prompt list | ground_truth string |
|---|---|
[
{
"content": "On segment $AC$, point $B$ is such that $AB = 14$ and $BC = 28$. Semicircles are constructed on segments $AB$, $BC$, and $AC$ as diameters in the same half-plane relative to line $AB$. Determine the radius of the circle that is tangent to all three semicircles.",
"role": "user"
}
] | 6 |
[
{
"content": "Find the limit of the sequence \\(x_n\\) defined by \\(x_1=1\\) and \\(x_{n+1}=2+\\frac{3}{x_n}+\\frac{1}{n}\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Suppose $f$ and $g$ are entire functions and $g(z) \\neq 0$ for all $z \\in \\mathbb{C}$. If $|f(z)| \\leq |g(z)|$, which of the following must be true about $f$? (a) $f(z) \\neq 0$ for all $z \\in \\mathbb{C}$ (b) $f$ is a constant function (c) $f(0) = 0$ (d) For some $c \\in \\mathbb{C}$, $f(z) ... | 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... | t_(1/2) = ln2/k |
[
{
"content": "Determine the number of zeros of the function \\( f(z) = z^2 + \\cosh(iz) \\) within the region \\( |z| < \\frac{1}{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... | 6.23 |
[
{
"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... | 17059 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... | 1050 L/(mol*cm) |
[
{
"content": "Show that the order of the automorphism group of \\(Z_3 \\times Z_3\\) is 48.",
"role": "user"
}
] | 48 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.05e+1 |
[
{
"content": "Consider the function \\(f(x)=\\int_x^{2x}\\frac{\\sin t}{t}dt\\). Determine the value of \\(x\\) on the interval \\((0,\\frac{3\\pi}{2})\\) at which this function attains a local maximum.",
"role": "user"
}
] | \dfrac{\pi}{3} |
[
{
"content": "Let $(n_k)$ be an increasing sequence of positive integers. Define $E$ as the set of all $x$ such that the sequence $\\sin(n_k x)$ converges. Prove that the Lebesgue measure of $E$, $\\mu(E)$, is zero.",
"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... | 0.46 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... | -5.81e4 J |
[
{
"content": "Is there a Hausdorff compactification of the irrational numbers with its usual topology such that the remainder is a discrete space?",
"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... | The total momentum after a collision is equal to the total momentum before the collision. For two objects in an elastic collision: p_{1i} + p_{2i} = p_{1f} + p_{2f}. For an inelastic collision: p_{1i} + p_{2i} = p_f, where p_f is the momentum of the combined object. |
[
{
"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... | Changes in the policies of large open economies and changes in current account balances or terms of trade can cause changes in the world interest rate. |
[
{
"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... | bowling ball > penny > hollow globe > wedding ring |
[
{
"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... | 8.45% |
[
{
"content": "Let $X \\subset \\mathbb{R}^n$ be a bounded and connected set, and let $f: X \\to \\mathbb{R}$ be a polynomial function. Consider a point $x \\in X$ and the nearest global maximum $m \\in X$ to $x$. Suppose that in every neighborhood of $x$, there exists a point $y$ and a path $p_t$ such that $p_0... | 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... | $61,235.23 |
[
{
"content": "Let $n$ be an odd integer with exactly 11 positive divisors. Find the number of positive divisors of $8n^3$.",
"role": "user"
}
] | 124 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 21.4 seconds |
[
{
"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... | 313656.6105 |
[
{
"content": "Suppose $f(x, y)$ is defined on some neighbourhood of $(0, 0)$. If $u(x, y) = \\sin f(x, y)$ is continuous at $(0, 0)$, does it imply that $f(x, y)$ is continuous at $(0, 0)$? Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "Determine whether the set \\(I_3 = \\{[a,b) \\times [c,d) \\times [e,f) \\mid a,b,c,d,e,f\\in\\Bbb{R}\\}\\) forms a semiring over \\(\\Bbb{R}^3\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Given the quadratic equations \\(x^2 - 10ax - 11b = 0\\) with roots \\(c\\) and \\(d\\), and \\(x^2 - 10cx - 11d = 0\\) with roots \\(a\\) and \\(b\\), find the value of \\(a + b + c + d\\).",
