prompt list | ground_truth string |
|---|---|
[
{
"content": "Triangle $ABC$ is an isosceles right triangle with the measure of angle $A$ equal to 90 degrees. The length of segment $AC$ is 6 cm. What is the area of triangle $ABC$, in square centimeters?",
"role": "user"
}
] | 18 |
[
{
"content": "Find $a+b+c$, given that $x+y\\neq -1$ and \\begin{align*}\nax+by+c&=x+7,\\\\\na+bx+cy&=2x+6y,\\\\\nay+b+cx&=4x+y.\n\\end{align*}",
"role": "user"
}
] | 7 |
[
{
"content": "Let \\((M, \\omega)\\) be a closed symplectic manifold, and let \\(f, g \\in C^\\infty(M)\\) be two smooth functions. Consider the Hamiltonian flows \\(\\phi_t\\) and \\(\\psi_t\\) in \\(Ham(M, \\omega)\\) generated by the Hamiltonian vector fields \\(X_f\\) and \\(X_g\\), respectively. Prove that... | \{f, g\} = 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... | 5.139 W |
[
{
"content": "Does a maximal ideal of $\\mathbb{Z}[X,Y]$ contain a prime number? Let $\\mathfrak{m}$ be a maximal ideal of $\\mathbb{Z}[X,Y]$. Determine if $\\mathfrak{m}$ contains a prime number.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the number of solutions to the quadratic congruence $x^2 \\equiv 182 \\pmod{727}$. Note that $727$ is a prime number.",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the integral: $$\\int_2^4 \\frac{\\sqrt{\\ln(9-x)}}{\\sqrt{\\ln(9-x)} + \\sqrt{\\ln(3+x)}} \\space\\mathrm{d}x$$",
"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... | 9/2 |
[
{
"content": "Find the maximum likelihood estimate (MLE) of $m$ in the linear model $y = mx + e$, where $m$, $x$, and $e$ are independent and identically distributed as $N(0,1)$, given a single observation $(x, y)$. Assume $x = 1$ and $y = 3$. Provide the analytical expression for the MLE of $m$.",
"role": ... | 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... | 0.89m |
[
{
"content": "Calculate the variance of the sum of two random variables, $X$ and $Y$, where $X$ is a binomial random variable with parameters $n$ and $p$, and $Y$ is a Bernoulli random variable with parameter $p$. Given that $X$ and $Y$ are dependent and $E(Y|X=x) = \\frac{x}{n}$, find $\\text{Var}(X + Y)$. \\(... | (n + 3)p(1 - p) |
[
{
"content": "Calculate the integral \\( I = \\int_0^1 |f(x) - f^{-1}(x)| \\, dx \\), where \\( f(x) = x e^{x^2 - 1} \\) and \\( f: [0, 1] \\to [0, 1] \\).",
"role": "user"
}
] | \dfrac{1}{e} |
[
{
"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... | Volume of vinegar = 91.16 mL; Volume of oil = 104.84 mL |
[
{
"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... | 694400 |
[
{
"content": "Let \\( F(\\alpha)/F \\) be a Galois extension where \\( \\alpha \\notin F \\). Suppose there exists an element \\( f \\) in \\( \\operatorname{Gal}(F(\\alpha)/F) \\) such that \\( f(\\alpha) = \\alpha^{-1} \\). Prove that the degree \\([F(\\alpha):F]\\) is even.",
"role": "user"
}
] | even |
[
{
"content": "Let \\(x, y, z\\) be positive real numbers such that \\(x + y + z = 3\\). Find the maximum value of the expression:\n\\[\nP = \\frac{2}{3 + xy + yz + zx} + \\left(\\frac{xyz}{(x+1)(y+1)(z+1)}\\right)^{\\frac{1}{3}}.\n\\]",
"role": "user"
}
] | \dfrac{5}{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... | 20.83 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... | 0.604*10^8 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... | -4.41e-9 C |
[
{
"content": "An ellipse has foci at $(9, 20)$ and $(49, 55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?",
