prompt list | ground_truth string |
|---|---|
[
{
"content": "Let $X$ and $Y$ be topological spaces, and let $q: Y \\rightarrow X$ be a covering map. If the induced map $H_1(q, \\mathbb{Q})$ is surjective, is the cardinality of the fiber $|q^{-1}(x)|$ constant for all $x \\in X$?",
"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... | $0.25 |
[
{
"content": "Solve the equation \\(x^2 + \\ln(x) = 0\\) for \\(x\\).",
"role": "user"
}
] | 0.6529 |
[
{
"content": "Find the value of $a$ where the functions $\\sin(x)$ and $\\ln(x)$ intersect, i.e., solve $\\sin(x) = \\ln(x)$ for $x$. This value of $a$ is used in setting up the integral for the volume of the solid created by revolving the region bounded by $\\sin(x)$, $\\ln(x)$, and the $x$-axis about the line... | 2.219 |
[
{
"content": "Find the value of \\( f\\left(\\frac{1}{4}\\right) \\) if \\( f:[0,1] \\to \\mathbb{R} \\) is a continuous function satisfying \\( \\int_{0}^{1} f(x) \\, dx = \\frac{1}{3} + \\int_{0}^{1} \\left(f\\left(x^{2}\\right)\\right)^{2} \\, dx \\).",
"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... | 2756.74 N |
[
{
"content": "Determine the largest constant \\( C \\) such that \\( \\sum_{i=1}^{4}\\left(x_{i}+\\frac{1}{x_{i}}\\right)^{3} \\geq C \\) for all positive real numbers \\( x_{1}, x_{2}, x_{3}, x_{4} \\) satisfying \\( x_{1}^{3}+x_{3}^{3}+3 x_{1} x_{3}=x_{2}+x_{4}=1 \\).",
"role": "user"
}
] | \dfrac{125}{2} |
[
{
"content": "Find all pairs of positive integers \\((a, b)\\) such that \\(\\frac{a b^{2}}{a+b}\\) is a prime number, given that \\(a \\neq b\\).",
"role": "user"
}
] | (6, 2) |
[
{
"content": "Find the area enclosed by the line $x + y = a$ and the curve $\\sqrt{x} + \\sqrt{y} = \\sqrt{a}$, given that $a > 0$. What could be the issue with the initial integration approach that resulted in a negative area?",
"role": "user"
}
] | \dfrac{a^2}{3} |
[
{
"content": "The radius of the inscribed circle is 6 cm. What is the number of centimeters in the length of $\\overline{AB}$? Express your answer in simplest radical form. [asy]\nimport olympiad; import geometry; size(150); defaultpen(linewidth(0.8));\ndraw((sqrt(3),0)--origin--(0,1)--cycle);\nreal r1 = (sqrt(... | AB=12+12\sqrt{3} |
[
{
"content": "Evaluate the definite integral: \\[ \\int _{-1}^{3} \\left[\\arctan \\left(\\dfrac{x}{x^2+1}\\right) + \\arctan \\left(\\dfrac{x^2 + 1}{x}\\right)\\right]dx. \\]",
"role": "user"
}
] | \pi |
[
{
"content": "Determine the number of roots of the polynomial \\( f(x) = x^{12} + x^8 + x^4 + 1 \\) in the finite field \\( \\mathbb{F}_{121} \\).",
"role": "user"
}
] | 4 |
[
{
"content": "A $1 \\times n$ rectangle ($n \\geq 1$) is divided into $n$ unit ($1 \\times 1$) squares. Each square of this rectangle is colored red, blue, or green. Let $f(n)$ be the number of colorings of the rectangle in which there are an even number of red squares. Determine the largest prime factor of $\\... | 37 |
[
{
"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... | \frac{\pi}{75} 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... | 4.899 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... | 5.12e-2 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... | $13,957.20 |
[
{
"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... | $1030.41 |
[
{
"content": "Given a random variable $X$ with an unconditional mean of 0, and the conditions $E[XY]=E[X|Y]=0$ and $E[XZ] \\neq 0$, determine whether it follows that $E[XYZ]=0$. Justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Suppose $f$ is a function defined on a subset $D$. If for any sequence of rational numbers $(a_n)$ and any sequence of irrational numbers $(b_n)$ converging to $0$, both $f(a_n)$ and $f(b_n)$ converge to $l$, does it follow that the limit of $f$ at $0$ is $l$? Answer yes or no and justify your ans... | Yes |
[
{
"content": "Calculate the sum of the squares of the cosines of 180 angles in arithmetic progression, starting from the angle \\( \\alpha \\).",
"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... | $22,522.81 |
[
{
