prompt list | ground_truth string |
|---|---|
[
{
"content": "Let $E \\subset \\mathbb{R}^n$ be a set with positive Lebesgue measure. Determine whether there always exists a point $x \\in E$ such that for any $r > 0$, the intersection $B(x,r) \\cap E$ has positive Lebesgue measure in $\\mathbb{R}^n$. Here, $B(x,r)$ denotes the ball of radius $r$ centered at ... | Yes |
[
{
"content": "Determine whether the field extension $K=\\mathbb{Q}(\\sqrt{2},\\sqrt{5})$ has a power integral basis using Dedekind's Theorem and the prime ideal decomposition in $\\mathbb{F}_3[X]$. Provide your answer as either 'Yes' or 'No'.",
"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... | -16.3i - 8.2j |
[
{
"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... | 18.4 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... | 8.01e-9 C/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... | Zn(II) + 2H2O -> Zn(OH)2 +H2; This reaction increases pH through the addition of base, making conditions unfavorable for moss growth. |
[
{
"content": "Determine the values of \\(a\\) and \\(b\\) for the hyperbola \\(\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1\\) such that it is congruent to the rectangular hyperbola \\(xy = 1\\).",
"role": "user"
}
] | \sqrt{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... | 80.25 nm |
[
{
"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... | 1.1009e+7 Nm^(-2) |
[
{
"content": "If an integral domain $R$ is a free $\\mathbb{Z}$-module, and $I$ is a non-zero ideal of $R$, do $R$ and $I$ have the same rank as $\\mathbb{Z}$-modules? Answer 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... | There is not much time gain by the reduction of the length of the pendulum |
[
{
"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... | 188742 J |
[
{
"content": "Let $M$ be an $n \\times n$ matrix consisting entirely of 1's. Compute the operator norm $\\|M\\|_{op}$ and show that $\\|M\\|_{op} = n$. ",
"role": "user"
}
] | 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... | 0.09 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 4.106 |
[
{
"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... | Y = 0.129 - 0.259(M) |
[
{
"content": "Consider an ellipse with foci at points \\(A(-2, 0)\\) and \\(B(2, 0)\\). This ellipse intersects a circle of radius 1 centered at the origin. Determine the maximum possible sum of distances from a point on the circle to the foci of the ellipse.",
"role": "user"
}
] | 2\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... | 8.66% |
[
{
"content": "If $\\log_9 (x-2)=\\frac{1}{2}$, find $\\log_{625} x$.",
"role": "user"
}
] | \frac14 |
[
{
"content": "In trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\\angle A = 37^\\circ$, $\\angle D = 53^\\circ$, and $M$ and $N$ be the midpoints of $\\overline{BC}$ and $\\overline{AD}$, respectively. Find the length $MN$.",
"role": "user"
}
] | 504 |
[
{
"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... | 363160 |
[
{
"content": "Determine the number of ways to distribute 80 candies to 3 kids such that:\n- The first kid receives any number of candies (including 0).\n- The second kid receives an even positive number of candies.\n- The third kid receives 0, 2, or 5 candies.\n- All candies are distributed.",
"role": "user... | 116 |
[
{
"content": "Verify that the Fourier transform of the constant function \\( f \\equiv 1 \\) is the Dirac delta function \\( \\delta(x) \\) by showing that the integral \\( \\int_\\mathbb{R} \\int_\\mathbb{R} \\exp(i\\lambda x)\\phi(x)\\, dx \\, d\\lambda \\) equals \\( \\phi(0) \\) for any Schwartz function \\... | \phi(0) |
