(4,C)?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the coefficient of $\\frac{x^3}{3!}$ in the series expansion of the exponential generating function $e^{e^x-1}$ for Bell numbers. Verify that this coefficient corresponds to the Bell number $B_3$.",
"role": "user"
}
] | 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... | 28.79 m/s |
[
{
"content": "Prove using the definition of limit that \\(\\lim_{(x,y)\\to(0,0)}\\frac{(1-\\cos(xy))\\sin y}{x^2+y^2} = 0\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfies the functional equation \\( f\\left(x^2+f(y)\\right) = y + f(x)^2 \\) for all \\( x, y \\in \\mathbb{R} \\).",
"role": "user"
}
] | f(x) = x |
[
{
"content": "Does there exist a set that cannot be equipped with a distance function, making it impossible to be a metric space?",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit \\( \\lim_{h \\to 0} \\frac{f(3+3h+h^2) - f(3)}{f(2+h-h^2) - f(2)} \\) given that \\( f(x) \\) is a differentiable function with \\( f'(2) = 5 \\) and \\( f'(3) = 7 \\).",
"role": "user"
}
] | \dfrac{21}{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... | C_12 H_22 O_11 + H_2 O -> 4 C_2 H_5 OH + 4 CO_2 |
[
{
"content": "Find the product of all divisors of \\(420^4\\).",
"role": "user"
}
] | 420^{2250} |
[
{
"content": "Determine the Lebesgue measure of the set of numbers in the interval (0,1) that contain 100 consecutive 4's in their decimal expansion.",
"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... | 2.05E8 |
[
{
"content": "Let \\( f \\) be a map from an open subset \\( U \\) of a normed vector space \\( E \\) to another normed vector space \\( F \\). Suppose that \\( f \\) is differentiable at a point \\( a \\) in \\( U \\). Is \\( f \\) continuous in a neighborhood of \\( a \\)? Provide a justification for your ans... | No |
[
{
"content": "Given that $x_n$ converges to a positive value $a$ and there exists $0 < M < a$ such that $x_n$ is eventually lower-bounded by $M$, can it be concluded that $\\sqrt{x_n}$ is eventually lower-bounded by $\\sqrt{M}$? If so, explain how this fact can be used to prove that $\\lim_{n \\to \\infty} \\sq... | \sqrt{a} |
[
{
"content": "Evaluate the integral \\( \\int_0^{\\frac{\\pi}{2}} \\frac{\\sin^3(t)}{\\sin^3(t)+\\cos^3(t)}dt \\).",
"role": "user"
}
] | \dfrac{\pi}{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... | $352,575 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 78.7 micro-tesla |
[
{
"content": "The graph of the parabola defined by the equation $y=(x-2)^2+3$ is rotated 180 degrees about its vertex, then shifted 3 units to the left, then shifted 2 units down. The resulting parabola has zeros at $x=a$ and $x=b$. What is $a+b$?",
"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... | -6000(Decrease in bank balance)=-10000(Decrease in bank loan)+4000(Increase in equity) |
[
{
"content": "Let \\(f(x) = a_0 + a_1x + a_2x^2 + \\ldots + a_nx^n\\), where \\(a_i\\) are non-negative integers. Given that \\(f(1) = 21\\) and \\(f(25) = 78357\\), find the value of \\(\\frac{f(10) + 3}{100}\\).",
"role": "user"
}
] | 51 |
[
{
"content": "Determine the Galois group of the field extension \\(\\mathbb{F}_3(x^4) \\subset \\mathbb{F}_{3^2}(x)\\).",
"role": "user"
}
] | D_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... | 725 |
[
{
"content": "Consider a Markov Chain with the transition matrix:\n\n$$M = \\left( \\begin{array}{ccc} b_0 & b_1 & b_2 \\\\ 0 & b_0+b_1 & b_2 \\\\ b_2 & b_1 & b_0 \\\\ \\end{array} \\right)$$\n\nwhere all $b_i >0$ and $b_0+b_1+b_2 = 1$. If the Markov chain starts from state $0$, calculate the expected time unti... | \dfrac{1}{b_2} |
[
{
