prompt list | ground_truth string |
|---|---|
[
{
"content": "Determine the dimension of the complex vector space generated by the set \\( E_n = \\{ e^{2 \\pi i k x} \\mid -n \\leq k \\leq n \\} \\).",
"role": "user"
}
] | 2n + 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0^+} \\frac{\\ln (x)+e^{-1/x}}{1/x^2 + x\\sin (-1/x^2)} \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Find all points of continuity for the function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) defined by \\[ f(x) = \\begin{cases} 2x & \\text{if } x \\in \\mathbb{Q} \\\\ x + 3 & \\text{if } x \\in \\mathbb{R} - \\mathbb{Q} \\end{cases} \\]",
"role": "user"
}
] | 3 |
[
{
"content": "Let \\( B = (B_t)_{t \\geq 0} \\) be a standard Brownian motion and define \\( M_t = 4B_t^2 + e^{4B_t -8t} -4t \\). Given that \\( M_t \\) is a martingale, find \\( \\mathbb{E}(M_\\sigma) \\) for \\( \\sigma = \\inf\\{t \\geq 0 : |B_t| = 1\\} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the range of values for the parameter \\(a\\) such that the cubic equation \\(2x^3 - 3x^2 - 12x + a = 0\\) has all roots real and distinct.",
"role": "user"
}
] | (-7, 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... | 46.5 cm^3 |
[
{
"content": "Two farmers agree that pigs are worth $\\$300$ and that goats are worth $\\$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $\\$390$ debt could be paid with two pigs, with one goat re... | \$30 |
[
{
"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... | $0.63 |
[
{
"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... | 3.61e-8 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... | L + (mg + sqrt(mg * (mg + 2k_s * L))) / k_s |
[
{
"content": "Determine the values of $n$ for which the Galois field with $2^n$ elements has a non-cyclic additive group.",
"role": "user"
}
] | n \geq 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... | 35.3 degrees |
[
{
"content": "Evaluate the integral \\( \\iint\\limits_R \\min\\{\\cos x, \\cos y\\} \\, dx \\, dy \\) over the region \\( R = \\{0 \\le x \\le \\frac{\\pi}{2},\\ 0 \\le y \\le \\frac{\\pi}{2}\\} \\).",
"role": "user"
}
] | \pi - 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... | $56,533.42 |
[
{
"content": "Consider the measure space $(\\mathbb{R},\\mathcal{B}(\\mathbb{R}),\\lambda)$ and a continuous function $g: \\mathbb{R} \\rightarrow \\mathbb{R}$. For $n \\in \\mathbb{N}$, define $g_n: \\mathbb{R} \\rightarrow \\mathbb{R}$ as $g_n(x) = g(x^n)\\mathbb{1}_{[0,1]}(x)$. Determine the limit $\\lim_{n\... | g(0) |
[
{
"content": "Find the radius of convergence for the power series \\( \\sum_{j=1}^\\infty \\frac{j!}{j^j}z^j \\).",
"role": "user"
}
] | e |
[
{
"content": "Determine the number of homomorphisms from the dihedral group \\(D_5\\) to the symmetric group \\(S_3\\).",
"role": "user"
}
] | 4 |
[
{
"content": "In $\\triangle XYZ$, we have $\\angle X = 90^\\circ$ and $\\tan Z = 7$. If $YZ = 100$, then what is $XY$?",
"role": "user"
}
] | 70\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... | 142 |
[
{
"content": "Evaluate the limit: $$ \\lim_{x \\to 0^+} \\frac{\\log x}{\\frac{1}{x+x^{x+1}}} $$",
"role": "user"
}
] | 0 |
[
{
"content": "Let \\(X\\) be a continuous random variable with the probability density function \\(f(x) = ke^{-x^2-7x}\\). Find the expected value \\(E[X^2]\\).",
