prompt list | ground_truth string |
|---|---|
[
{
"content": "Given a natural number $N_0$ and a sequence of complex numbers $\\{F_N\\}_{N \\in \\mathbb{N}}$ that satisfies the following conditions: \\( \\lim_{N \\rightarrow \\infty} |F_N|^{1/N}=0 \\) and \\( |F_N|\\leq N\\sum_{k=N+1}^{\\infty} |F_k| \\) for all $N \\geq N_0$. Prove or disprove that there ex... | True |
[
{
"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... | 64 ft/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... | 1,381,124.20 |
[
{
"content": "Consider the function:\n\n$$G(x)=\\begin{cases}\\displaystyle\\int_{2x}^{x}\\cos t^3\\,\\mathrm dt,&x\\neq0\\\\0,&x=0\\end{cases}$$\n\nIs $G$ continuous and differentiable at $x=0$? If it is differentiable, find $G'(0)$. Provide your answer with justification.",
"role": "user"
}
] | -1 |
[
{
"content": "If $\\lim \\limits_{n \\to \\infty} a_n = \\infty$, does $\\lim \\limits_{n \\to \\infty} \\sqrt[n]{a_n}$ necessarily converge? Answer yes or no and justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the power series \\( \\sum_{n = 1}^{\\infty}\\dfrac{n}{9^{n}} \\) without using the formula for infinite geometric series.",
"role": "user"
}
] | \dfrac{9}{64} |
[
{
"content": "Square $ABCD$ is inscribed in the region bound by the parabola $y = x^2 - 8x + 12$ and the $x$-axis, as shown below. Find the area of square $ABCD.$\n\n[asy]\nunitsize(0.8 cm);\n\nreal parab (real x) {\n return(x^2 - 8*x + 12);\n}\n\npair A, B, C, D;\nreal x = -1 + sqrt(5);\n\nA = (4 - x,0);\nB ... | 24 - 8 \sqrt{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... | 214 m/s |
[
{
"content": "Is it possible to construct an infinite group where every element has order 3?",
"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... | Cutter A |
[
{
"content": "Evaluate the limit: $$\\lim\\limits_{x\\to 0} \\frac{x^3\\sin(x)}{[\\ln(1+x)-x]^2}$$ without using L'Hopital's rule or series expansions.",
"role": "user"
}
] | 4 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -2.80% |
[
{
"content": "Determine the values of \\( a \\) for which the integral \\( \\int _0^{\\infty }\\:\\frac{\\sin\\left(x\\right)}{\\left(1+x\\right)^a} dx \\) converges.",
"role": "user"
}
] | a > 0 |
[
{
"content": "Determine whether all Steiner Triple Systems (STS) are regular hypergraphs, meaning that every vertex has the same degree. Answer with 'Yes' or 'No'.",
"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... | 8850 |
[
{
"content": "Does $SO(7)$ have a universal covering by $SU(6)$?",
"role": "user"
}
] | No |
[
{
"content": "Given the matrix \\( A = \\begin{bmatrix} 2 & -2 & -4 \\\\ -1 & 3 & 4 \\\\ 1 & -2 & -3 \\end{bmatrix} \\), where \\( A = B + C \\), \\( B \\) is symmetric \\( (B = B^T) \\), and \\( C \\) is skew-symmetric \\( (C = -C^T) \\). If \\( (CB)^T = D \\) and \\( D = [ d_{ij} ]_{3 \\times 3} \\), find the... | 9 |
[
{
"content": "Let $M$ be an oriented, compact, connected $n$-dimensional smooth manifold with boundary. Determine whether the $n$-th singular homology of $M$, denoted by $H_n(M)$, is always zero. Provide a justification for your answer.",
"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... | 60263.01 |
[
{
"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... | 35.3 Vm |
[
{
"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... | 8.686 |
[
{
"content": "Determine the order of the quotient ring $\\mathbb{Z}[i]/\\langle a+ib \\rangle$ in terms of $a$ and $b$.",
"role": "user"
}
] | a^2 + b^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... | 4.86 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... | 2509.6 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 cm |
[
{
"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... | spring |
[
{