[
{
"content": "Use Taylor series to show that the expression \\( P(Dc|-) = \\frac{0.98-0.98p}{0.98-0.93p} \\) can be approximated by \\( 1 - 0.056p \\) in the interval \\([0, 0.1]\\).",
"role": "user"
}
] | 1 - 0.056p |
[
{
"content": "For a village with $M$ villagers, a virus is introduced by a wanderer. Each week, every infected villager coughs on $n$ random villagers, infecting them if they are not immune, and then becomes immune. Determine the value of $n$ that maximizes the expected survival time of the virus.",
"role":... | 1 |
[
{
"content": "Evaluate \\( \\lim_{x \\to 0} \\frac{P_n}{x} \\) where \\( P_n = a^{P_{n-1}} - 1 \\) for all \\( n = 2, 3, 4, \\ldots \\) and \\( P_1 = a^x - 1 \\), with \\( a \\in \\mathbb{R}^+ \\).",
"role": "user"
}
] | (\ln a)^n |
[
{
"content": "Evaluate $b+c+d$ given the integral:\n$$\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\frac{\\left(\\frac{\\pi}{2}-x\\right)\\cos x}{3+\\sin^2 x}\\mathrm{d}x= \\frac{\\pi^a}{b\\sqrt c+d}$$\nwhere $a$, $c$, and $d$ are positive integers.",
"role": "user"
}
] | 9 |
[
{
"content": "What is the probability that a group generated by two random invertible $2 \\times 2$ matrices $A$ and $B$ over the integers is isomorphic to the free group $F_2$?",
"role": "user"
}
] | 1 |
[
{
"content": "Suppose $P$ is an $n \\times n$ matrix such that $P^T P^3 = I$, where $I$ is the $n \\times n$ identity matrix. Determine all possible values of $\\det(P)$. \\( \\text{(A) } 1 \\quad \\text{(B) } -1 \\quad \\text{(C) } 0 \\quad \\text{(D) } \\text{None of these} \\)",
"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... | viroid |
[
{
"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... | 1.817 m |
[
{
"content": "Given a continuous function $f: [0, 1] \\rightarrow X$ with $f(0) = f(1)$, determine if there exists a continuous function $g: S^1 \\rightarrow X$ that represents the loop defined by $f$ but parameterized using the circle $S^1$ instead of the interval $[0, 1]$. Provide a justification for your ans... | 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... | 1885 N |
[
{
"content": "What is the sum of the values of $x$ that satisfy the equation $x^2-5x+5=9$?",
"role": "user"
}
] | 5 |
[
{
"content": "Determine whether the following statement is true or false: If $K$ is the algebraic closure of $F$, then as a ring, $K$ is integral over $F$. Provide a justification for your answer.",
"role": "user"
}
] | True |
[
{
"content": "Evaluate the line integral \\( \\oint_C \\left(x^2y - xy^2\\right) \\, dx + \\left(xy^2 - x^2y\\right) \\, dy \\), where \\( C \\) is the circle \\( x^2 + y^2 = a^2 \\), oriented counterclockwise.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{\\binom{n^2}{n}}{\\binom{2n^2}{n}} \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Let $f$ be an entire function such that $24 \\le |f'''(z)| \\le 30$ and $f(0) = f(1) = f(2) = 3$. Evaluate the integral \\( \\int_{|z|=5} \\frac{zf'(z)}{f(z)} \\, dz. \\)",
"role": "user"
}
] | 6\pi i |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 102.9 Hz |
[
{
"content": "The length of the diameter of this spherical ball is equal to the height of the box in which it is placed. The box is a cube and has an edge length of 30 cm. How many cubic centimeters of the box are not occupied by the solid sphere? Express your answer in terms of $\\pi$. [asy]\ndraw(Circle((7.5,... | 27000-4500\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... | 3.78e5 Pa |
[
{
"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... | 31 kJ/mol |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.316 nm |
