prompt list | ground_truth string |
|---|---|
[
{
"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... | 21.49 |
[
{
"content": "Find the smallest positive integer \\( n \\) such that the sum of the squares of its first four divisors equals \\( n \\).",
"role": "user"
}
] | 130 |
[
{
"content": "Real numbers $a$ and $b$ are chosen with $1<a<b$ such that no triangle with positive area has side lengths $1, a,$ and $b$ or $\\tfrac{1}{b}, \\tfrac{1}{a},$ and $1$. What is the smallest possible value of $b$?",
"role": "user"
}
] | \frac{3 + \sqrt{5}}{2} |
[
{
"content": "Determine whether there exist non-homeomorphic closed subsets \\(A\\) and \\(B\\) of \\(\\mathbb{R}^2\\) such that their complements, \\(\\mathbb{R}^2 \\setminus A\\) and \\(\\mathbb{R}^2 \\setminus B\\), are homeomorphic.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the minimum value of $y$ for $n \\geq 2$, where $y$ is given by the expression: $$y = \\frac{1}{2}(e^x - e^{-x}) + \\frac{n}{2}(e^x + e^{-x}).$$",
"role": "user"
}
] | \sqrt{n^2 - 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... | $167,888 |
[
{
"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 m/s^2 |
[
{
"content": "Calculate the limit \\( \\lim _{n \\rightarrow \\infty} \\frac{a_{n}}{n} \\), where \\( a_{n} \\) denotes the number of numbers among \\( 2^{1}, 2^{2}, \\ldots, 2^{n} \\) that begin with the digit 1.",
"role": "user"
}
] | \log_{10} 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... | 13784.25 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... | 84% |
[
{
"content": "Find the coefficient of $x^6$ in the expansion of the expression \\( \\left[(1+x)(1+x^2)^2(1+x^3)^3 \\cdots (1+x^n)^n\\right] \\).",
"role": "user"
}
] | 28 |
[
{
"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... | 1.46e-3 N/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... | 0.02177 |
[
{
"content": "Determine whether the fact that a singleton \\( \\{x\\} \\) is a strong deformation retract of a topological space \\( X \\) implies that \\( X \\) is locally path connected at \\( x \\).",
"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... | 1.979 m/sec |
[
{
"content": "For positive integers $n$ and $k$, consider the function $f: \\Bbb{Z}^+ \\to \\Bbb{Z}^+$ such that:\n\\[f\\left(\\sum_{i=1}^{n} a_{i}^{n}\\right) =\\frac{1}{k}\\sum_{i=1}^{n} \\left(f(a_{i})\\right)^{n}\\]\nFor $n=10$ and $k=10^9$, determine the number of functions $f$ that satisfy this condition.... | 1 |
[
{
"content": "Determine whether there exist six natural numbers such that none of them divides any of the others, but the square of each of them is divisible by each of the others.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit \\( \\lim_{q\\rightarrow 1} \\frac{\\phi^5(q)_{\\infty}}{\\phi(q^5)_{\\infty}} \\), where the Euler function is defined as \\( \\phi(q)=\\prod_{n = 1}^{\\infty}(1-q^{n}) \\) and \\( q=\\exp(2\\pi i \\tau) \\) with \\( |q|<1 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether a space $(X, \\tau)$ with $\\tau$ being the finite-closed topology has the Fixed Point Property, meaning that every continuous mapping $f: (X, \\tau) \\to (X, \\tau)$ has a fixed point.",
"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... | \int_{a(t_0)}^{a(x)}\frac1{a'}[1+w(a^{-1}(a'))]\mathrm da' |
[
{
"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... | 418.92 Hz |
[
{
"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... | 5.01% |
[
{
"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... | $40.197 |
[
{
"content": "Determine which integers between 00 and 99, inclusive, occur as the last two digits in the decimal expansion of infinitely many terms of the sequence \\( \\{a_n\\} \\) defined by \\( a_1 = 3 \\) and \\( a_{i+1} = 3^{a_i} \\) for \\( i > 0 \\).",
"role": "user"
}
] | 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... | $120,954.54 |
[
{
"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... | 194870 |
[
{
"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... | {eq}\mathbf {r}(t) = \displaystyle \left(9-\frac {35}{\sqrt{101}}t-3.5t^2\right)\mathbf{i}\,+\,\left(9-\frac {30}{\sqrt{101}}t-3t^2\right)\mathbf{j}\,+\,\left(4+\frac {20}{\sqrt{101}}t+2t^2\right)\mathbf{k}{/eq} |
[
{
"content": "Find the covariance of $X_1^2$ and $Y_1^2$ for a bivariate normal distribution with zero means, unit variances, and covariance $\\theta$. Express your answer in terms of $\\theta$. \\( \\text{Cov}(X_1^2, Y_1^2) = ? \\)",
"role": "user"
}
] | 2\theta^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... | 7769 |
[
{
"content": "Determine if there exists a sequence of prime numbers of the form $p_1$, $2p_1+1$, $2(2p_1+1)+1$, and so on, that contains more than four terms.",
"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... | Preston Company
Balance Sheet
As of December 31, 19XX
Assets
Current Assets:
Cash $17,000
Accounts Receivable $24,000