"content": "Four distinct integers $a$, $b$, $c$ and $d$ have the property that when added in pairs, the sums 16, 19, 20, 21, 22, and 25 are obtained. What are the four integers in increasing order? (place a comma and then a space between each integer)",
"role": "user"
}
] | 7,9,12,13 |
[
{
"content": "Find the exact value of the limit as \\( n \\) approaches infinity:\n\n\\[ \\lim_{n\\to\\infty}\\prod_{k=1}^{\\lceil\\alpha\\sqrt{n}\\rceil-1}\\frac{2n-2k}{2n-k} \\]\n\nfor any \\( \\alpha>0 \\).",
"role": "user"
}
] | e^{-\alpha^2/4} |
[
{
"content": "Let $k$ be a field that is not algebraically closed, and let $n > 1$ be an integer. Does there exist a polynomial $f \\in k[X_1, \\ldots, X_n]$ such that the zero set $Z(f) = \\{(a_1, \\ldots, a_n) \\in k^n : f(a_1, \\ldots, a_n) = 0\\} = \\{0\\}$?",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the integral of the scalar field \\( \\phi(x,y,z) = xyz \\) over the surface of the unit sphere. Express the integral in spherical coordinates and evaluate it: \n\\[ I = \\int_{S} \\phi \\, dA \\]\nwhere \\( S \\) is the surface of the unit sphere.",
"role": "user"
}
] | 0 |
[
{
"content": "Find a curve passing through the points (0,0) and (1,1) that is an extremal of the functional \\( J(x,y,y') = \\int_{0}^{1} \\left[ (y')^2 + 12x y \\right] \\, dx \\).",
"role": "user"
}
] | y = x^3 |
[
{
"content": "Does there exist a closed, smooth planar curve and a specific direction such that, when a finite opening is cut in the curve and a photon is launched through this opening into the interior along the specified direction, the photon will perpetually bounce off the curve's boundary, remaining confine... | Yes |
[
{
"content": "Find the closed-form solution for the following limit involving a sum:\n\\[ \\lim_{n\\to\\infty}\\sum_{k=0}^n\\binom{n}{k}\\frac{3k}{2^n(n+3k)} \\]",
"role": "user"
}
] | \dfrac{3}{5} |
[
{
"content": "Given an arithmetic sequence $\\{a_{n}\\}$, the sequence $\\{b_{n}\\}$ is defined by: $b_{1}=a_{1}$, $b_{2}=a_{2} + a_{3}$, $b_{3}=a_{4} + a_{5} + a_{6}$, etc. If $\\lim _{n \\rightarrow \\infty} \\frac{b_{n}}{n^{3}} = 2$, determine the common difference $d$ of the sequence $\\{a_{n}\\}$. Choose f... | 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... | Tt |
[
{
"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... | asset |
[
{
"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... | 0.73m |
[
{
"content": "Evaluate the integral \\( \\int_{a}^{b}\\left(\\left\\lfloor x+\\frac{1}{2}\\right\\rfloor-\\left\\lfloor x-\\frac{1}{2}\\right\\rfloor\\right) dx \\).",
"role": "user"
}
] | b - a |
[
{
"content": "Find the limit of the sequence \\((z_n)\\) defined by \\[ z_n = \\frac{(1+2)^{p+1}+(1+2+3)^{p+1}+\\cdots+(1+2+\\cdots+n)^{p+1}}{n^{p+2}} \\] using the Stolz-Cesaro theorem.",
"role": "user"
}
] | \infty |
[
{
"content": "Find the value of $x$ that satisfies the equation: $$\\frac{a^{1/x}}{1/x} + \\frac{a^x}{x} = 2a$$ assuming $a$ is a positive constant.",
"role": "user"
}
] | 1 |
[
{
"content": "Solve the equation \\(\\sqrt{x} = 1 + \\ln(5 + x)\\) for \\(x\\).",
"role": "user"
}
] | 16.58 |
[
{
"content": "Given point $A(-3, 1)$, the equation of the median through $B$ as $L_1: 2x + y - 3 = 0$, and the equation of the angle bisector of angle $C$ as $L_2: 7x - 4y - 1 = 0$, find the equation of side $AB$ in the triangle.",
"role": "user"
}
] | 4x + 7y + 5 = 0 |
[
{
"content": "Evaluate the double integral \\( \\int \\int_R xy \\, dA \\), where \\( R \\) is the square with vertices \\((0,0), (1,1), (2,0), (1,-1)\\), using the change of variables formula and an appropriate transformation.",
"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... | c(w1, w2, y) = (w1 + w2)y^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... | 1.007e5 N/C, outward |
[
{
"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... | 15.48% |
[
{