"content": "A particle starts from the origin and moves equally likely in any of the four directions (up, down, left, right) with each move being independent of the previous ones. Let $X_i$ denote the number of moves in direction $i$ for $i = 1, 2, 3, 4$, where $1$ corresponds to up, $2$ to down, $3$ to left,... | n |
[
{
"content": "Suppose $(T_n)$ is a sequence in the space of compactly supported distributions $\\mathcal{E}'(\\mathbb{R})$ that converges to $T$ in the weak topology $\\sigma(\\mathcal{E}',\\mathcal{E})$. Does this imply that $(T_n)$ also converges to $T$ in the strong dual topology $\\beta(\\mathcal{E}',\\math... | Yes |
[
{
"content": "Evaluate the integral \\( \\int_\\gamma \\frac{|z|e^z}{z^2}dz \\), where \\( \\gamma \\) is the circle of radius 2 centered at the origin in the complex plane.",
"role": "user"
}
] | 4\pi i |
[
{
"content": "Let \\(n \\ge 2\\) be an integer. Determine the number of pairs of integers \\((x,y)\\) with \\(0 \\le x,y \\le n-1\\) such that \\(x^n+y^n\\) is congruent to a number between \\(0\\) and \\(n^2-1\\) modulo \\(n^2\\).",
"role": "user"
}
] | n^2 |
[
{
"content": "Find the codimension of the subspace \\( A = \\overline{\\operatorname{span}}\\{S^n(1,4,-1,0,0,\\ldots):n=0,1,2,\\ldots\\} \\) in \\( \\ell_2 \\), where \\( S \\) is the right shift operator defined by \\( Se_i = e_{i+1} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the infinite series: $$\\sum_{n=1}^\\infty\\frac{n}{1\\cdot3\\cdot5\\cdots(2n+1)}$$ What is its value?",
"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... | 2.94 ft |
[
{
"content": "A number with a sum of digits equal to 2021 is divided by 7, resulting in a number composed entirely of the digit 7. Determine the sum of all possible numbers of digits 7 that this resulting number can have.",
"role": "user"
}
] | 503 |
[
{
"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... | 61.5 % |
[
{
"content": "Evaluate the limit \\( \\lim\\limits_{n \\rightarrow +\\infty} \\dfrac{ (\\ln(n))^{an} }{ (an)! } \\) for \\( a > 0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Compute the operator norm of the integral operator \\( T \\) defined by \\( Tf(x) = \\int_0^1 \\frac{f(y)}{1+x^2+y^2} \\, dy \\), where the norm is given by \\( \\|f\\|_\\infty = \\sup_{x \\in [0,1]} |f(x)| \\).",
"role": "user"
}
] | \dfrac{\pi}{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... | 1187.9kg |
[
{
"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... | 1.8 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... | 313.92 J |
[
{
"content": "Find the Fréchet derivative of the functional $\\|Au-f\\|^2$, where $A$ is a bounded linear operator on an infinite-dimensional Hilbert space. Express your answer in terms of $A$, $u$, and $f$.",
"role": "user"
}
] | 2A^*(Au - f) |
[
{
"content": "Calculate the line integral \\( \\oint_{C} F \\cdot d\\vec{r} \\) for the vector field \\( F(x,y)=\\left(\\frac{-y}{(x+1)^2+y^2}, \\frac{x+1}{(x+1)^2+y^2}\\right) \\), where \\( C \\) consists of the boundary of the square with vertices at (4,4), (-4,4), (-4,-4), and (4,-4) (oriented counterclockw... | 0 |
[
{
"content": "Given that $k$ is a positive integer less than 6, how many values can $k$ take on such that $3x \\equiv k \\pmod{6}$ has no solutions in $x$?",
"role": "user"
}
] | 4 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -$111 million |
[
{
"content": "Determine whether every subset of $\\mathbb{R}$ with Lebesgue measure zero is a member of $K_{\\sigma}$, the class of sets that are countable unions of compact sets.",
"role": "user"
}
] | No |
[
{
"content": "Prove that if $f: \\mathbb{R} \\to \\mathbb{R}$ is an odd convex function, then $f(x) = ax$ for some $a \\in \\mathbb{R}$.",
"role": "user"
}
] | f(x) = ax |
[
{