"content": "Given $x_1 > y_1 > 0$, consider the recursively defined sequences $(x_n)_{n=1}^{\\infty}$ and $(y_n)_{n=1}^{\\infty}$ with the following update rules:\n$$x_{n+1} = \\frac{x_n + y_n}{2}$$\n$$y_{n+1} = \\frac{2x_ny_n}{x_n + y_n}$$\nBoth sequences are decreasing, $x_n > y_n$ for all $n \\in \\mathbb{... | \sqrt{x_1 y_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... | 1.02e7 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... | 17.6 kJ |
[
{
"content": "Consider the operator $T:C[0, 1] \\rightarrow C[0, 1]$ defined by $Tf(x) = \\int_{0}^{x} f(t)dt$, where $C[0, 1]$ consists of all real-valued continuous functions on the interval $[0, 1]$ with the norm $||f||_{\\infty} = \\sup_{t \\in [0, 1]} |f(t)|$. Determine which of the following statements ab... | A |
[
{
"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... | 319.5804 N |
[
{
"content": "Find all pairs of prime numbers \\(p\\) and \\(q\\) such that \\(1+\\frac{p^q-q^p}{p+q}\\) is a prime number.",
"role": "user"
}
] | (2, 5) |
[
{
"content": "Find the ordinal $b$ such that $b = \\bigcup 2$, where $2 \\in \\mathbb{N}$. Express your answer as an ordinal.",
"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... | 284.68 degrees |
[
{
"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... | 231.5 |
[
{
"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... | 67.8 N/C |
[
{
"content": "Find an upper bound for the recurrence relation \\( T(x) \\leq 4 T(\\left \\lfloor{\\frac{x}{2}} \\right \\rfloor) + x \\) using the master theorem, where \\( x \\in \\mathbb{N} \\).",
"role": "user"
}
] | O(x^2) |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to\\infty}x\\sin\\left[\\pi\\left(\\sqrt[3]{x^3+4x^2}-\\sqrt[3]{x^3+x^2}\\right)\\right]$$ without using L'Hopital's rule.",
"role": "user"
}
] | \dfrac{5\pi}{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... | Machine
2,100,000
Depreciation Expense (2,000,000/10)200,000
Accumulated Depreciation
600,000
Retained Earnings
1,700,000 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.6 m/s^2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $613.43 |
[
{
"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... | 78.25 s |
[
{
"content": "Is it true that for a non-amenable discrete group \\(\\Gamma\\), there exists a generating set \\(S\\) such that the critical probability for percolation \\(p_c(\\Gamma,S)<\\frac{1}{2}\\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the determinant of the matrix \\(X\\), where \\(X = A + A^2B^2 + A^3 + A^4B^4 + \\cdots\\) for 100 terms. The matrices \\(A\\) and \\(B\\) are given by:\n\\[A = \\begin{bmatrix} 2 & -2 & -4 \\\\ -1 & 3 & 4 \\\\ 1 & -2 & -3 \\end{bmatrix}\\]\n\\[B = \\begin{bmatrix} -4 & -3 & -3 \\\\ 1 & ... | 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... | $77,355 |
[
{
"content": "Determine whether the limit inferior of the sequence $H_n - \\log(n)$, where $H_n$ is the $n$-th harmonic number, is equal to the Euler-Mascheroni constant $\\gamma$. Provide a justification for your answer.",
"role": "user"
}
] | \gamma |
[
{
"content": "Find the area of a right triangle if the ratio of its legs is 1.05 and the difference between the radii of the inscribed and circumscribed circles is 17.",
"role": "user"
}
] | 840 |
[
{
"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_s} = {\mu_{s\;max}} |
[
{
"content": "Compute the limit: \\[ \\lim_{n\\rightarrow \\infty} \\frac{1+\\sqrt{2}+\\sqrt[3]{3}+\\cdots+\\sqrt[n]{n}}{n} \\]",
"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... | 16.05 g Fe |
[
{
"content": "In a 3x3x3 tic-tac-toe game with three players A, B, and C taking turns in order, how many unique board states are possible after 6 moves? Assume that each player places their mark in an empty cell during their turn.",
"role": "user"
}
] | 26640900 |
[
{
"content": "Compute the degree of the field extension \\( \\mathbb{Q}(\\sqrt{2}, \\sqrt[3]{2}, w) \\) over \\( \\mathbb{Q} \\), where \\( w = -\\frac{1}{2} + i\\frac{\\sqrt{3}}{2} \\).",
"role": "user"
}
] | 12 |
[
{
"content": "Consider a $2015 \\times 2015$ board colored in a checkerboard pattern with black corner cells. A piece starts on one black cell, and another black cell is marked. Can the piece tour all cells of the board, visiting each cell exactly once, and end on the marked cell? (Adjacent cells share a common... | 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... | $1,020.03 |