"role": "user"
}
] | \dfrac{51}{4} |
[
{
"content": "Let \\( P \\) be an arbitrary point on the graph of the function \\( y = x + \\frac{2}{x} \\) for \\( x > 0 \\). From point \\( P \\), perpendiculars are drawn to the line \\( y = x \\) and to the y-axis, with the feet of these perpendiculars labeled as \\( A \\) and \\( B \\), respectively. Calcu... | -1 |
[
{
"content": "Let \\( f : \\mathbb{R} \\to (0, \\infty) \\) be a continuous function. Prove that there exists a real number \\( \\alpha \\) such that \\( f(\\alpha) = \\frac{\\alpha}{1-\\alpha} \\).",
"role": "user"
}
] | \alpha |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 0}\\frac{\\tan x-\\sin x}{x^3}$$",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Find the value of \\( \\lim_{n\\to\\infty}\\inf \\ f_n \\) where \\( f_n = \\mathbb{1}_{[n,n+1]} \\) is the indicator function that takes the value 1 on the interval \\([n,n+1]\\) and 0 elsewhere.",
"role": "user"
}
] | 0 |
[
{
"content": "Find the necessary and sufficient condition for the real numbers $a_1, a_2, \\dots, a_n$ such that the expression \\( \\sum_{k=1}^{n} a_k \\cot(kx) \\) admits a finite limit as \\( x \\to 0 \\). Compute this limit.",
"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... | 2.9666 |
[
{
"content": "Find the function \\( g(x) \\) that is implicitly defined by the relation: \\[ \\sqrt{x^2 - y^2} + \\arccos\\left(\\frac{x}{y}\\right) = 0, \\quad y \\neq 0. \\]",
"role": "user"
}
] | x |
[
{
"content": "An insurance policy covers two employees of a company, reimbursing no more than one loss per employee per year. The policy reimburses the full amount of the loss up to a company-wide maximum of $8000. The probability of an employee incurring a loss is 40%, independent of the other employee. Each l... | \dfrac{2}{51} |
[
{
"content": "Compute the limit: \\[ \\lim_{x\\rightarrow 0} \\left( \\frac{1}{x}\\ln (x!)\\right) \\]",
"role": "user"
}
] | -\gamma |
[
{
"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... | 8.043 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... | 7300 |
[
{
"content": "Let $A$ and $B$ be $n \\times n$ skew-symmetric matrices. Suppose there exist permutations $g, h: \\{1, \\ldots, n\\} \\to \\{1, \\ldots, n\\}$ such that $A_{i,j} = B_{g(i),h(j)}$ for all $1 \\leq i, j \\leq n$. Must there exist a permutation $f: \\{1, \\ldots, n\\} \\to \\{1, \\ldots, n\\}$ such ... | Yes |
[
{
"content": "Given that the alternating subgroup \\(\\mathbb{A}_n\\) has index 2 in the symmetric group \\(\\mathbb{S}_n\\), does it follow that every element \\(\\sigma \\in \\mathbb{S}_n\\) can be expressed as \\(\\sigma = \\widetilde{\\sigma}\\alpha_{\\sigma}\\), where \\(\\widetilde{\\sigma}\\) is a fixed ... | No |
[
{
"content": "Is the conditional expectation \\( E(X|Y) \\) \\( \\sigma(X) \\)-measurable, given that it is \\( \\sigma(Y) \\)-measurable?",
"role": "user"
}
] | No |
[
{
"content": "Find the element in the quotient ring $R/(x^3+x^2+x+1)$ whose image in $R/(x^2+1) \\times R/(x+1)$ is $(2x+1+(x^2+1), 3+(x+1))$. Use the Chinese remainder theorem for polynomials to determine this element.",
"role": "user"
}
] | 2x^2 + 2x + 3 |
[
{