"role": "user"
}
] | 1210 |
[
{
"content": "Determine the value of $r$ that maximizes the expression: $$\\sum_{k=0}^{r} \\binom{20}{r-k} \\binom{20}{k}.$$",
"role": "user"
}
] | 20 |
[
{
"content": "Christine must buy at least $45$ fluid ounces of milk at the store. The store only sells milk in $200$ milliliter bottles. If there are $33.8$ fluid ounces in $1$ liter, then what is the smallest number of bottles that Christine could buy? (You may use a calculator on this problem.)",
"role":... | 7 |
[
{
"content": "Consider 9 distinct apples and 7 distinct bananas to be placed into 6 identical bins. Determine the number of ways to configure the bins such that exactly one bin contains 5 apples, exactly one bin contains 3 bananas, and the remaining 4 bins each contain 1 apple and 1 banana. Provide your answer ... | 105840 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 12 mg |
[
{
"content": "Let \\( S = \\{ x \\in \\mathbb{R} : 2x^2 \\cos\\frac{1}{x} = 1 \\} \\) and \\( T = \\{ x \\in \\mathbb{R} : 2x^2 \\cos\\frac{1}{x} \\leq 1 \\} \\cup \\{0\\} \\). Under the usual metric on \\( \\mathbb{R} \\), which of the sets \\( S \\) or \\( T \\) is complete?",
"role": "user"
}
] | T |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 21.34% |
[
{
"content": "If $\\sin x + \\sin y = \\frac{96}{65}$ and $\\cos x + \\cos y = \\frac{72}{65}$, then what is the value of $\\tan x + \\tan y$?",
"role": "user"
}
] | \frac{507}{112} |
[
{
"content": "Calculate the limit \\( \\lim_{n \\to \\infty}\\frac{\\sqrt[n]{(2n)!}}{2n} \\) using Stirling's approximation.",
"role": "user"
}
] | \infty |
[
{
"content": "Evaluate the contour integral \\( \\oint_{C_3 (0)} \\frac{z+7}{z^4 + z^3 - 2 z^2}\\,dz \\) using the Residue Theorem, where \\(C_3(0)\\) is the circle of radius 3 centered at the origin, oriented counter-clockwise.",
"role": "user"
}
] | 0 |
[
{
"content": "What is the largest $4$ digit integer congruent to $15 \\pmod{22}?$",
"role": "user"
}
] | 9981 |
[
{
"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... | Partial Balance Sheet
Intangible Assets:
Patents and Trademarks (at cost) $115.5
Goodwill (at cost) $130.8
Less: Accumulated Amortization $47.1
Total Intangible Assets $199.2
Plant Assets:
Other Plant Assets $767.2
Land $193.8
Machinery and Equipment $1,817.2
Buildings $840.9
Less: Accumulated Depreciation $1,940.8
Total Plant Assets $678.3
Total Plant and Intangible Assets $1,877.5 |
[
{
"content": "Evaluate the limit: $$\\lim_{u\\to 0}\\frac{(u+1)^2\\sin(3u-3)}{\\cos(u^3+u-1)\\sin^2(u^2-u)}.$$",
"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... | 3102.67 mole |
[
{
"content": "Let \\( a_{n} \\) be the coefficient of the \\( x \\) term in the expansion of \\( (3-\\sqrt{x})^{n} \\) for \\( n=2,3,4, \\cdots \\). Find \\( \\lim _{n \\rightarrow \\infty}\\left(\\frac{3^{2}}{a_{2}}+\\frac{3^{3}}{a_{3}}+\\cdots+\\frac{3^{n}}{a_{n}}\\right) \\).",
"role": "user"
}
] | 18 |
[
{
"content": "Suppose $A \\subset \\mathbb{R}^2$ is an open subset such that there exist infinitely many homeomorphisms that, when restricted to $A$, give self-maps that are also homeomorphisms. Does it follow that $A$ is an open disc?",
"role": "user"
}
] | No |
[
{
"content": "Given a matrix where the value at each index is determined by the XOR operation between the column and row indices, and a fixed amount is subtracted from each element (ensuring non-negative results), calculate the sum of all elements in the matrix. For example, consider a matrix with 5 rows and 8 ... | 105 |
[
{
"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... | $135,048 |
[
{
"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... | $35,700 |
[
{
"content": "In an equilateral triangle \\(\\triangle ABC\\) with side length 4, point \\(D\\) is located on side \\(BC\\) such that \\(BD = 1\\). Let \\(r\\) and \\(s\\) be the radii of the inscribed circles of triangles \\(\\triangle ADB\\) and \\(\\triangle ADC\\), respectively. Calculate the product \\(rs\... | 4 - \sqrt{13} |
[
{
"content": "Given the polynomial \\(x^4 + px^3 + qx^2 + rx + s = 0\\) with rational coefficients and four complex roots, two of the roots add up to \\(3 + 4i\\) and the other two have a product of \\(13-i\\). Compute the value of \\(q\\).",