"content": "Determine the range of the real number \\( a \\) such that the function \\( f(x) = a^x (a^x - 3a^2 - 1) \\) is increasing on the interval \\([0, +\\infty)\\). Choose the correct option:\n\n(A) \\(\\left(0, \\frac{2}{3}\\right]\\)\n(B) \\(\\left[\\frac{\\sqrt{3}}{3}, 1\\right)\\)\n(C) \\((0, \\sqrt... | B |
[
{
"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... | 0.8 |
[
{
"content": "Calculate the derivative \\( G'(2) \\) of the function \\( G(x) \\) defined by the integral \\[ G(x) = \\int_{\\pi/x}^{\\pi x} \\cos^{15} t \\, dt \\] for \\( x > 0 \\).",
"role": "user"
}
] | \pi |
[
{
"content": "Evaluate the sum \\( \\sum_{n=1}^{\\infty}\\frac{(-1)^{n+1}}{2n-1} \\) using the Maclaurin series expansion of a known function.",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | non-current asset |
[
{
"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... | 3.76e11 |
[
{
"content": "In a right-angled triangle with a 15-degree angle and a hypotenuse of length 7, a point is randomly chosen on the hypotenuse. Two perpendicular lines are drawn from this point to the other two sides of the triangle. Calculate the probability that the area of the rectangle formed by these perpendic... | \dfrac{1}{7} |
[
{
"content": "Simplify the expression: \\[ \\int_{-\\infty}^{+\\infty} f(r-r')\\,\\Delta_{r'}\\delta(r')\\,\\mathrm{d^3r'} \\] where \\( \\Delta \\) denotes the Laplacian operator.",
"role": "user"
}
] | \Delta f(r) |
[
{
"content": "Let $\\{B_t\\}_{t\\ge 0}$ be a standard Brownian motion, and let $\\mu$ and $a$ be given real numbers with $\\mu \\neq 0$. Define $X_t = \\mu t + B_t$. Consider the set $L = \\{\\tau \\in \\Re_+ \\mid X_\\tau = a\\}$. Is it true that for every $T \\in \\Re_+$, the set $[0,T] \\cap L$ is almost sur... | 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... | $470K |
[
{
"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... | $6,708.90 |
[
{
"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... | Profit Margin = (Total revenue - Total expenses) / Total revenue |
[
{
"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... | 1.9e7 J |
[
{
"content": "Find the minimum value of the function \\( f(x) = (x+a+b)(x+a-b)(x-a+b)(x-a-b) \\) without using differential calculus.",
"role": "user"
}
] | -4a^2b^2 |
[
{
"content": "Simplify the expression \\(\\frac{1}{4(2+m\\pi i)} \\left( e^{-(2+m\\pi i )}- e^{2+m\\pi i }\\right)\\) and find the values of \\(m\\) for which the expression is real.",
"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... | 0.311 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... | 1.183 |
[
{
"content": "If \\(x = (9+4\\sqrt{5})^{48} = [x] + f\\), where \\([x]\\) is the integer part of \\(x\\) and \\(f\\) is the fractional part, find the value of \\(x(1-f)\\). Choose the correct answer from the options below:\n\n\\(\\text{a.) } 1 \\\\\n\\text{b.) less than } 1 \\\\\n\\text{c.) more than } 1 \\\\\n... | 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... | 3.354 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... | 4.24 m |
[
{
"content": "For a set of independent and identically distributed continuous random variables $X_1, X_2, \\\\ldots, X_n$, calculate the probability $P[X_1 > \\max(X_2, X_3, \\\\ldots, X_j)]$, where $j$ is an integer such that $3 \\leq j \\leq n$. Express your answer in terms of $n$ and $j$.",
"role": "user... | \dfrac{1}{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... | $18,024 |
[
{
"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... | E = -dV/dr |
[
{
"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... | 0.03434 amu |
[
{
"content": "Consider a connected undirected simple non-planar graph $G$ with 15 vertices. If removing any edge from $G$ results in a planar graph, how many edges does $G$ have?",
"role": "user"
}
] | 40 |
[
{
"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... | 0.74 |
