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... | 35.10 Ns |
[
{
"content": "Calculate the value of \\( \\sqrt{M - 2} + 1 \\), where \\( M = \\lim\\limits_{x\\to 0 } \\left[\\dfrac{1 - f(x)^3}{5 \\tan^2x}\\right] \\), \\( f(x) = \\cos(2x) \\cos(4x) \\cos(6x) \\cos(8x) \\cos(10x) \\), and \\( M \\) is finite.",
"role": "user"
}
] | 9 |
[
{
"content": "Given a $4\\times 4$ real matrix $T$ such that $T^4=0$, determine which of the following sequences $k_1,\\ k_2,\\ k_3,\\ k_4$ is NOT a possible combination for the nullity of $T^i$:\n\n1. $k_1 = 3,\\ k_2 = 4,\\ k_3 = 4,\\ k_4 = 4$\n2. $k_1 = 1,\\ k_2 = 3,\\ k_3 = 4,\\ k_4 = 4$\n3. $k_1 = 2,\\ k_2 ... | 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... | 75 m |
[
{
"content": "Given that \\( x, y, z \\) are positive real numbers such that \\( x + y + z = 1 \\), find the minimum value of the function \\( f(x, y, z) = \\frac{3x^2 - x}{1 + x^2} + \\frac{3y^2 - y}{1 + y^2} + \\frac{3z^2 - z}{1 + z^2} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Compute $\\gcd(6^210^2,15^4)$.",
"role": "user"
}
] | 225 |
[
{
"content": "Let \\( f : \\mathbb{R} \\to \\mathbb{R} \\) be a continuous function such that \\( x \\leq f(x) \\leq x+1 \\) for all \\( x \\in \\mathbb{R} \\). Determine the set \\( f(\\mathbb{R}) \\).",
"role": "user"
}
] | \mathbb{R} |
[
{
"content": "Find the last three digits of the product of all odd numbers from 1 to 2005.",
"role": "user"
}
] | 375 |
[
{
"content": "Determine whether the following statement is correct: \"The limit superior of $x_{n}$ (if finite) is the smallest real number $b$ such that, for any positive real number $\\varepsilon$, there exists a natural number $N$ such that $x_n < b + \\varepsilon$ for all $n > N$.\" Provide a justification ... | Correct |
[
{
"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... | 2.3e8 |
[
{
"content": "Find the eigenvalues and their corresponding eigenvectors of the linear operator $T: P(\\mathbb{R}) \\rightarrow P(\\mathbb{R})$ defined by $Tp = p'$, where $P(\\mathbb{R})$ is the space of polynomials with real coefficients.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\int_{B(0,r)} \\frac{1}{|x|^{n-1}} \\, dx \\) and determine if it can be expressed as a constant times the radius \\( r \\).",
"role": "user"
}
] | \frac{2\pi^{n/2}}{\Gamma(n/2)} 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... | 353 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... | 176.773 |
[
{
"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. Add dilute HCl to precipitate Ag+ as AgCl.
2. Add H2S, ammonia, NaOH, and ammonium sulfide to precipitate Cr3+ as Cr(OH)3.
3. Add ammonium carbonate to precipitate Ba2+ as BaCO3. |
[
{
"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... | 4.58e-20 J |
[
{
"content": "In a regular pentagon \\(ABCDE\\), point \\(K\\) is on side \\(AE\\) and point \\(L\\) is on side \\(CD\\). Given that \\(\\angle LAE + \\angle KCD = 108^\\circ\\) and the ratio \\(AK: KE = 3:7\\), determine the ratio \\(CL: AB\\).",
"role": "user"
}
] | \dfrac{7}{10} |
[
{
"content": "Does the operation \\(\\delta_{G}\\) on the underlying set of a group \\(G\\) uniquely determine the group structure of \\(G\\) if \\(G\\) is an elementary abelian 2-group?",
"role": "user"
}
] | Yes |
[
{
"content": "How many $4 \\times 4$ matrices can be formed using only $1$ and $-1$ as elements such that the sum of all elements in any row and column is $0$?",
"role": "user"
}
] | 90 |
[
{
"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... | 248.80 s or 4.15 minutes |
[
{
"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... | $781,584.15 |
[
{
"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... | $13,635.68 |
[
{
"content": "If $X$ is a compact Hausdorff space, is every isomorphism from ${\\mathcal C}(X)$ onto ${\\mathcal C}(X)$ continuous?",
"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... | 22 J/K.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... | 7.42kJ/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... | The monthly operating income will increase by $2,100 if Dong Fang decides to outsource. |
