prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\sum_{k=1}^{\\infty}\\frac{1}{k^{2}\\sqrt[k]{n}}\\sin^{2}\\left(\\frac{n \\pi}{k}\\right) \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\int_1^3 e^{2x}\\left(1+\\{2x\\}+\\{2x\\}^2\\right)dx \\) using the properties of the fractional part function.",
"role": "user"
}
] | e^6 - e^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... | 3.78*10^{6} N/C |
[
{
"content": "Find a value of $R > 0$ such that for all complex numbers $z$ with $|z| > R$, the inequality $|z^{10} + 100z + 9| > 1$ holds.",
"role": "user"
}
] | 2 |
[
{
"content": "The summary of a survey of 100 students listed the following totals:\n\n59 students did math homework\n\n49 students did English homework\n\n42 students did science homework\n\n20 students did English and science homework\n\n29 students did science and math homework\n\n31 students did math and Eng... | 18 |
[
{
"content": "Construct a triangle using a compass and a ruler, given one side and the medians to the other two sides.",
"role": "user"
}
] | \triangle ABC |
[
{
"content": "Evaluate the double integral \\( \\int\\int_R \\frac{1}{x} \\, dx \\, dy \\) over the region \\( R \\), which is a circular disc in \\( \\mathbb{R}^2 \\) with radius \\( a \\) and center \\( (a,0) \\). Use polar coordinates to express the integral and determine the appropriate limits for \\( \\the... | 2\pi a |
[
{
"content": "In triangle $ABC,$ $E$ lies on $\\overline{AC}$ such that $AE:EC = 2:1,$ and $F$ lies on $\\overline{AB}$ such that $AF:FB = 1:4.$ Let $P$ be the intersection of $\\overline{BE}$ and $\\overline{CF}.$\n\n[asy]\nunitsize(0.8 cm);\n\npair A, B, C, D, E, F, P;\n\nA = (1,4);\nB = (0,0);\nC = (6,0);\n... | \left( \frac{4}{13}, \frac{1}{13}, \frac{8}{13} \right) |
[
{
"content": "Let $f:\\mathbb{R}^n\\to \\mathbb{R}$ be a convex function with a unique minimum $x^*$, and let $g:\\mathbb{R}^n\\to \\mathbb{R}$ be a convex function such that $f(x)=g(x)$ if $\\|x-x^*\\|\\geq 1$, and $f(x)\\leq g(x)$ if $\\|x-x^*\\|\\leq 1$ where $\\|\\cdot\\|$ is a norm. Determine whether there... | Yes |
[
{
"content": "Evaluate the limit: $$ \\lim_{x \\to \\infty} \\left(1- \\frac{1}{x}\\right)^{-x^2}. $$",
"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... | 1.85% |
[
{
"content": "Calculate the limit: $$\\lim_{n\\to\\infty}\\frac{[x] + [2x] + \\dots + [nx]}{n^2}$$ where $[x]$ denotes the floor function.",
"role": "user"
}
] | \dfrac{x}{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... | 1388 N |
[
{
"content": "Let $X$ be a separable space and $x^{**} \\in X^{**}$. If $x^{**}(x^{*}_{n}) \\rightarrow 0$ for every weak*-null sequence $(x^{*}_{n})_{n}$ in $X^{*}$, is $x^{**}$ in $X$? 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... | 2.95 m/s |
[
{
"content": "Find all natural numbers \\( n \\) that satisfy the inequality:\n\\[\n\\frac{4}{5} + \\frac{4}{45} + \\frac{4}{117} + \\cdots + \\frac{4}{16n^{2} - 8n - 3} > n - 5\n\\]\nCalculate the sum of all such solutions.",
"role": "user"
}
] | 15 |
[
{
"content": "Given the recursive sequence defined by $a_0 = y$ and $a_n = \\left \\lfloor{ \\frac{5 a_{n-1}}{2}} \\right \\rfloor$, determine if the sequence will eventually contain an even number for any odd natural number $y$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Given the polynomial \\( P(x) = x^{3} - 2x^{2} - x + 1 \\) with real zeros \\( a > b > c \\), find the value of the expression:\n\\[ a^{2} b + b^{2} c + c^{2} a. \\]",
"role": "user"
}
] | 4 |
[
{
"content": "Find the remainder when the sum of all positive integers that can be expressed in the form \\(2^a \\cdot 3^b \\cdot 5^c\\), where \\(a, b, c\\) are positive integers satisfying \\(a+b+c=10\\), is divided by 1001.",
"role": "user"
}
] | 34 |
[
{