[
{
"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_2H_3O_2^-(aq) + H^+(aq) = HC_2H_3O_2(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... | -0.60 |
[
{
"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... | 0.91 |
[
{
"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... | $3.61 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\rightarrow \\infty }\\sum_{k=n}^{5n}\\binom{k-1}{n-1}\\left( \\frac{1}{5} \\right )^n\\left( \\frac{4}{5} \\right )^{k-n} \\]",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Find the positive integer \\( n \\) such that \\[ \\lfloor\\log_2{1}\\rfloor+\\lfloor\\log_2{2}\\rfloor+\\lfloor\\log_2{3}\\rfloor+\\cdots+\\lfloor\\log_2{n}\\rfloor=1994. \\]",
"role": "user"
}
] | 312 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\rightarrow 1^+} \\left (1 - \\frac{1}{x}\\right)^x \\left( \\log\\left(1 - \\frac{1}{x}\\right) + \\frac{1}{x - 1}\\right) \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Solve the equation \\(\\frac{x}{2+\\frac{x}{2+\\frac{x}{2+\\ddots \\frac{x}{2+\\frac{x}{1+\\sqrt{1+x}}}}}}=1\\) where there are 1985 terms of 2 in the expression.",
"role": "user"
}
] | 3 |
[
{
"content": "Let $A$ and $B$ be two metric spaces, and let $f: A \\to B$ be an isometry. If $C \\subset A$ is dense in $A$, is $f(C)$ dense in $B$? 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... | U/k |
[
{
"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... | $O(n\sqrt{n} \cdot \log n)$ |
[
{
"content": "Determine whether the sequence \\( a_n = \\frac{\\sqrt{b^n}}{\\sqrt{n!}} \\) converges or diverges, where \\( b > 1 \\) is a real number.",
"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... | -381 kJ |
[
{
"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... | 1601.3 kJ |
[
{
"content": "You have collected sand from the beach 100 times, with the weight of sand collected on each trip following a uniform distribution $\\mathrm{U}[10-\\sqrt{48}/2, 10+\\sqrt{48}/2]$. Assuming each collection is independent, estimate the probability that the total weight of sand collected is between 98... | 0.7745 |
[
{
"content": "Find the value of the infinite series \\( \\sum_{k=1}^{\\infty}\\frac{k^n}{k!} \\) for a given integer \\( n \\). For example, when \\( n = 6 \\), the sum is \\( 203e \\).",
"role": "user"
}
] | B_n e |
[
{
"content": "Find the Bayes estimator for the parameter \\( \\theta \\) given the loss distribution \\( l(a,\\theta)=\\frac{(a-\\theta)^2}{\\theta(1-\\theta)} \\), a prior distribution \\( \\pi(\\theta) \\) that is Uniform from 0 to 1, and a likelihood function \\( f(x|\\theta) \\) for a sequence of \\( n \\) ... | \dfrac{X}{n} |
[
{
"content": "Let $K(x)$ be a bounded and continuous function, and let $\\phi_n \\to \\delta$ weakly in the space of tempered distributions, where $\\phi_n$ is a Schwartz function. Determine whether the following limit holds:\n\\[\n\\lim_{n \\to \\infty} \\int_{\\mathbb{R}} \\phi_n(x) (K(x) - K(0)) \\, dx = 0.\... | 0 |
[
{
"content": "Find the partial limit of the function \\( f(x,y) = e^{\\frac{2x}{\\sqrt{x^2+y}}} \\) at the point \\((0,0)\\) by the set \\( A = \\{(x,x): x > 0\\} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "The ratio of the areas of two squares is $\\frac{32}{63}$. After rationalizing the denominator, the ratio of their side lengths can be expressed in the simplified form $\\frac{a\\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. What is the value of the sum $a+b+c$?",
"role": "user"
}
] | 39 |
[
{
"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... | 10.0 |
[
{
"content": "Determine whether the limit \\( \\lim_{(x,y)\\to (0,0)} \\frac {x^3y^2}{x^4+y^6} \\) exists.",
"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... | 40.41 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... | 1.02e23 formula units |
[
{
"content": "Determine the set of values $L \\in [0,1]$ for which there exists a subsequence of the sequence $J = \\left\\{\\frac{1}{2}, \\frac{1}{3}, \\frac{2}{3}, \\frac{1}{4}, \\frac{2}{4}, \\frac{3}{4}, \\frac{1}{5}, \\frac{2}{5}, \\ldots\\right\\}$ that converges to $L$. Provide your answer as a set of va... | [0,1] |