"content": "Let \\( S = \\{1, 2, 3, 4\\} \\). The sequence \\( a_{1}, a_{2}, \\cdots, a_{n} \\) of \\( n \\) terms has the following property: for any non-empty subset \\( B \\) of \\( S \\) (denote the number of elements in \\( B \\) as \\( |B| \\)), there are exactly \\( |B| \\) consecutive terms in the seq... | 8 |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{2\\pi}|\\sin x - \\cos x|\\,dx. \\)",
"role": "user"
}
] | 4\sqrt{2} |
[
{
"content": "Determine whether there exist real numbers $a$, $b$, and $c$ such that the equation \\(a^{2n+1} + b^{2n+1} = c^{2n+1}\\) holds for every odd positive integer \\(2n+1\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Find the minimum value of\n\\[f(x) = x + \\frac{1}{x} + \\frac{1}{x + \\frac{1}{x}}\\]for $x > 0.$",
"role": "user"
}
] | \frac{5}{2} |
[
{
"content": "Given the equations:\n\\[-29x^3 + 15y^3 = -61209\\]\n\\[-60x - 90y = -3420\\]\nand the restriction \\(x < y\\), find the value of:\n\\[-50x^2 + 70y^2\\]",
"role": "user"
}
] | 18270 |
[
{
"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... | 965 m/s |
[
{
"content": "Determine whether the infimum of the Riemann Upper Sum is equal to the limit of the Upper Sum as the partition size approaches zero. Specifically, consider the Riemann upper sum \\( U_\\Gamma = \\sum_{k=1}^N[\\sup_{x\\in I_k}f(x)]v(I_k) \\), where \\( \\Gamma = \\{I_k\\} \\) is a partition of \\( ... | Yes |
[
{
"content": "Coplanar squares $ABGH$ and $BCDF$ are adjacent, with $CD = 10$ units and $AH = 5$ units. Point $E$ is on segments $AD$ and $GB$. What is the area of triangle $ABE$, in square units? Express your answer as a common fraction.\n\n[asy]\ndraw((0,0)--(5,0)--(5,5)--(0,5)--(0,0)--cycle,linewidth(2));\nd... | \frac{25}{3} |
[
{
"content": "Evaluate the infinite series: \\[ \\sum_{n=1}^\\infty \\arctan\\left(\\frac{1}{n^2+n+1}\\right) \\]",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Determine the set of points where the function \\( f(x) = \\sum_{n=1}^\\infty \\frac{1}{x^2+n^2} \\) is differentiable for \\( x \\in \\mathbb{R} \\).",
"role": "user"
}
] | \mathbb{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... | 1.99 ft/s |
[
{
"content": "In an abelian group \\( G \\), generators \\( x \\) and \\( y \\) have orders 16 and 24, respectively, with the relation \\( x^2 = y^3 \\). Determine the order of \\( G \\).",
"role": "user"
}
] | 48 |
[
{
"content": "Compute:\n\\[\n\\sin \\frac{\\pi}{12} + \\sin \\frac{3\\pi}{12} + \\sin \\frac{5\\pi}{12} + \\sin \\frac{7\\pi}{12} + \\sin \\frac{9\\pi}{12} + \\sin \\frac{11\\pi}{12}.\n\\]",
"role": "user"
}
] | \sqrt{2} + \sqrt{6} |
[
{
"content": "In parallelogram ABCD, points P, Q, R, and S are located on sides AB, BC, CD, and DA, respectively, such that AP = BQ = CR = DS = \\( \\frac{1}{3} \\) of the length of AB. Determine the ratio of the area of quadrilateral PQRS to the area of parallelogram ABCD.",
"role": "user"
}
] | \dfrac{5}{9} |
[
{
"content": "Evaluate the line integral \\( \\oint_C \\frac{dz}{z-2} \\) where \\( C \\) is the circle \\( |z-1|=5 \\).",
"role": "user"
}
] | 2\pi i |
[
{
"content": "Find the maximum value of \\(M = a^2 + b^2 + a - 3b\\) given that \\(a, b \\in \\mathbb{R}\\) satisfy the inequality \\(a^2 + b^2 \\leq 40\\).",
"role": "user"
}
] | 60 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 10.8 / r radians per second |
[
{
"content": "Find the limit: \\[ \\lim_{n \\to \\infty} \\frac{n^n}{e^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... | 154.73 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... | mu = (2 sin((3pi)/8))/(1 + 2 cos((3pi)/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... | $177120.7 |
[
{
"content": "Evaluate the closed complex integral \\( \\oint\\frac{1}{(z+1)(1-z)}dz \\) where the contour is defined by \\(|z|=3\\). Can this integral evaluate to zero even though it includes poles within the contour?",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\int_0^{100}\\frac{e^{-\\sqrt{x}}}{\\sqrt{x}+100}dx \\) and determine if the result is greater than 0.005.",