[
{
"content": "Let $f$ be a continuous function on $\\Bbb{R}$. If $\\lim_{n \\to \\infty } \\int _0 ^ 1 f(nx) \\, dx = a$, does it follow that $\\lim_{n \\to \\infty } \\int _0 ^ 1 f(n+x) \\, dx = a$? Answer yes or no and justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "What is the greatest three-digit number \"abc'' such that $4,a,b$ forms a geometric sequence and $b,c,5$ forms an arithmetic sequence?",
"role": "user"
}
] | 697 |
[
{
"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_6H_4OCOCH_3COOH_(aq) + OH^(-)(aq)-> C_6H_4OCOCH_3COO^(-)(aq) + H_2O_(l) |
[
{
"content": "Do all matrices with a common eigenvector necessarily commute? 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... | 8.09% |
[
{
"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... | 34.53*10^3 W |
[
{
"content": "Evaluate the summation \\( S = \\sum_{n=1}^{\\infty} n^2 e^{in\\theta} \\) where \\( \\theta = \\frac{\\pi}{3} \\).",
"role": "user"
}
] | -i\sqrt{3} |
[
{
"content": "Let \\( f(x)=e^{x+1}-1 \\) and \\( g(x)=100|f(x)|-\\sum_{k=1}^n|f(x^k)| \\), where \\( n \\in \\mathbb{N} \\). It is given that \\( g(x) \\) is a differentiable function over the complete real numbers. Find the sum of all possible values of \\( n \\).",
"role": "user"
}
] | 39 |
[
{
"content": "Evaluate the limit: \\( \\lim_{x \\to 0} x^{x^x} \\)",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the axiom of regularity can fail in set theory.",
"role": "user"
}
] | Yes |
[
{
"content": "A calculator displays the number 0 initially and has digits from 0 to 9, along with two operation signs. When a sequence of keys is pressed, the calculator performs operations in the order of the keystrokes, ignoring repeated operation signs in a row except for the last one. If a random sequence o... | \dfrac{11}{23} |
[
{
"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... | $218,427 |
[
{
"content": "Let $V$ be a finite-dimensional vector space. Let $f=(f_1,f_2,\\dots)\\colon V\\to \\mathbb{R}^{\\mathbb{N}}$ be an injective linear map. Do the functions $f_1,f_2,\\dots$ span the dual space $V^*$? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the limit of the sequence \\((x_n)_{n\\geq 1}\\) defined by \\[ x_n = \\left(1 + \\frac{1}{3} + \\frac{1}{5} + \\cdots + \\frac{1}{2n-1}\\right) - \\frac{1}{2} \\left(1 + \\frac{1}{2} + \\frac{1}{3} + \\cdots + \\frac{1}{n}\\right). \\] Find \\( \\lim_{n\\to\\infty} x_n \\).",
"role"... | \ln 2 |
[
{
"content": "Compute \\((101^{24} \\times 97^{25}) \\mod 493\\).",
"role": "user"
}
] | 345 |
[
{
"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... | $4010 F |
[
{
"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... | 515.2 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{z \\to 0} \\frac{z^5}{1+z-e^z} \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Given that we have \\( n \\) variables, each with a pairwise correlation of 1, and the variance of their sum is 1, find the lower bound on the sum of the individual variances.",
"role": "user"
}
] | \dfrac{1}{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... | 3.35kJkg^-1K^-1 |
[
{
"content": "Consider a $22 \\times 22$ board filled with numbers from 1 to $22^2$. Can you always find two adjacent cells, either sharing a side or a vertex, such that the sum of the numbers in these cells is divisible by 4?",
"role": "user"
}
] | Yes |
[
{
"content": "Given the function $f(x) = \\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\sum_{k=1}^n(b_k)$, where $\\sum_{k=1}^n(b_k)$ denotes the count of $1$s in the binary representation of a dyadic number $x$, suppose $f(x) = \\frac{1}{2}$. Is the sequence $F_n = \\sum_{k=1}^n(1-2b_k) = \\sum_{k=1}^n (-1)^{b_k}$ ... | No |
[
{
"content": "Three vertices of a cube are $P=(7,12,10)$, $Q=(8,8,1)$, and $R=(11,3,9)$. What is the surface area of the cube?",
"role": "user"
}
] | 294 |
[
{
"content": "Evaluate the series: $$\\sum_{n=0}^{\\infty} \\frac{n^{2}-5n+2}{n!}$$",
"role": "user"
}
] | -e |
[
{