"role": "user"
}
] | 85 |
[
{
"content": "Let $A$ be a bounded self-adjoint semi-positive linear operator on a Hilbert space $X$, such that $(Ax,x) \\ge 0$ for all $x \\in X$. It is known that $(I+A)^{-1}$ is well-defined with $\\|(I+A)^{-1}\\| \\le 1$. Show that the operator $(I+A)^{-1}A$ satisfies $\\|(I+A)^{-1}A\\| \\le 1$. Does this i... | \| (I + A)^{-1} A \| \le 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... | 0.019 s |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty} e^{-2n} \\sum_{k=0}^{2n} \\frac{(2n)^k}{k!} \\]",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Find the Taylor series and the radius of convergence for the function \\( \\frac{1+z}{1-z} \\) centered at \\( z_0 = i \\).",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "Find the function $f(x)$ that satisfies the condition $f(x) = f'(x) + f''(x) + f'''(x) + f''''(x) + \\ldots$, where $f(x)$ is an indefinitely differentiable function, and $f(0) = 1$. Determine $f(x)$. \n",
"role": "user"
}
] | e^{x/2} |
[
{
"content": "Find the greatest common divisor (gcd) of all numbers of the form \\((a-b)(b-c)(c-d)(d-a)(a-c)(b-d)\\), where \\(a, b, c, d\\) are integers.",
"role": "user"
}
] | 12 |
[
{
"content": "Find the remainder when \\(4^{101}\\) is divided by 101, without using Fermat's Little Theorem.",
"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... | 0.74 m/s |
[
{
"content": "Determine whether the following statement is true or false:\n\nIf $C$ is the circle in $\\mathbb{C}$ of radius $10$ centered at $z=2i$ with positive orientation and\n$$f(z) = \\frac{\\cos(4z)}{z},$$\nthen\n$$ \\oint_{C} f(z) \\, \\mathrm{d}z = 2\\pi i.$$",
"role": "user"
}
] | 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... | 1.72 |
[
{
"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... | 1200 tires weekly |
[
{
"content": "In rectangle $ABCD$, points $F$ and $G$ lie on $\\overline{AB}$ so that $AF = FG = GB$ and $E$ is the midpoint of $\\overline{DC}$. Also, $\\overline{AC}$ intersects $\\overline{EF}$ at $H$ and $\\overline{EG}$ at $J$. The area of rectangle $ABCD$ is 70. Find the area of triangle $EHJ$.\n\n[asy]\n... | 3 |
[
{
"content": "Consider the continuous function $f:\\Bbb{R}\\to\\Bbb{R}$ defined by $xf(x) = e^x-1$. Determine the limit \\[\\lim _{n\\to \\infty }nf^{\\left(n\\right)}\\left(x\\right)\\] where $f^{\\left(n\\right)}\\left(x\\right)$ denotes the $n$th derivative of $f$. Provide your answer as a single value or ex... | 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... | 106500 |
[
{
"content": "Evaluate the limit \\( \\lim_{x \\to 0^+} \\left( \\frac{\\ln(1+x)}{3(1+x)^{1/3}-1}\\right)^{\\frac{x}{\\sin^2x} } \\) using L'Hopital's rule.",
"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... | 9000 |
[
{
"content": "Find the largest integer \\( m \\) such that \\( a^{12} \\equiv 1 \\pmod{m} \\) for all integers \\( a \\) that are relatively prime to \\( m \\).",
"role": "user"
}
] | 65520 |
[
{
"content": "Evaluate the definite integral: $$\\int\\limits_{0}^{2}\\left(\\sqrt{1+x^3}+\\sqrt[3]{x^2+2x}\\:\\right)dx$$",
"role": "user"
}
] | 6 |
[
{
"content": "Find the limit of \\(x^{-4}\\left(1-e^{x^2/2}\\cos \\left(x\\right)\\right)\\) as \\(x\\to0\\) using the expansions of \\(\\exp\\) and \\(\\cos\\).",
"role": "user"
}
] | \dfrac{1}{12} |
[
{
"content": "Can a set $B$ of real numbers, which does not include 0 and 1, contain exactly 1000 elements if it satisfies the following conditions: for any $b \\in B$, both $\\frac{1}{b}$ and $1-\\frac{1}{b}$ are also in $B$?",