[
{
"content": "Determine the coefficient \\(a\\) in the function \\(a\\cos(x)\\) such that the arc length of one complete cycle from \\(0\\) to \\(2\\pi\\) is equal to \\(2\\pi\\), given that \\(b < 1\\) or \\(b = 1\\).",
"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... | break-even point = fixed cost / (price per unit - variable cost per unit) |
[
{
"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... | 50mL |
[
{
"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 = -v0^2/(2D) |
[
{
"content": "Find the variance of the random variable \\( Z = Y^2 + Y + 2017 \\), where \\( X \\) is uniformly distributed over \\([-1, 3]\\) and \\( Y \\) is the distance from \\( X \\) to the nearest endpoint of \\([-1, 3]\\).",
"role": "user"
}
] | \dfrac{139}{45} |
[
{
"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... | 41.8 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... | {
"April": 52500,
"May": 76500,
"June": 89000,
"Total": 218000
} |
[
{
"content": "Suppose that $\\sec x+\\tan x=\\frac{22}7$ and that $\\csc x+\\cot x=\\frac mn,$ where $\\frac mn$ is in lowest terms. Find $m+n.$",
"role": "user"
}
] | 44 |
[
{
"content": "Solve for integer values of \\( x \\) in the equation: \\[ 3^{3x} - 3^x = (3x)! \\]",
"role": "user"
}
] | 2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Amortization Table:
| Date | Interest Payable | Amortization of Premium | Interest Expense | Carrying Amount |
|------------|-----------------|-----------------------|-----------------|----------------|
| Jan 1, 2014 | | | | $3,475,220 |
| Dec 31, 2014 | ... |
[
{
"content": "We have triangle $\\triangle ABC$ where $AB = AC$ and $AD$ is an altitude. Meanwhile, $E$ is a point on $AC$ such that $AB \\parallel DE.$ If $BC = 12$ and the area of $\\triangle ABC$ is $180,$ what is the area of $ABDE$?",
"role": "user"
}
] | 135 |
[
{
"content": "Find the expectation \\( E[e^{iu(X_{4/5}-\\frac{1}{2} X_{3/5})}|X_{3/5}] \\) where \\( X_t \\) is a Brownian bridge.",
"role": "user"
}
] | e^{-u^2/20} |
[
{
"content": "Let $a,b \\in \\mathbb{R}_+$ be real positive numbers with $\\frac{1}{2}<a<1$ and let $I=[0,b]$ be a closed real interval. Define the sequence of functions $f_n(x) : I \\to \\mathbb{R}$ by \\[ f_n(x) = { \\dfrac{n}{(2n-1)^{a+x}} - \\dfrac{n}{(2n)^{a+x}} } \\] and the sequence of functions $g_n(x) ... | 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... | 2.68e-7 keV |
[
{
"content": "Determine whether the sequence of integrals \\( \\int_{0}^{\\infty} g_n(x) \\, dx \\) converges, where \\( g_{n}(x) \\) is defined as follows: \\[ g_{n}(x)=\\begin{cases} \\frac{1}{n^2} & \\text{if } x \\in [0, n) \\\\ \\frac{-1}{n^2}x+\\frac{1}{n} & \\text{if } x \\in [n, n+1) \\\\ 0 & \\text{if ... | 0 |
[
{
"content": "Acme Corporation has released an alphabet soup in which each of the vowels (A, E, I, O, U) of the English alphabet appears five times (and the consonants do not appear at all). How many five-letter words can be formed from a bowl of Acme Vowel Soup? (Note: The words do not have to be actual words ... | 3125 |
[
{
"content": "Evaluate the limit: $$\\lim\\limits_{h\\rightarrow 0}{\\frac{1}{h} \\int\\limits_{3}^{3+h}{e^{t^2} dt}}$$",
"role": "user"
}
] | e^9 |
[
{
"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... | 0.130 |
[
{
"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... | 6.96 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... | 1752 kg |
[
{
"content": "Find the minimum value of the function \\( f(x) = 3x^2 - 6x + 2\\sqrt{x} + \\sqrt{3-2x} \\) without using differentiation.",