[
{
"content": "Simplify the expression: $$\\lim_{x\\to0}\\left(\\prod^{\\frac{1}{x}-1}_{i=1}\\frac{1}{\\sec\\frac{xi\\pi}{2}}\\right)^x$$",
"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... | Jan: $9,240
Feb: $9,840
Mar: $13,920
Apr: $10,800
May: $9,720
Jun: $15,270 |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty} \\int_1^\\infty \\frac{2n}{e^{2x}+n^2x}dx$$",
"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... | 450.76 K |
[
{
"content": "Determine the Galois group $Gal(F/\\mathbb{Q})$, where $F$ is the splitting field of the polynomial $X^4 - 2X^3 - 8X - 3$ over $\\mathbb{Q}$.",
"role": "user"
}
] | S_3 |
[
{
"content": "Determine the values of \\( \\alpha > 0 \\) for which the series \\[ \\sum_{n_{1}=1}^{\\infty}\\cdots\\sum_{n_{k}=1}^{\\infty}\\frac{1}{\\left(n_{1}^{2}+\\dots+n_{k}^{2}\\right)^{\\alpha/k}} \\] converges, where \\( k > 0 \\) is an integer.",
"role": "user"
}
] | \alpha > \dfrac{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... | 2.1e-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... | The NPV of bond refunding of Bond A and Bond B is $2,587,200 and $4,281,942, respectively. |
[
{
"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... | 4200/m m/s |
[
{
"content": "Calculate the limit of the function \\( \\frac{(1.000002)^n}{n^{1.000002}/\\log n} \\) as \\( n \\to \\infty \\).",
"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... | 0.182 |
[
{
"content": "The parabolas defined by the equations $y=x^2+4x+6$ and $y=\\frac{1}{2}x^2+x+6$ intersect at points $(a,b)$ and $(c,d)$, where $c\\ge a$. What is $c-a$?",
"role": "user"
}
] | 6 |
[
{
"content": "Determine whether there exist prime numbers that can be expressed in the form $p \\equiv a \\pmod{n}$ for any positive integer $n$ and non-negative integer $a < n$, where $(a, n) = 1$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. The aquarium is filled with water to a depth of 37 cm. A rock with volume $1000 \\text{cm}^3$ is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise? Exp... | 0.25\,\text{ cm} |
[
{
"content": "Let \\(A\\) be a skew-symmetric matrix such that \\(A^TA = I\\). Determine the value of \\(A^{4n-1}\\) for \\(n \\in \\mathbb{N}\\). Choose from the following options:\n\n(A) \\(-A^T\\) \n(B) \\(I\\) \n(C) \\(-I\\) \n(D) \\(A^T\\)",
"role": "user"
}
] | D |
[
{
"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... | 2965 N |
[
{
"content": "Determine the maximum possible error in approximating \\( f(x) = x\\ln(x) \\) using the third-degree Taylor polynomial about \\( x=1 \\) on the interval \\([2,4]\\).",
"role": "user"
}
] | \dfrac{27}{4} |
[
{
"content": "Determine whether the polynomial \\(15x^5-25x^3+10=0\\) is solvable by radicals over \\(\\mathbb{Q}\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Are the braid groups $B_n$ conjugacy separable? A group $G$ is called conjugacy separable if for every two elements $x, y \\in G$ that are not conjugate in $G$, there exists a finite index normal subgroup $N$ such that $xN$ and $yN$ are not conjugate in $G/N$. Determine if this property holds for ... | Yes |
[
{
"content": "Given that $x$, $y$, and $z$ satisfy the following equations:\n\n$$x+y+z=1$$\n\n$$x^2+y^2+z^2=2$$\n\n$$x^3+y^3+z^3=3$$\n\nFind the value of $x^4+y^4+z^4$. \\(\\boxed{\\ }\\)",
"role": "user"
}
] | \dfrac{25}{6} |
[
{
"content": "Evaluate the integral \\( I = \\int_0^1 \\frac{dx}{\\sqrt{(1-x^2)(1-x^2k^2)}} \\) for \\( 0 < k < 1 \\).",
"role": "user"
}
] | K(k) |
[
{
"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... | 4.516 eV |
[
{
"content": "Let $f(x)$ be a function such that \\( f(x) + 2f\\left(\\frac{x+2010}{x-1}\\right) = 4020 - x \\) for all \\( x \\neq 1 \\). Find the value of \\( f(2012) \\).",
"role": "user"
}
] | \dfrac{6028}{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... | -0.32 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to-\\infty}\\left(x(\\log(1+\\sqrt{1+x^2})-\\log(-x))\\right)$$",
"role": "user"
}
] | -1 |
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