"content": "Determine the number of homomorphisms from the symmetric group $S_n$ to the cyclic group $\\mathbb{Z}_2$ for $n \\geq 5$. Provide your answer as a single integer.",
"role": "user"
}
] | 2 |
[
{
"content": "Let $V$ be a finite-dimensional vector space over a field $F$. Suppose $B$ is a basis of $V$ such that \\( \\left\\{\\begin{bmatrix} w \\\\ w \\end{bmatrix} \\mid w \\in B \\right\\} \\) is a basis for $V^2$. Determine the dimension of $V$. \\( \\boxed{} \\)",
"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... | 242 pm |
[
{
"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... | L=\frac{\frac{1}{2}kx_c^2-mg\sin\theta x_c}{\mu mg\cos\theta+mg\sin\theta} |
[
{
"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... | thorium-227 |
[
{
"content": "Find the limit \\( \\lim\\limits_{n \\to \\infty} S^n(x) \\), where \\( S^n(x) = \\sin(\\sin(\\ldots\\sin(x)\\ldots)) \\) is the composition of the sine function with itself \\( n \\) times, starting with \\( S(x) = \\sin(x) \\).",
"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... | -1007.66 |
[
{
"content": "Calculate the line integral \\( \\int_{C} \\vec{F} \\cdot \\vec{n} \\, ds \\) where \\( \\vec{F}(x, y) = x^{10} \\vec{i} + (3x - 10x^9y) \\vec{j} \\) and \\( \\vec{n} \\) is the unit normal vector to the curve \\( C \\). The curve \\( C \\) is defined by \\( x^2 + y^2 = 1 \\) and is restricted to ... | \dfrac{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... | $5,000,000 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\rightarrow\\infty}\\frac{\\cos\\left(\\frac{1}{x}+x\\right)-\\cos\\left(x\\right)}{\\cos\\left(\\frac{1}{x}+x\\right)}$$",
"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... | W = (-3.20*10^-20 Cm)E |
[
{
"content": "Suppose $X$ is a topological space, $C$ is a closed subset of $X$, and $U$ is an open subset of $X$ that is dense in $X$. Is $U \\cap C$ necessarily dense in $C$? Provide a justification for your answer.",
"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... | $57,866.58 |
[
{
"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 monthly payment is $144.33. The following shows the principal and interest portions of the first six payments:
Payment 1: Principal = $74.33, Interest = $70.00
Payment 2: Principal = $74.69, Interest = $69.64
Payment 3: Principal = $75.06, Interest = $69.27
Payment 4: Principal = $75.43, Interest = $68.90
Payment 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... | 31590 kJ/hr |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty} \\frac{x^2}{(1+x^2)^2} \\, dx \\).",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Does there exist a simple graph, called \"prime,\" which has no non-trivial automorphisms, meaning it only has the identity automorphism?",
"role": "user"
}
] | Yes |
[
{
"content": "Suppose that \\( \\int_1^\\infty |f(x)|dx \\) converges and \\( \\lim \\limits_{x \\to \\infty} f(x) = L \\). What is the value of \\( L \\)? Justify your answer.",
"role": "user"
}
] | 0 |
[
{
"content": "Given a sequence \\(\\{a_n\\}\\) with the sum of the first \\(n\\) terms \\(S_n\\), where\n\\[ a_1 = 3, \\quad S_n = 2a_n + \\frac{3}{2}((-1)^n - 1). \\]\n\nIf there exist three terms \\(a_1, a_p, a_q\\) (\\(p, q \\in \\mathbb{Z}_+\\), \\(1 < p < q\\)) that form an arithmetic sequence, find \\( q ... | 1 |
[
{
"content": "Let \\( f : [0, 1] \\to \\mathbb{R} \\) be a differentiable function such that \\( f'(x) = \\sqrt{1 + f^2(x)} \\) for all \\( x \\in [0, 1] \\). Given that \\( f(0) = 1 \\) and \\( f(1) = -1 \\), find the value of the integral \\( \\int_{0}^{1} f'(x) \\, dx \\).",
"role": "user"
}
] | -2 |
[
{
"content": "Calculate the integral part of the number \\( T = \\frac{2}{1} \\cdot \\frac{4}{3} \\cdot \\frac{6}{5} \\cdot \\frac{8}{7} \\cdots \\frac{2016}{2015} \\). Given that \\( T = 2017\\int_{0}^{\\frac{\\pi}{2}} \\sin^{2017}(x)dx \\), and the identity \\( \\int_0^{\\frac{\\pi}{2}} \\sin^{2n+1}(x) dx = \... | 56 |