"role": "user"
}
] | 0 |
[
{
"content": "Solve the nonlinear ordinary differential equation \\( y''(x) + x^2 y'(x) + y(x)^3 = 0 \\) with the initial conditions \\( y(0) = 0 \\) and \\( y'(0) = 0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the value of $a$ such that the series \\( \\sum_{n=1}^{\\infty} n^{\\frac{1}{3}} \\left| \\sin\\left(\\frac{1}{n}\\right) - \\frac{1}{n^a} \\right| \\) converges.",
"role": "user"
}
] | 1 |
[
{
"content": "Let $K$ be a nonarchimedean local field with residue characteristic $p \\neq l$, and let $I_K$ be the inertia subgroup of its absolute Galois group. Consider an irreducible representation $V$ of $I_K$ over $\\overline{\\mathbb{F}}_l$ with finite image $I$. Suppose $V$ is primitive, meaning it is n... | No |
[
{
"content": "Find the minimum value of \\((\\tan C - \\sin A)^2 + (\\cot C - \\cos B)^2\\) given that \\((\\sin A, \\cos B)\\) lies on a unit circle centered at the origin, and \\(\\tan C\\) and \\(\\cot C\\) are defined.",
"role": "user"
}
] | 3 - 2\sqrt{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... | 2.4*10^-19 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... | 1.81% |
[
{
"content": "Let $A$ and $B$ be $C^*$-algebras that admit faithful representations on separable Hilbert spaces. Does the $C^*$ tensor product $A \\otimes_\\beta B$ also admit a faithful separable representation?",
"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... | 11.79% |
[
{
"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... | 1.81e7 |
[
{
"content": "Let $T$ be a self-adjoint trace-class operator on $L^2(\\mathbb{R})$. Can $T$ be represented as an integral operator with the kernel given by \\( k_T(x,y) = \\sum_{i=1}^\\infty \\lambda_i \\phi_i(x) \\bar\\phi_i(y) \\), where \\( \\{\\phi_i\\} \\) is an eigenbasis of $T$ and \\( T=\\sum_i \\lambda... | 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... | Electric Co.: 3 years
Water Works: 3 years |
[
{
"content": "Find the recurrence relation connecting $k(z+1)$ and $k(z)$ for the function $k(z) = \\Gamma\\left(\\frac{1}{2}+z\\right)\\Gamma\\left(\\frac{1}{2}-z\\right)\\cos{\\pi z}$. Evaluate $k(z)$ for a specific integer value to determine its value for any real integer $z$.",
"role": "user"
}
] | \pi |
[
{
"content": "Calculate the conditional expectation \\( E(Y|X) \\) given the joint probability density function \\( f_{XY}(x,y) = 2e^{-(x+y)} \\) for \\( 0 < x < y \\). The conditional density function is \\( f_{Y|X}(y|x) = e^{x-y} \\) for \\( 0 < x < y \\).",
"role": "user"
}
] | X + 1 |
[
{
"content": "Determine whether the following statement is true or false: If $X$ is a separable metric space, then the space of all continuous functions on $X$ with the supremum metric is also separable.",
"role": "user"
}
] | B |
[
{
"content": "Let $f(z)$ be an analytic function in the open unit disk $D = \\{z \\in \\mathbb{C} : |z| < 1\\}$. Given that $f(z) = 1$ for all $z$ with $\\text{Im}(z) = 0$ and $\\text{Re}(z) \\in [-\\frac{1}{2}, \\frac{1}{2}]$, determine the value of $f\\left(\\frac{1}{2}+\\frac{i}{2}\\right)$. Use the identity... | 1 |
[
{
"content": "Does there exist a Banach space \\((C,\\|.\\|)\\) with \\(C\\subset\\mathbb{R}^{\\mathbb{Z}}\\) such that for each \\(\\varepsilon>0\\), there exists \\(k\\in\\mathbb{N}\\) where if \\(v=(v_n)_{n\\in\\mathbb{Z}}\\in C\\) has \\(\\|v\\|\\leq1\\), then \\(|v_n|\\leq\\varepsilon\\) whenever \\(|n|\\g... | 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... | The Lewis structures for fulminic acid (HCNO) are shown below.