"role": "user"
}
] | No |
[
{
"content": "Given \\( \\phi = \\frac{1}{|\\underline{r}|} \\) and \\( E = -\\operatorname{grad} \\phi \\), show that \\( \\operatorname{div} E = 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... | 1.98 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... | [x,y] = meshgrid(0:0.01:3);z = your_function(x,y).*((y > 0.5*x + 1) & (y > 6*x - 2));surf(z) |
[
{
"content": "Is the bounded component of the complement of a Jordan curve in the plane always simply connected?",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( x, y, z \\) be positive numbers satisfying the following system of equations:\n\n\\[\n\\left\\{\n\\begin{array}{l}\nx^{2} + xy + y^{2} = 75 \\\\\ny^{2} + yz + z^{2} = 36 \\\\\nz^{2} + xz + x^{2} = 111\n\\end{array}\n\\right.\n\\]\n\nFind the value of the expression \\( xy + yz + xz \\).",
... | 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... | a = 2i + 3j + 0k |
[
{
"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... | Proceeds for the issuer: $68,700,000
Proceeds for the underwriter: $15,300,000 |
[
{
"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... | 4.29 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... | This year: 11.15%
Last year: 7.45% |
[
{
"content": "Determine the limiting behavior of the sequence \\( \\left\\langle \\frac{f(n)}{n^2} \\right\\rangle \\), where \\( f(n) \\) is the number of ordered pairs \\((a, b)\\) such that the least common multiple (lcm) of \\(a\\) and \\(b\\) is \\(n\\), with \\(a, b \\in \\{1, 2, \\ldots, n\\}\\).",
"... | 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... | $17,045,648.25 |
[
{
"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... | DKI=KC-CD |
[
{
"content": "Find the derivative of the trace of the product of a matrix by its transpose, where \\( A \\) is a \\( p \\times n \\) matrix. Specifically, compute \\( \\frac{d}{dA} \\text{tr}(AA^T) \\).",
"role": "user"
}
] | 2A |
[
{
"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... | 429 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... | 26.212 degrees Celcius |
[
{
"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... | $325 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... | $8750.60 |
[
{
"content": "Determine whether the differential equation \\( f''(x) - f'(x) - f(x)^3 = 0 \\) can be reduced to a first-order equation.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit \\( \\lim_{r \\to 0} \\frac{2r^2}{\\sin(2\\theta)} \\) given that the \\(x\\) and \\(y\\) axes are not in the domain of the function \\(\\frac{(x^2+y^2)^2}{xy}\\), where \\(x=r\\cos(\\theta)\\) and \\(y=r\\sin(\\theta)\\).",
"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... | <6/5, 3/5> |
[
{
"content": "Compute the limit as $n$ approaches infinity of the integral \\( \\int_0^1 \\frac{e^{-(n + x^2)}}{n^2 + x} \\, dx \\). Determine whether it is sufficient to compute the integral of \\( f_n(x) \\) and then take the limit, or if a more intricate approach is required.",
"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... | 2.44 m/s^2 |
[
{
"content": "The area bounded by the curve \\( y = e^{x^2} \\), the x-axis, and the lines \\( x = 1 \\) and \\( x = 2 \\) is given to be \\( a \\) square units. Find the area bounded by the curve \\( y = \\sqrt{\\ln x} \\), the y-axis, and the lines \\( y = e \\) and \\( y = e^4 \\) in terms of \\( a \\).",