"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... | 25 m/s |
[
{
"content": "Determine whether the series \\( S_n = \\sum_{k=1}^n \\binom{n}{k}\\frac{1}{k}\\left(-\\frac{1}{1-a} \\right)^k\\left(\\frac{a}{1-a} \\right)^{n-k} \\) converges as \\( n \\to \\infty \\), where \\( 0 < a < 1 \\). If it converges, find the limit of \\( S_n \\) as \\( n \\to \\infty \\).",
"rol... | 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... | 0.730 |
[
{
"content": "Given a deck of $n$ cards numbered $1$ to $n$, calculate the probability of forming the maximum possible number of piles when drawing cards according to a specified rule.",
"role": "user"
}
] | \dfrac{1}{n!} |
[
{
"content": "Find the limit of the sequence \\( a_n = \\sqrt[n!]{\\frac{1}{2^{(n!)}}-\\frac{1}{3^{(n!)}}} \\) as \\( n \\to \\infty \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Determine whether the set of proper fractions \\( \\frac{p}{q} \\), where \\( p \\) and \\( q \\) are prime numbers, is dense in the interval \\([0,1]\\). In other words, does the union of this set with its limit points equal the entire interval \\([0,1]\\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine if the Diophantine equation \\(-900x^2 + y^2 + 1842300 = 0\\) has any integer solutions. Use the hint to check the equation modulo the prime divisors of 900: \\(2^2\\), \\(3^2\\), and \\(5^2\\).",
"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... | A country can impose a fixed exchange rate regime through two primary mechanisms: 1. Open market operations: The government buys or sells its own currency in the foreign exchange market to maintain the fixed rate. If the value of its currency falls, the government buys its currency and sells foreign reserves. Conversely, if its currency rises, it sells its currency and buys foreign reserves. 2. Legal restrictions: The government can make it illegal to exchange its currency at a rate other than the fixed rate. This can involve restrictions on currency trading or capital controls to prevent speculation and maintain the fixed exchange rate. |
[
{
"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... | 12.1525 g / equivalence |
[
{
"content": "Determine whether a countable group can have an uncountable number of distinct subgroups.",
"role": "user"
}
] | Yes |
[
{
"content": "Solve the inequality \\(\\sqrt{x^{2}-3 x-54}-\\sqrt{x^{2}-27 x+162}<8 \\sqrt{\\frac{x+6}{x-9}}\\) and find the sum of its integer solutions within the interval \\([-25, 25]\\).",
"role": "user"
}
] | -290 |
[
{
"content": "Evaluate the integral \\( \\int_{\\gamma} \\tanh(z) \\, dz \\) where the contour \\( \\gamma(t) = e^{it} \\) for \\( 0 \\leq t \\leq 2\\pi \\). Use the residue theorem to determine the poles and compute the integral.",
"role": "user"
}
] | 0 |
[
{
"content": "Find the minimum value of the function \\( f(x, y) = \\frac{1}{x} + \\frac{1}{y} + 2x + 2y \\) for \\( x, y > 0 \\) using the theorem that states: If \\( f:A\\rightarrow\\mathbb{R} \\) is convex and \\( \\xi \\) has a directional derivative in every direction with \\( \\nabla f(\\xi)=0 \\), then \... | 4\sqrt{2} |
[
{
"content": "Is the set of real numbers uncountable?",
"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... | 3.4e8 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, -55, -83), (0, -17, -25) |
[
{
"content": "Calculate the product of all primitive n-th roots of unity for each integer \\( n \\geq 3 \\). Specifically, evaluate the expression:\n\\[ \\prod_{w \\in G_n^*} w \\]\nwhere \\( G_n^* \\) denotes the set of primitive n-th roots of unity.",
"role": "user"
}
] | 1 |
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