"role": "user"
}
] | 51 |
[
{
"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 kgm^2/s |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $0.42 |
[
{
"content": "Determine the values of $p \\geq 0$ for which the integral \\( \\int_0^{\\infty} \\frac{dx}{x^{p}+x^{-p}} \\) converges.",
"role": "user"
}
] | p > 1 |
[
{
"content": "Find the maximum volume of a rectangular solid in the first octant ($x \\geq 0$, $y \\geq 0$, $z \\geq 0$) with one vertex at the origin and the opposite vertex lying on the plane $x + y + z = 1$. Use the method of Lagrange multipliers to derive the system of equations needed to solve for the maxi... | \dfrac{1}{27} |
[
{
"content": "Find the orthogonal complement of the subset \\( C = \\{ u \\in L^2(0,2) : \\int_0^2 u(t) \\, dt = 1 \\} \\) in the space \\( L^2(0,2) \\).",
"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... | 2.60e+05 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... | 0.820 mol/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... | Critical biodiversity hotspots are located across the globe, spanning Africa, Asia-Pacific, Europe and Central Asia, North and Central America, and South America. Examples include the Cape Floristic Region, Madagascar and the Indian Ocean Islands, Japan, the forests of Eastern Australia, the Mediterranean Basin, Madrean pine-oak woodlands, the Atlantic forest, and the tropical Andes. |
[
{
"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... | 7200 C |
[
{
"content": "Find the Fourier series representation of the function \\(W(t)\\) given by:\n\\[\nW(t) = \\begin{cases} 2, & \\text{for } -2 \\leq t < 4 \\\\ 0, & \\text{otherwise} \\end{cases}\n\\]\nUse the formulas for \\(a_n\\) and \\(b_n\\) extended for an interval of length \\(2l\\).",
"role": "user"
}... | 2 |
[
{
"content": "Solve the inequality $$-13(r+5) + 25 > 4(r-10)$$for $r$. Express your answer in interval notation.",
"role": "user"
}
] | (-\infty,0) |
[
{
"content": "Determine whether there exists a Lebesgue measurable set $D \\supset [0,1]$ such that $D \\neq [0,1]$ and $\\lambda^{*}(D) = 1$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the spectrum of the $C^*$-algebra formed by the direct sum $C(K) \\oplus C(K')$, where $K$ and $K'$ are compact Hausdorff spaces.",
"role": "user"
}
] | K \sqcup K' |
[
{
"content": "Let $X$ be a metric space and let $U, V \\subset X$ be open and bounded subsets. Is the Hausdorff metric \\( h(U,V) = \\max(\\sup_{x\\in U}\\inf_{y\\in V}d(x,y),\\sup_{x\\in V}\\inf_{y\\in U}d(x,y)) \\) defined for all possible choices of $U$ and $V$?",
"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... | 2898 m from origin at an angle 233.4 degrees from the positive x-axis |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 21 m |
[
{
"content": "Find \\(\\lim\\limits_{x\\to +\\infty} \\frac{x^{2}e^{-x}}{x^{2}+1}\\sin(xe^{x^{2}})\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Does the uniform convergence of a sequence of Lipschitz continuous functions imply that the limit function is also Lipschitz continuous?",
"role": "user"
}
] | No |
[
{
"content": "Suppose we have \\(n\\) independent variables from \\(U[0,1+\\theta]\\) and let \\(\\lambda=1+\\theta\\). The estimator of \\(\\theta\\) is given by \\(\\bar{\\theta}=2 \\bar{X} - 1\\). Find the bias of \\(\\bar{\\theta}\\) given that the distribution of \\(\\bar{X}\\) for \\(n\\) variables from \... | 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... | $230,000 |
[
{
"content": "Simplify the expression: \\[ \\frac{\\left(a^{2} b \\sqrt{b}-6 a^{5 / 3} b^{5 / 4}+12 a b \\sqrt[3]{a}-8 a b^{3 / 4}\\right)^{2 / 3}}{a b \\sqrt[3]{a}-4 a b^{3 / 4}+4 a^{2 / 3} \\sqrt{b}} \\]",
"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... | 0.968 s |
[
{
"content": "Find the Shnirelman density of the set \\( A = \\{ \\lfloor n \\ln (n) \\rfloor : n = 1,2,3,... \\} \\).",
"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... | 1.073 |
[
{