"content": "Given three sets $t_i$, $t_j$, and $t_k$, suppose that for any element $x$, if $x$ is in the symmetric difference of $t_i$ and $t_j$, then $x$ is not in the symmetric difference of $t_i$ and $t_k$. Similarly, if $x$ is in the symmetric difference of $t_i$ and $t_k$, then $x$ is not in the symmetri... | 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... | 22.88 ft/s |
[
{
"content": "Find the exponential generating function for the sequence \\( t_n \\), where \\( t_n \\) represents the number of ways a teacher can divide \\( n \\) students into groups, assigning one student as president and another as vice president within each group.",
"role": "user"
}
] | e^{x^2 e^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 rad/s |
[
{
"content": "Determine all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that for all \\( x, y \\in \\mathbb{R} \\), the equation \\( f(x - f(x - y)) + x = f(x + y) \\) holds.",
"role": "user"
}
] | f(x) = 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... | 1.343*10^8 Newton |
[
{
"content": "Let $X$ and $Y$ be topological spaces, and let $f: X \\rightarrow Y$ be a function. Suppose $V \\subset f(X)$ and $f^{-1}(V) = U$. Is it true that $V = f(U)$? Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the norm of the linear functional \\( \\phi(x_n) = \\sum_{n\\ge 1} (-2)^{1-n}x_n \\), where \\( \\phi: c_0 \\to \\mathbb{R} \\) and \\( c_0 \\) is the set of sequences converging to 0.",
"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... | 3.37e-3 s^{-1} |
[
{
"content": "Determine whether there exists a real number $a > 1$ such that the sequence $\\lfloor a^n \\rfloor$ consists entirely of composite numbers.",
"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... | 88.5 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... | 2.14e13 electrons must be added to the bead. |
[
{
"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... | $121378.60 |
[
{
"content": "What is the remainder when $5^{207}$ is divided by 7?",
"role": "user"
}
] | 6 |
[
{
"content": "Determine the Zariski closure of the set \\( X = \\{(x, \\sin(x)): x \\in \\mathbb{A}^{1}\\} \\) in \\( \\mathbb{A}^{2} \\).",
"role": "user"
}
] | \mathbb{A}^{2} |
[
{
"content": "Determine all real values of \\(\\lambda\\) such that the inequality \\(x + y + z \\leq \\frac{3}{2}\\) holds for all non-negative real numbers \\(x, y, z\\) satisfying \\(x^2 + y^2 + z^2 + 2\\lambda xyz = 1\\).",
"role": "user"
}
] | [1, \infty) |
[
{
"content": "Find the Taylor series expansion of the operator \\( D = e^{bx \\frac{d}{dy}} \\) when it operates on an infinitely differentiable function \\( g(y) \\).",
"role": "user"
}
] | \sum_{n=0}^{\infty} \frac{(bx)^n}{n!} g^{(n)}(y) |
[
{
"content": "Determine the greatest common divisor of all numbers of the form \\(2020^n + 1\\) for positive integers \\(n\\), that are less than \\(2020^{2020} + 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... | d) 36% |
[
{
"content": "Determine the exact value of the norm \\( \\Vert f \\Vert \\) for the functional \\( f \\) defined on the normed space \\( C[-1,1] \\) by:\n\\[ f(x) = \\int_{-1}^0 x(t) \\, \\mathrm{d}t - \\int_0^1 x(t) \\, \\mathrm{d}t \\]\nfor all \\( x \\in C[-1,1] \\), where \\( C[-1,1] \\) is the space of all... | 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... | a) methylidenecyclohexane
b) 1-methylcyclohex-1-ene |
[
{
"content": "Given that $f$ is a well-defined piecewise differentiable function on $\\mathbb{R}$ and as $x \\to \\infty$, $$f(x) = x\\log x - x + O(\\log x),$$ determine if $f(x)$ is a monotone increasing function for sufficiently large $x$.",
"role": "user"
}
] | Yes |
[
{
"content": "Given the polynomials $p(x)$ and $q(x)$ with $p(2)=5$, $p(3)=12$, and the relationship $p(q(x))=p(x)q(x)-p(x)$, determine the value of $q(10)$. \\( \\boxed{} \\)",
"role": "user"
}
] | 114 |
[
{
"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... | -24 |
[
{
"content": "How many two-digit prime numbers can be formed by choosing two different digits from the set $\\{2, 7, 8, 9\\}$ to be used as the tens digit and units digit?",
"role": "user"
}
] | 4 |
[
{
"content": "What is the derivative of the fractional part of $x$, denoted as $\\{x\\}$?",