[
{
"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... | 32 g/mol |
[
{
"content": "Is it possible to find two functions $h: \\mathbb{R}^3 \\to \\mathbb{R}$ and $k: \\mathbb{R} \\to \\mathbb{R}^3$ such that both $h$ and $k$ have all partial derivatives at the origin, but the composition $h \\circ k$ does not have partial derivatives at the origin?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the minimum value of $|\\sin z|$ on the circle centered at the origin with radius $(2N + 1)\\pi$, where $N$ is a non-negative integer.",
"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... | 8693.79 |
[
{
"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... | 0.0464 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... | 2g/3r |
[
{
"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... | 210 kg |
[
{
"content": "Evaluate the limit: $$ \\lim_{(x,y)\\to(1,1)} \\frac{(x-1)^{4/3}-(y-1)^{4/3}}{(x-1)^{2/3}+(y-1)^{2/3}} $$",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Cc |
[
{
"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... | R = \frac{\rho}{4 \pi} (\frac{1}{r_2} - \frac{1}{r_1}) |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 0} \\left( \\frac{e\\cos(x)-e}{\\ln(1+\\ln(2-e^{\\sin^2(x)}))} \\right) \\] without using L'Hôpital's rule or infinite series.",
"role": "user"
}
] | \dfrac{e}{2} |
[
{
"content": "Calculate the expected number of collisions when hashing 80,000 unique entries using a 36-bit hash function.",
"role": "user"
}
] | 0.0466 |
[
{
"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... | $256,717.50 |
[
{
"content": "For a positive integer \\( n \\) where \\( n \\geq 2 \\), define \\( a_n = \\sum_{k=1}^{n-1} \\frac{n}{(n-k) 2^{k-1}} \\). Determine the maximum value of the sequence \\( \\{a_n\\} \\).",
"role": "user"
}
] | \dfrac{10}{3} |
[
{
"content": "Compute the integral of \\( z^2 + z \\overline{z} \\) over the contour \\( C = \\{ z \\mid |z| = 1 \\} \\), where \\( \\overline{z} \\) is the complex conjugate of \\( z \\).",
"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... | 4.1 |
[
{
"content": "Determine the radius of convergence for the infinite series: $$\\sum_{n=0}^\\infty\\frac{2^n}{2^n+1}x^{n^2}$$",
"role": "user"
}
] | 1 |
[
{
"content": "Determine whether a homeomorphism on \\(\\bar{\\mathbb{R}}:=\\mathbb{R}\\cup\\{\\pm\\infty\\}\\) or \\(\\bar{\\mathbb{Z}}:=\\mathbb{Z}\\cup\\{\\pm\\infty\\}\\) has zero topological entropy.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit \\( \\lim_{x \\to 0}\\frac{\\tan^{2}x - x^{2}}{x^{4}}.\\)",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "Find all complex numbers $z$ such that $|z| + \\overline{z} = 9 - 3i$. \\( \\overline{z} \\) denotes the complex conjugate of \\( z \\).",
"role": "user"
}
] | 4 + 3i |
[
{
"content": "Evaluate the determinant of the matrix:\n\\[\n\\begin{vmatrix}\n p^2 & (p+2)^2 & (p+4)^2 & (p+6)^2 \\\\\n q^2 & (q+2)^2 & (q+4)^2 & (q+6)^2 \\\\\n r^2 & (r+2)^2 & (r+4)^2 & (r+6)^2 \\\\\n s^2 & (s+2)^2 & (s+4)^2 & (s+6)^2 \\\\\n\\end{vmatrix}\n\\]\nwithout expanding it.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\frac{1}{2\\pi i}\\int_\\mathcal{C} |1+z+z^2+\\cdots+z^{2n}|^2~dz \\) where \\(\\mathcal{C}\\) is the unit circle.",
"role": "user"
}
] | 2n |
[
{
"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... | 0.0625 |
[
{
"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 (y, t) = \frac{50}{\zeta} \cos(2\pi \times 10^5t - 0.5y) \mathbf 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... | 2.5e+22 atoms of O |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 28.01 m/s |
[
{