Total Current Assets $41,000
Non-Current Assets:
Building $44,000
Land $21,000
Total Non-Current Assets $65,000
Total Assets $106,000
Liabilities and Owner's Equity
Liabilities:
Accounts Payable $11,000... |
[
{
"content": "Calculate $\\sqrt{30p} \\cdot \\sqrt{5p} \\cdot \\sqrt{6p}$ . Express your answer in simplest radical form in terms of $p$.",
"role": "user"
}
] | 30p \sqrt{p} |
[
{
"content": "Let the radius of convergence of the power series \\( \\sum_{n=0}^\\infty a_n z^n \\) be \\( R_1 \\) and that of \\( \\sum_{n=0}^\\infty b_n z^n \\) be \\( R_2 \\). Consider the power series \\( \\sum_{n=0}^\\infty a_n b_n z^n \\). If \\[ \\lim\\sup \\left| \\frac{a_{n+1}b_{n+1}}{a_n b_n} \\right|... | R_1 R_2 \leq 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... | 10.4 eV |
[
{
"content": "Let $(X,m)$ be an outer measure space with finite measure. Suppose that for any $A \\subseteq X$, there exists a Caratheodory measurable subset $B \\supseteq A$ such that $m(A) = m(B)$. If $F \\subseteq X$ satisfies $m(F) + m(X \\setminus F) = m(X)$, is $X \\setminus F$ necessarily Caratheodory me... | Yes |
[
{
"content": "Let \\( f(x) \\) be a four-times differentiable function such that \\( f(2x^2-1)=2xf(x) \\). Find the value of \\( f''''(0) \\).",
"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... | 1/6 L |
[
{
"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... | 2.2e7 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... | 5.87e7 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... | AsCl_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... | 6.6e15 m/s^2 |
[
{
"content": "Solve for $x$: $2^{x-3}=4^{x+1}$",
"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... | $12,709.76 |
[
{
"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... | 18.98 |
[
{
"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... | -12.4 W |
[
{
"content": "Determine whether the product of all primitive $n$th roots of unity is equal to 1 for $n > 2$. Provide a justification for your answer.",
"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... | 3.60e-21 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... | 22.7432503 m/s |
[
{
"content": "Evaluate the limit \\(\\lim_{x \\to 0^+}\\left(\\frac{1}{x} - \\frac{1}{\\arctan(x)}\\right).\\)",
"role": "user"
}
] | 0 |
[
{
"content": "Let $P(x,y)$ be a point on the hyperbola $3x^2-4y^2=36$ that is nearest to the line $3x+2y=1$. Find the value of $\\sqrt{2}(y-x)$.",
"role": "user"
}
] | -9 |
[
{
"content": "Given 100 lines on a plane, determine the maximum number of right triangles that can be formed by selecting any three of these lines.",
"role": "user"
}
] | 62500 |
[
{
"content": "In a metric space, determine whether it is possible for a sequence in a closed and bounded set to lack a convergent subsequence. Answer with 'Yes' or 'No' and justify your response.",
"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... | 42.08 kg/min |
[
{
"content": "On Black Friday, 225 iPhones are available for sale, and there are 100 customers in line. Each customer's desired number of iPhones follows a Poisson distribution with a mean of 2. What is the approximate probability that every customer can purchase the number of iPhones they want?",
"role": "... | 0.964 |
[
{
"content": "Evaluate the limit as \\( n \\) approaches infinity of the series: \\[ \\sum_{r =1}^{n}\\frac{6n}{9n^2-r^2}. \\]",
"role": "user"
}
] | \ln 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... | $40.56 million |
[
{
"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... | 1.06 m |
[
{
"content": "Let $(a_n)$ be a sequence of rational numbers such that the series $\\sum_{n=1}^\\infty v_p(a_n)$ converges for all prime numbers $p$. If $\\prod_{n=1}^\\infty a_n$ converges to a real number $b$, does it follow that $v_p(b) = \\sum_{n=1}^\\infty v_p(a_n)$ for all primes $p$? Answer yes or no and ... | Yes |
[
{
"content": "Evaluate the definite integral \\( \\int_0^\\pi \\frac{\\sin(\\frac{21}{2}x)}{\\sin(\\frac{1}{2}x)} \\, dx \\).",
"role": "user"
}
] | \pi |
[
{
"content": "For a fixed positive integer $k$, let $X_i$ be a sequence of i.i.d. discrete random variables with mean $\\mu$ and variance $\\sigma^2$. Define $Z_n = \\frac{\\sum_{j=1}^{n-k+1} \\prod_{i=j}^{j+k-1} X_i}{n}$. Calculate $\\lim_{n \\to \\infty} n^{\\frac{5}{6}} \\text{Var}(Z_n)$. \\( \\text{Var} \\)... | 0 |
[
{
"content": "Determine the 2019th positive integer \\( n \\) such that \\( \\binom{2n}{n} \\) is not divisible by 5.",
"role": "user"
}
] | 37805 |
[
{
"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... | Calcium gluconate |
[
{
"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... | 466.7 cm^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... | 55,000 |
[
{
"content": "Evaluate the limit: $$ \\lim_{n\\rightarrow+\\infty}n\\int_{0}^{1}\\frac{x^{n}}{x^{n}+x+1}\\,dx $$",
"role": "user"
}