<img>/cimages/multimages/16/resonance_structures_of_fulminic_acid3616242726877843039.jpg</img> |
[
{
"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... | 9.5% |
[
{
"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... | -40 rad/s^2 |
[
{
"content": "For a given constant $b > 10,$ there are two possible triangles $ABC$ satisfying $AB = 10,$ $AC = b,$ and $\\sin B = \\frac{3}{5}.$ Find the positive difference between the lengths of side $\\overline{BC}$ in these two triangles.",
"role": "user"
}
] | 16 |
[
{
"content": "Calculate the Fourier coefficient $c_1$ for the function $f(x) = x \\cos(x)$ on the interval $(-\\pi, \\pi)$. The Fourier coefficients are given by the formula:\n\\[ c_n = \\frac{1}{2 \\pi} \\int_{-\\pi}^{\\pi} x \\cos(x) e^{-inx} \\, dx \\]",
"role": "user"
}
] | \dfrac{i}{4} |
[
{
"content": "Consider disks $s_1, \\cdots, s_n$ in the plane, and let $a_{ij}$ be the area of $s_i \\cap s_j$. Is the matrix $A = (a_{ij})$ always positive semidefinite? In other words, is it true that for any real numbers $x_1, \\cdots, x_n$, we have \\( \\sum_{i,j=1}^n x_ix_j a_{ij} \\geq 0 \\)?",
"role"... | Yes |
[
{
"content": "Find the largest constant \\( K > 0 \\) such that for any triangle with sides \\( a, b, c \\) and semiperimeter \\( s \\), the inequality \\( l_a^2 + l_b^2 + l_c^2 > K s^2 \\) holds, where \\( l_a, l_b, l_c \\) are the lengths of the angle bisectors of the triangle.",
"role": "user"
}
] | \dfrac{8}{9} |
[
{
"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... | discourage |
[
{
"content": "In a soccer match between Team A and Team B, the final score is a 9-9 tie. During the match, Team A had the lead at one point, and later, Team B also had the lead. How many different sequences of 18 goals can represent the course of the match, considering the order in which the goals were scored?"... | 38896 |
[
{
"content": "Find the least possible sum of the distances from a point E inside a rectangle ABCD to its vertices, given that these distances are all distinct integers.",
"role": "user"
}
] | 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... | $5,773.54 |
[
{
"content": "Find the minimum value of \\(a^2 + b^2\\) given that all roots of the polynomial \\(x^4 + ax^3 + bx^2 + ax + 1 = 0\\) are real.",
"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... | 1783 N |
[
{
"content": "Find the remainder when \\((177 + 10^{15})^{166}\\) is divided by 1003.",
"role": "user"
}
] | 234 |
[
{
"content": "What is the cardinality of a single equivalence class of Cauchy sequences in the rational numbers, corresponding to a single real number?",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "Given the character table for $S_3$:\n\n$$\\begin{array}{c|c|c|} \n & \\text{1} & \\text{(12)} & \\text{(123)} \\\\ \\hline\n\\text{$X_1$} & 1 & 1 & 1 \\\\ \\hline\n\\text{$X_2$} & 1 & -1 & 1 \\\\ \\hline\n\\text{$X_3$} & 2 & 0 & -1 \\\\ \\hline\n\\end{array}$$\n\nand the class function $f$ with v... | 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... | 1.14 * 10^24 |
[
{
"content": "Show that the infinite series \\( \\sum\\limits_{n=-\\infty}^{\\infty}\\frac{-1}{x+\\pi n + \\frac{\\pi}{2}} \\) is equivalent to \\( \\tan(x) \\).",
"role": "user"
}
] | \tan(x) |
[
{
"content": "Determine the last digit of the following expression for any natural numbers \\(n\\) and \\(m\\):\n\\[\n({}_{4n+1} C_0 )^{4m+1} + ({}_{4n+1} C_1 )^{4m+1} +({}_{4n+1} C_2 )^{4m+1} + \\cdots + ({}_{4n+1} C_{4n+1} )^{4m+1}.\n\\]",
"role": "user"
}
] | 2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | v_r = (v-v_0) -2((v-v_0).v_n)v_n, where v_r is the reflected ball velocity after collision, v is the ball velocity, v_0 is the obstacle velocity, and v_n is the impact normal vector. |
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