"content": "Determine whether any integer can be the smallest number in more than one primitive Pythagorean triple. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether a number ending in 222 can be expressed as a non-trivial power of a natural number, i.e., in the form \\(x^y\\) where \\(x, y > 1\\) are natural numbers.",
"role": "user"
}
] | No |
[
{
"content": "Determine the range of values for \\(c \\in \\mathbb{R}^+\\) such that the Banach fixed-point theorem's condition is satisfied for the function \\(g(x_1, x_2) = \\left(\\frac{1}{6}(x_1x_2 + cx_2 - 1), \\frac{1}{6}(x_1^2 - x_2 + 1)\\right)\\) on the set \\(M = [-1,1]\\times [-1,1]\\).",
"role":... | (0, 4) |
[
{
"content": "Let $G$ be a profinite group, which is a compact and totally disconnected topological group, with the property that all of its normal subgroups of finite index are open sets. Does this imply that all of its subgroups of finite index are open sets?",
"role": "user"
}
] | Yes |
[
{
"content": "Given a probability space $(X, A, P)$ and a sequence of events $(A_n)_{n\\in \\mathbb{N}}$ such that $P(A_n) = \\frac{1}{7^n}$ for $n \\in \\mathbb{N}$, determine the probability of the event $\\limsup A_n$.",
"role": "user"
}
] | 0 |
[
{
"content": "For a constant $c$, a sequence $a_n$ is defined by \\( a_n=\\int_c^1 nx^{n-1}\\left(\\ln \\left(\\frac{1}{x}\\right)\\right)^n dx \\) for \\( n=1, 2, 3, \\ldots \\). Find \\( \\lim_{n\\to\\infty} a_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... | Date | Cash Interest | Interest Expense | Premium Amortization | Book Value of Bond
Jan. 1, year 1 | | | | 20,534.20
Dec. 31, year 1 | 1,400.00 | 1,232.05 | 167.95 | 20,366.25
Dec. 31, year 2 | 1,400.00 | 1,221.98 | 178.02 | 20,188.23
Dec. 31 year 3 | 1,400.00 | 1,211.29 | 188.71 | 19,999.52 |
[
{
"content": "Determine the maximal possible dimension of an abelian Lie subalgebra of the Heisenberg Lie algebra \\( \\mathfrak{h}_n \\) of dimension \\( 2n+1 \\). The Heisenberg Lie algebra \\( \\mathfrak{h}_n \\) has a basis \\( x_1, \\dots, x_n, y_1, \\dots, y_n, c \\) with the Lie bracket defined by \\([x_... | n + 1 |
[
{
"content": "In the predator-prey system given by the equations \\( \\frac{dR}{dt}=6R-2RW \\) and \\( \\frac{dW}{dt}=-4W+5RW \\), find the value of \\( RW \\) when the system is at equilibrium with \\( W \\neq 0 \\) and \\( R \\neq 0 \\).",
"role": "user"
}
] | \dfrac{12}{5} |
[
{
"content": "Calculate the greatest common divisor (GCD) of the numbers $5^{2^{303} - 1} - 1$ and $5^{2^{309} - 1} - 1$. Provide your answer as a single number.",
"role": "user"
}
] | 78124 |
[
{
"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... | Frequency of recessive allele (q): 0.5657
Frequency of dominant allele (p): 0.4343
Frequency of dominant homozygous trait (p^2): 0.1886
Frequency of dominant heterozygous trait (2pq): 0.4914
Frequency of recessive homozygous trait (q^2): 0.3200 |
[
{
"content": "Let $a_1, a_2, \\ldots, a_{100}$ be non-zero real numbers such that \\(a_1 + a_2 + \\cdots + a_{100} = 0\\). Determine which of the following statements is true:\n\nA. \\(\\sum_{i=1}^{100} a_i 2^{a_i} \\ge 0\\) and \\(\\sum_{i=1}^{100} a_i 2^{-a_i} \\ge 0\\)\n\nB. \\(\\sum_{i=1}^{100} a_i 2^{a_i} ... | B |
[
{
"content": "Let \\(X\\) be a metric space and let \\(U,V\\subset X\\) be nonempty open and bounded subsets. Determine whether the Hausdorff metric \\( h(U,V)=\\max(\\sup_{x\\in U}\\inf_{y\\in V}d(x,y),\\sup_{x\\in V}\\inf_{y\\in U}d(x,y)) \\) is defined for all possible choices of \\(U\\) and \\(V\\).",
"... | 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... | Cs(s) + 1/2Br_2(l) rarr CsBr(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... | 0.00727 atm |
[
{