"content": "Calculate the circulation of the vector field \\( \\vec{F}(x, y, z) = \\langle \\sin(x^2+z)-2yz, 2xz + \\sin(y^2+z), \\sin(x^2+y^2)\\rangle \\) along the circle parameterized by \\( \\vec{r}(t)=\\langle\\cos(t), \\sin(t), 1\\rangle \\) for \\( t \\in [0, 2\\pi] \\). Use Stoke's Theorem to evaluate... | 4\pi |
[
{
"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 cm^3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1229534 ft-lb |
[
{
"content": "If $\\frac{1}{x} + \\frac{1}{y} = 3$ and $\\frac{1}{x} - \\frac{1}{y} = -7$ what is the value of $x + y$? Express your answer as a common fraction.",
"role": "user"
}
] | -\frac{3}{10} |
[
{
"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... | 385.635 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... | 1.85 * 10^6 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... | $1.01 |
[
{
"content": "Twenty-seven solid gold spherical beads each of radius 3 are melted down and recast into a larger, solid gold sphere. How many units are in the radius of this larger gold sphere?",
"role": "user"
}
] | 9 |
[
{
"content": "An $\\textit{annulus}$ is the region between two concentric circles. The concentric circles in the figure have radii $b$ and $c$, with $b>c$. Let $\\overline{OX}$ be a radius of the larger circle, let $\\overline{XZ}$ be tangent to the smaller circle at $Z$, and let $\\overline{OY}$ be the radius ... | \pi a^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... | 897.63 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... | $97,469.33 |
[
{
"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... | $17.63 |
[
{
"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... | modern economics |
[
{
"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... | 1.54 g |
[
{
"content": "Let $X_1, X_2, X_3$ be independent and identically distributed random variables, each following a uniform distribution on the interval (0, 1). Calculate the expected value of the expression $E\\left(\\frac{X_1 + X_2}{X_1 + X_2 + X_3}\\right)$. What is the value?",
"role": "user"
}
] | \dfrac{2}{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... | 18.62% |
[
{
"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... | 923.05K |
[
{
"content": "Determine the possible leading digits that both $2^n$ and $5^n$ can have for certain natural numbers $n$. (The 14th All-Russian Mathematical Olympiad, 1988)",
"role": "user"
}
] | 3 |
[
{
"content": "Solve the congruence \\(x^3 + 2x + 2 \\equiv 0 \\pmod{125}\\) using Hensel's Lemma.",
"role": "user"
}
] | 113 |
[
{
"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... | covalent |
[
{
"content": "Calculate the average arc length between two randomly selected points on a sphere with a radius of 2.",
"role": "user"
}
] | \pi |
[
{
"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... | 3.22 kJ |
[
{
"content": "Stuart has drawn a pair of concentric circles, as shown. He draws chords $\\overline{AB}$, $\\overline{BC}, \\ldots$ of the large circle, each tangent to the small one. If $m\\angle ABC=75^\\circ$, then how many segments will he draw before returning to his starting point at $A$? [asy]\nsize(100... | 24 |
[
{
"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... | $7,863.38 |
[
{
"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... | q1 = 80/3
q2 = 80/3
P = $46.67
Profit1 = $711.2
Profit2 = $711.2 |
[
{
"content": "Evaluate the Fourier coefficient \\( \\hat{f}_k \\) for the function \\( f(t) = \\frac{1-r^2}{1+r^2-2r\\cos t} \\) where \\( t \\in \\mathbb{R} \\) and \\( |r| < 1/2 \\). Verify that the correct result is \\( \\hat{f}_k = r^{|k|} \\).",
"role": "user"
}
] | r^{|k|} |
[
{
"content": "Determine the range of values for \\( k \\) such that the equation \\( x^4 - 14x^2 + 24x - k = 0 \\) has four real and unequal roots.",
"role": "user"
}
] | (8, 11) |
[
{
"content": "In triangle $ABC$ with sides $AB=1$, $BC=\\sqrt{3}$, and $AC=2$, points $D$, $E$, and $F$ are located on $AB$, $BC$, and $AC$ respectively, such that $\\angle EDF = \\angle DFA = 90^{\\circ}$. If the maximum possible value of $[DEF]^2$ can be expressed as $\\frac{a}{b}$, where $a$ and $b$ are posi... | 67 |
[
{
"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... | 0.204 m |
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