... | 2e^4 - e - a |
[
{
"content": "The graph below shows a portion of the curve defined by the quartic polynomial $P(x)=x^4+ax^3+bx^2+cx+d$.\n\n[asy]\nunitsize(0.8 cm);\n\nint i;\n\nreal func (real x) {\n return(0.5*(x^4/4 - 2*x^3/3 - 3/2*x^2) + 2.7);\n}\n\ndraw(graph(func,-4.5,4.5));\ndraw((-4.5,0)--(4.5,0));\ndraw((0,-5.5)--(0,5... | \text{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... | 6.7626e-3 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... | 0.335 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... | \frac{ 490\left< 1, 12 \right>}{\sqrt{145}} |
[
{
"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... | 92% |
[
{
"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... | 13,100 units, $235,800 |
[
{
"content": "Let \\( f(z) = z + \\frac{1}{z} \\) for \\( z \\in \\mathbb{C} \\) with \\( z \\neq 0 \\). Which of the following is always true? \n\n(a) \\( f \\) is analytic on \\( \\mathbb{C} \\setminus \\{0\\} \\) \n(b) \\( f \\) is a conformal mapping on \\( \\mathbb{C} \\setminus \\{0\\} \\) \n(c) None of... | 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... | $23,260.70 |
[
{
"content": "Determine whether $\\frac{1}{4}$ is an element of the Cantor set.",
"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... | $711227 |
[
{
"content": "Suppose \\( x \\) is a positive real number such that \\( \\{x\\}, [x] \\), and \\( x \\) are in a geometric progression. Find the least positive integer \\( n \\) such that \\( x^n > 100 \\). Here, \\([x]\\) denotes the integer part of \\( x \\) and \\( \\{x\\} = x - [x] \\).",
"role": "user"... | 10 |
[
{
"content": "Find the position of the fraction \\( \\frac{2018}{2187} \\) in the sequence: \\( \\frac{2}{3}, \\frac{2}{9}, \\frac{4}{9}, \\frac{6}{9}, \\frac{8}{9}, \\frac{2}{27}, \\frac{4}{27}, \\ldots, \\frac{26}{27}, \\ldots, \\frac{2}{3^{n}}, \\frac{4}{3^{n}}, \\ldots, \\frac{3^{n}-1}{3^{n}}, \\ldots \\)."... | 1552 |
[
{
"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... | 53.3 m |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty} e^{-n} \\sum_{k=0}^{n} \\frac{n^k}{k!} \\]",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Evaluate the limit of the sequence \\( P_n \\) defined by \\( P_n = \\frac{(2^3 - 1)}{(2^3 + 1)} \\cdot \\frac{(3^3 - 1)}{(3^3 + 1)} \\cdot \\ldots \\cdot \\frac{(n^3 - 1)}{(n^3 + 1)} \\) as \\( n \\to \\infty \\).",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "An equilateral triangle $ABC$ has a side length of $10$. A square $PQRS$ is inscribed in the triangle such that $P$ is on $AB$, $Q$ and $R$ are on $BC$, and $S$ is on $AC$. The area of the square $PQRS$ is expressed as $m + n\\sqrt{k}$, where $m$ and $n$ are integers, and $k$ is a prime number. Ca... | 10 |
[
{
"content": "If a group \\( G \\) has order \\( 7^3 \\), must \\( G \\) contain an element of order 7?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the number of derangements of the multiset \\( \\{1, 1, 2, 2, 14\\} \\), where a derangement is defined as a permutation in which no element appears in its original position or the position of any of its identical elements.",
"role": "user"
}
] | 4 |
[
{
"content": "There are two concentric spheres of radii 3 units and 6 units. What is the volume, in cubic units, of the region within the larger sphere and not within the smaller sphere? Express your answer in terms of $\\pi$.",
"role": "user"
}
] | 252\pi |
[
{