"content": "Determine the smallest number of times $n$ a fair coin must be tossed so that the probability of the proportion of heads falling within the interval $[0.49, 0.51]$ is at least 0.95.",
"role": "user"
}
] | 9604 |
[
{
"content": "Let \\((a_k)_{k\\geq 1}\\) be a positive sequence such that the series \\(\\sum_{k\\geq 1} a_k\\) converges to \\(L\\). Define \\(f(n):=\\sum_{k=1}^n \\frac{1}{a_k}\\). Prove or disprove that there exists an \\(n_0\\) such that for all \\(n \\geq n_0\\), \\(f(n) > n^2\\ln(n)\\).",
"role": "use... | True |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty} \\frac{\\sin^2 x}{x^2} \\, dx \\) using the given Fourier transform properties. The function \\( f(x) = \\Theta(x + 1) - \\Theta(x - 1) \\) has a Fourier transform \\( \\widetilde{f}(k) = 2\\frac{\\sin k}{k} \\), where \\( \\Theta(x) \\) is the ... | \pi |
[
{
"content": "In a ring with unity \\((R, +, \\times, 0, 1)\\), determine if a non-zero nilpotent element \\(a \\in R\\setminus \\{0\\}\\) can also be an idempotent element. An element \\(a\\) is nilpotent if there exists a natural number \\(n\\) such that \\(a^n = 0\\), and it is idempotent if \\(a^2 = a\\). P... | No |
[
{
"content": "Let \\( f: \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R} \\) be a function such that for every \\( x, y \\in \\mathbb{R} \\), the functions \\( f(x, \\cdot): \\mathbb{R} \\to \\mathbb{R} \\) and \\( f(\\cdot, y): \\mathbb{R} \\to \\mathbb{R} \\) are continuous. Is it true that \\( f \\) is cont... | No |
[
{
"content": "Given a smooth closed curve $\\mathcal{C}$ in the plane $\\mathbb{R}^2$, with a curve $\\mathcal{C'}$ located at a constant distance $d > 0$ from $\\mathcal{C}$ and orthogonal to it at all points, if $\\mathcal{C}$ has length $L$, find the length $L'$ of $\\mathcal{C'}$. Express your answer in ter... | L + 2\pi d |
[
{
"content": "Determine the number of real solutions to the equation \\( e^{x^2} = ex \\) using calculus.",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the line integral \\( \\int_\\gamma \\frac{x\\,dy-y\\,dx}{x^2+y^2} \\), where \\( \\gamma \\) is the oriented piecewise \\( \\mathcal{C}^1 \\)-arc consisting of the line segment from \\((1,0)\\) to \\((0,1)\\), followed by the line segment from \\((0,1)\\) to \\((-1,0)\\).",
"role": "... | \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... | 340.4 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... | $7.00 |
[
{
"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... | 6.68 uT |
[
{
"content": "Determine whether a compact ordered topological space with the total order topology necessarily has a maximum and a minimum.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral \\( \\int_0^{\\pi/2} \\frac{1}{1+\\tan^{\\sqrt{2}}{x}}\\,\\mathrm{d}x \\).",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Determine the minimum value of \\( A \\) such that the inequality \\( A(1.05^{n}) > \\frac{10(1.05^{n+1}-(n+1)(1.05)+n)}{0.05^{2}} \\) holds for all natural numbers \\( n \\).",
"role": "user"
}
] | 4200 |
[
{
"content": "Find the value of \\(x\\) such that \\(2^{35} \\equiv x \\pmod{561}\\).",
"role": "user"
}
] | 263 |
[
{
"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... | $1,672.80 |
[
{
"content": "Determine the number of zeros of the polynomial \\( g(z) = z^4 + iz^3 + 1 \\) that lie in the region \\( \\{ z \\in \\mathbb{C} : \\text{Re}(z) > 0, \\text{Im}(z) > 0 \\} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine whether there exists a non-decreasing sequence of continuous functions \\( \\{f_n\\} \\) in \\( L_p(0,\\infty) \\) such that \\( f_n \\rightarrow f \\) pointwise for every \\( n \\), where \\( f \\in L_p(0,\\infty) \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral: $$\\int_1^3 \\frac{\\ln(x)}{\\ln(x)+\\ln(4-x)}dx.$$",
"role": "user"
}
] | 1 |
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