] | \ln \frac{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... | Lease Receivable 150,000
Equipment 150,000
Cash 30,677
Lease Receivable 30,677 |
[
{
"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... | $16,101.58 |
[
{
"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... | 4*10^-2 m^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... | $56,740 |
[
{
"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... | Q = 133.70, P = 0.48 |
[
{
"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... | $70.57 |
[
{
"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... | 1.544 J/K.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... | 0.02 H |
[
{
"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... | \mathbf{\psi(x)=C_2\psi_2(x)+C_4\psi_4(x)+C_6\psi_6(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... | 21000 |
[
{
"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... | 9.93 g |
[
{
"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... | 1.24e6 N/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... | 632 years |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 84.0848 degrees Celsius |
[
{
"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... | Collins Company
Balance Sheet
December 31
Assets:
Cash 59,700
Accounts Receivable 49,000
Prepaid Rent 21,000
Supplies 12,200
Total Assets $141,900
Liabilities:
Accounts Payable $28,100
Unearned Revenue 14,800
Total Liabilities $42,900
Stockholders' Equity:
Capital Stock 51,300
Ending Retained Earnings 47,700
Total 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... | 113,005.45 |
[
{
"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... | 0.77 rev/s |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3.75 N-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... | 18.3% |
[
{
"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... | CH_3CH_2OH_2Cl |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 11.42 |
[
{
"content": "Find \\( \\omega_0 \\) in the set of complex numbers such that the substitution \\( z = \\omega - \\omega_0 \\) transforms the cubic equation \\( z^3 + Az^2 + Bz + C = 0 \\) into \\( \\omega^3 - m\\omega - n = 0 \\). Assume the constants are real numbers.",
"role": "user"
}
] | \dfrac{A}{3} |
[
{
"content": "The line \\(\\frac{x}{4}+\\frac{y}{3}=1\\) intersects the ellipse \\(\\frac{x^{2}}{16}+\\frac{y^{2}}{9}=1\\) at points \\(A\\) and \\(B\\). Determine the number of points \\(P\\) on the ellipse such that the area of \\(\\triangle PAB\\) is equal to 3. Choose the correct answer from the following o... | B |
[
{
"content": "Evaluate the limit of the integral as $n$ approaches infinity:\n\n$$\\lim_{n \\to \\infty} \\int_0^3 \\sin\\left(\\frac{\\pi}{3} \\sin\\left(\\frac{\\pi}{3} \\sin \\left(\\frac{\\pi}{3} \\dots \\sin\\left(\\frac{\\pi}{3} x\\right) \\dots \\right)\\right)\\right)dx$$\n\nwhere the sine function is a... | \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... | c |
[
{
"content": "Consider $5 \\times 5$ matrices $A$ and $B$ such that $AB = 0$ and $\\text{rank}(B) = 3$. Determine if it is possible for the dimension of the solution space of $Ax = 0$ to be greater than 2. If it is possible, provide an example.",
"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... | 1271 g/mol |
[
{
"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... | 333855.25 |
[
{
"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... | 22.3 m |
[
{
"content": "Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1992$ are not factorial tails?",
"role": "user"
}
] | 396 |
[
{
"content": "Evaluate the line integral \\( \\int_{\\gamma} \\alpha \\) where \\( \\alpha = dx_1 - \\frac{1}{x_1} dx_2 \\) and \\( \\gamma(t) = (t^2, t^3) \\) for \\( t \\in [0, 1] \\). Determine if this integral can be evaluated solely based on the endpoint coordinates of \\( \\gamma \\).",
"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... | Assets
Cash$14,200
Accounts receivable$8,400
Supplies inventory $4,500
Equipment$35,500
Accumulated depreciation($14,400)
Total Assets$48,200
Liabilities
Accounts payable$2,100
Equity
Common stock$5,000
Retained earnings$41,100
Total Liabilities and Equity$48,200 |
[
{
"content": "Given the ellipse \\( C_1: \\frac{x^{2}}{4} + y^{2} = 1 \\) and the ellipse \\( C_2: (x - 2)^{2} + 4y^{2} = 1 \\), find the coordinates of the point where the common tangent to both ellipses intersects the x-axis.",
"role": "user"
}
] | (4, 0) |
[
{
"content": "Determine whether the manifold of all $2\\times 2$ Hermitian matrices is diffeomorphic to $\\mathbb{R}^4$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Provide an example of an ideal $A \\subseteq \\Bbb{C}[X]$ such that $A \\subsetneq I(Z(A))$, where $Z(A)$ is the set of zeroes of all polynomials in $A$ and $I(Z(A))$ is the ideal of polynomials that vanish on $Z(A)$. What is one such ideal $A$?",
"role": "user"
}
] | (X^2) |
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