"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... | 892.71 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... | 10694444.44 |
[
{
"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... | 6.41e-15 m |
[
{
"content": "Find the limit: \\[ \\lim_{n\\to +\\infty}\\left[\\frac{\\frac{a_n}{s_n}}{-\\ln(1-\\frac{a_n}{s_n})}\\right] \\] where \\(a_n=\\sqrt{n}\\) and \\(s_n=a_1+a_2+\\ldots+a_n\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Let $a,$ $b,$ $c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7 = 0.$ Given that $P(x)$ is a cubic polynomial such that $P(a) = b + c,$ $P(b) = a + c,$ $P(c) = a + b,$ and $P(a + b + c) = -16,$ find $P(x).$",
"role": "user"
}
] | 2x^3 + 6x^2 + 9x + 11 |
[
{
"content": "Find the remainder when $3^{3^{27}} - 1$ is divided by 23, given that $3^{27} \\equiv 6 \\pmod{22}$. Use this hint to simplify the computation.",
"role": "user"
}
] | 15 |
[
{
"content": "Find \\( \\limsup_{n\\to\\infty}\\left(\\frac{2\\cdot5\\cdot8\\cdot\\cdots\\cdot(3n-4)}{3^nn!}\\right)^{1/n} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Let $D=\\{z\\in \\mathbb{C} :|z| < 1\\}$ be the open unit disc in the complex plane, and let $f:D \\rightarrow D$ be a non-constant holomorphic function. Suppose that $f(1/3)=0$. Determine which of the following are possible values of $|f(1/2)|$: \\(\\frac{4}{5}\\), \\(\\frac{3}{5}\\), \\(\\frac{2... | \dfrac{1}{5} |
[
{
"content": "Let $f(x)$ be a polynomial with integer coefficients. If $f(r)$ is divisible by a prime number $p$ for some integer $r$, determine if there exists a polynomial $g(x)$ such that $f(x) - (x-r)g(x)$ is divisible by $p$.",
"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... | 6.6 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... | 496.31 Btu/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... | 2.2e-3M |
[
{
"content": "Evaluate the contour integral \\( \\oint_C (4z - 1)\\,dz \\) along the quarter circle \\(|z| = 1\\) from the point \\((0,-1)\\) to \\((1,0)\\) counterclockwise.",
"role": "user"
}
] | 3 - i |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 17.7 m/s |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $18,432 |
[
{
"content": "If $V$ is an irreducible representation of a semisimple Lie algebra with highest weight $\\lambda$, what is the highest weight of the dual representation $V^*$?",
"role": "user"
}
] | -w_0(\lambda) |
[
{
"content": "Determine the exponential generating function (EGF) for the sequence \\( \\{0, 1, 2a, 3a^2, 4a^3, \\ldots\\} \\).",
"role": "user"
}
] | x e^{a x} |
[
{
"content": "Consider a circle with radius 4 and center $O_1$, a circle with radius 2 and center $O_2$ that lies on the circumference of circle $O_1$, and a circle with radius 1 and center $O_3$ that lies on the circumference of circle $O_2$. The centers $O_1$, $O_2$, and $O_3$ are collinear in that order. Let... | \sqrt{6} |
[
{
"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... | $2,700 |
[
{
"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... | 4675.761 |
[
{
"content": "Evaluate the limit \\( \\lim_{x \\to \\infty} F(x) \\) where \\( F(x) = \\int_1^x \\frac{\\cos t + \\cosh t}{e^t} \\, dt \\).",
"role": "user"
}
] | \infty |
[
{
"content": "Let \\( a = (123)(456) \\) in \\( S_{10} \\). Determine the highest possible order of a permutation \\( b \\) in \\( S_{10} \\) such that \\( b^k = a \\) for some integer \\( k \\).",
"role": "user"
}
] | 12 |
[
{
"content": "Let \\( f \\in C^{\\infty}(\\mathbb{R}^2) \\) be a smooth function with compact support. Suppose \\( \\text{supp}(f) \\subseteq [a,b] \\times [c,d] \\) and \\( \\varepsilon > 0 \\). Can \\( f(x,y) \\) be approximated by a sum of the form \\( \\sum_{i=1}^m g_i(x) h_i(y) \\), where \\( g_i, h_i \\in... | Yes |
[
{
"content": "A police car is moving down a narrow street at 90 km/h, with a blue light on top rotating counterclockwise at one rotation per second. The buildings are 3 meters from the car on the right. Calculate the speed of the light beam on the wall when it is 6 meters ahead of its source.",
"role": "use... | 30\pi - 25 |
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