[
{
"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... | $930.11 |
[
{
"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 Fe_(s) + 3 MgCl_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... | 26496.85 |
[
{
"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... | FV = PV*(1+r)^n |
[
{
"content": "Determine the number of zeros of the function \\( f(z) = \\frac{1}{3}e^z - z \\) within the unit disc \\( |z| < 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... | 6.24 s |
[
{
"content": "Suppose $R$ is a ring and $a, b \\in R$ such that $Ra = R$ and $bR = R$. If there exists $x \\in R$ such that $ab = 1$ and $bx = 1$, does it follow that $a$ is invertible? 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... | 51.15 kg |
[
{
"content": "Find the Cauchy Principal Value of the integral \\( \\int_0^\\infty \\frac{x^{-p}}{x-1} \\, dx \\) for \\(|p|<1\\).",
"role": "user"
}
] | \pi \cot(\pi p) |
[
{
"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... | 4.82 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... | 2.047 |
[
{
"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... | 23.4 x 10^3 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... | 0.0907 m |
[
{
"content": "What is the expected number of fair coin flips needed to obtain the sequence HTT using the conditional expectation technique?",
"role": "user"
}
] | 8 |
[
{
"content": "Evaluate the infinite series $\\sum_{k = 1}^\\infty \\frac{k^2}{2^k}.$",
"role": "user"
}
] | 6 |
[
{
"content": "Let $X$ and $Y$ be jointly continuous random variables with the joint density function given by: $$f_{X,Y}(x,y) = \\begin{cases} \\frac{1}{3}(x+y), & \\text{for } x \\in (0,1) \\text{ and } y \\in (0,2) \\\\ 0, & \\text{otherwise}. \\end{cases}$$ Compute the expected value $\\mathbb{E}[\\sin(\\pi ... | \dfrac{1}{\pi} |
[
{
"content": "Evaluate the limit inferior: $$\\liminf_{x\\to\\infty}\\frac{f'^2}{ff''}$$ given that $f \\in C^3[0,+\\infty)$, $f, f', f'' > 0$, $f''' \\leq 0$, $\\frac{f'}{f''} \\geq \\frac{x}{2}$, and $\\frac{f}{f'} \\geq \\frac{x}{4}$.",
"role": "user"
}
] | 2 |
[
{
"content": "Determine the smallest degree of a monic polynomial $p(x)$ with integer coefficients such that $p(n)$ is divisible by 2021 for every positive integer $n$. Provide your answer as an integer.",
"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... | $45.64 |
[
{
"content": "If $X_n$ converges in distribution to a random variable $X$ with a uniform distribution on $[0,2]$, does $X_n^2$ converge in distribution? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a continuous function. Consider the set $S$ of points where $f$ has a local maximum. Determine whether $S$ is a closed set. Answer with \"True\" if it is closed, or \"False\" if it is not.",
"role": "user"
}
] | B |
[
{
"content": "Determine the number of solutions to the equation \\( \\{x\\}^{2}=\\left\\{x^{2}\\right\\} \\) in the interval \\([1, 100]\\), where \\(\\{u\\}\\) denotes the fractional part of \\(u\\).",
"role": "user"
}
] | 9901 |
[
{
"content": "Amy, Betty, Cathy, and Daisy have $36$ apples that they plan to split among themselves in a ratio of $2:3:3:4$, respectively. How many apples will Betty and Cathy have together?",
"role": "user"
}
] | 18 |
[
{
"content": "Let \\( P(x) \\) be a monic cubic polynomial with real roots such that \\( P(0) = 0 \\) and \\( P(1) = 1 \\). If \\( S \\) is the sum of the roots of \\( P(x) \\), how many integer values cannot be attained by \\( S \\)?",
"role": "user"
}
] | 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... | +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... | 126.7 mph |
[
{
"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 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... | 3.25 |
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