"content": "Evaluate the expression \\(((1 \\oplus 2) \\oplus (3 \\oplus 4)) \\oplus 5\\) given that \\(a \\oplus b = ab - a - b\\).",
"role": "user"
}
] | -41 |
[
{
"content": "My classroom has 11 rows of chairs, with 11 chairs in each row. The chairs in each row are numbered from 1 to 11. How many chairs have odd numbers?",
"role": "user"
}
] | 66 |
[
{
"content": "Consider a monic quartic polynomial with four distinct real roots. The turning points of this polynomial form an equilateral triangle. Find the area of this triangle.",
"role": "user"
}
] | 3^{\frac{5}{6}} |
[
{
"content": "Let $N$ be the product of all divisors of $2013^{2013}$. Determine the last two digits of $N$ in its decimal notation.",
"role": "user"
}
] | 41 |
[
{
"content": "Compute the integral \\( \\int_{0}^{\\infty} \\frac{\\log(x)}{x^2+1}\\, dx \\) using the function \\( f(z) = \\frac{(\\log z)^2}{z^2+1} \\) where \\( |z|>0 \\) and \\( - \\frac{\\pi}{2} < \\arg(z)< \\frac{3 \\pi}{2} \\).",
"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... | t_1 = \frac{\pi}{2}\sqrt{\frac{m}{k}} |
[
{
"content": "Given a non-empty set $A$ and a symmetric relation $R \\subset A \\times A$, if $(R \\circ R)$ is a function, determine whether $R$ is necessarily a function.",
"role": "user"
}
] | Yes |
[
{
"content": "For subsets $U, W, W'$ of a vector space $V$, if $U \\oplus W = U \\oplus W'$, does it necessarily follow that $W = W'$? Determine whether this statement is true or false.",
"role": "user"
}
] | False |
[
{
"content": "In triangle $ABC$, $\\angle BAC = 117^\\circ$. The angle bisector of $\\angle ABC$ intersects side $AC$ at $D$. Given that $\\triangle ABD \\sim \\triangle ACB$, find the measure of $\\angle ABC$ in degrees.",
"role": "user"
}
] | 42 |
[
{
"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... | 1.11 M |
[
{
"content": "Which function grows faster as $n$ becomes larger: $f(n)=100^{100^{100^n}}$ or $g(n)=2^{2^{2^{2^n}}}$?",
"role": "user"
}
] | g(n) |
[
{
"content": "Evaluate the definite integral \\( \\int_{0}^{\\infty} \\frac{\\log x}{1+x^2} \\, 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... | Common Size Analysis
Jeter
Net Sales: 100%
Cost of Goods Sold: 62.10%
Other Expenses: 31.71%
Net Income: 6.20%
Azul
Net Sales: 100%
Cost of Goods Sold: 72.70%
Other Expenses: 22.40%
Net Income: 4.90% |
[
{
"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... | 0.0417 kJ/K |
[
{
"content": "Find the degree of the splitting field of the polynomial $x^3 - 7$ over the rational numbers $\\mathbb{Q}$. The roots of the polynomial are $\\sqrt[3]{7}$, $\\sqrt[3]{7} \\omega$, and $\\sqrt[3]{7} \\omega^2$, where $\\omega$ is a primitive cube root of unity.",
"role": "user"
}
] | 6 |
[
{
"content": "Given a closed triangle \\( T \\), determine the greatest angle \\( \\alpha \\) such that any two points in \\( T \\) are contained in an \\( \\alpha \\)-set \\( S \\subset T \\).",
"role": "user"
}
] | \alpha |
[
{
"content": "Find the length $L$ of the catenary $C$ given by the equation $y = \\cosh(x)$ over the interval $\\log(2) \\leq x \\leq \\log(3)$. Express your answer in terms of known functions or constants.",
"role": "user"
}
] | \dfrac{7}{12} |
[
{
"content": "In triangle $\\Delta ABC$, $\\angle A = 40^\\circ$, $\\angle B = 20^\\circ$, and $AB - BC = 4$. Calculate the length of the angle bisector from $\\angle C$. \\(\\boxed{\\text{Length}}\\)",
"role": "user"
}
] | 4 |
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