"content": "In $\\triangle ABC$, given that $\\overrightarrow{AB} \\cdot \\overrightarrow{AC} = 7$ and $\\left|\\overrightarrow{AB} - \\overrightarrow{AC}\\right| = 6$, find the maximum area of $\\triangle ABC$. Express your answer as a single number.",
"role": "user"
}
] | 12 |
[
{
"content": "Determine the values of $n$ that satisfy the inequality:\n\n$$2(n-2) < Ap_n\\prod_{i=3}^n \\left(\\frac{p_i-2}{p_i}\\right)$$\n\nwhere $p_i$ denotes the $i$th prime number, and $A$ is defined by the relationship $p_n = 6A + r$ with $0 \\leq r < 6$. What are the possible values of $n$?",
"role"... | n \geq 8 |
[
{
"content": "Given the function \\( f(x) = ax + b \\) and the piecewise function \\( g(x) = \\begin{cases} \n f(x), & 0 \\leq x \\leq a \\\\\n f(f(x)), & x > a \n\\end{cases} \\), if for any positive number \\( a \\), the function \\( g(x) \\) is a strictly increasing function on the interval \\( (0, +\\infty)... | \dfrac{1}{4} |
[
{
"content": "Let \\((X, \\mathcal{T})\\) and \\((Y, \\mathcal{S})\\) be completely regular topological spaces. Suppose \\((X, \\mathcal{T})\\) is compact and \\(X \\cap Y\\) is dense in both \\(X\\) and \\(Y\\). Can we deduce that \\((Y, \\mathcal{S})\\) is compact?",
"role": "user"
}
] | No |
[
{
"content": "Is the function \\( Y(t) = |\\zeta(\\sigma_0 + it)| \\), where \\( \\sigma_0 > \\frac{1}{2} \\) and \\( \\zeta(s) \\) is the Riemann zeta function, an analytic function of the real variable \\( t \\)?",
"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... | 62.27% |
[
{
"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... | $709,789 |
[
{
"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... | $505.30 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\rightarrow\\infty} \\frac{\\int_0^x \\cos(t^2) dt - \\int_0^\\infty \\cos(t^2) dt}{x^3}$$",
"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... | 576.81 Hz |
[
{
"content": "Let \\( f_n: \\mathbb{R} \\to \\mathbb{R} \\) be a sequence of continuous functions converging uniformly on compact sets to \\( f: \\mathbb{R} \\to \\mathbb{R} \\). Let \\((X_n)_{n\\in\\mathbb{N}}\\) be a collection of independent and identically distributed random variables that are integrable. D... | 0 |
[
{
"content": "Evaluate the double integral \\( \\iint_D \\frac{1}{y}~dA \\), where the region \\( D \\) is bounded by the curves \\( y^3 = x^2 \\), \\( y^3 = 6x^2 \\), \\( y = 2x \\), and \\( y = 3x \\) in the first quadrant. Use the change of variables \\( u = \\frac{y^3}{x^2} \\) and \\( v = \\frac{y}{x} \\) ... | \dfrac{95}{648} |
[
{
"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... | 389.44 |
[
{
"content": "Determine the maximum value of \\( a \\) such that the function \\( a \\log(x+1) \\) does not exceed \\( \\sqrt{x} \\) for any \\( x \\geq 0 \\).",
"role": "user"
}
] | 1.242 |
[
{
"content": "Calculate the derivative of the scalar field \\(u=f(x, y, z)\\) at the point \\(M_{0}(x_{0}, y_{0}, z_{0})\\) along the line \\(l\\), defined by the system of equations:\n\n\\[\\begin{cases} f(x, y, z)=a & (a=\\text{const}) \\\\ \\varphi(x, y, z)=0 \\end{cases}\\]\n\nFind the derivative in the dir... | 0 |
[
{
"content": "Compute the limit \\( \\lim_{n \\rightarrow \\infty} \\binom{n}{n-k} p^n (1-p)^n \\) where \\( 0 \\leq p \\leq 1 \\) and \\( 0 \\leq n-k \\leq n \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the big O notation for the summation:\n\\[ \\sum_{i=j}^{n-1} \\left(\\frac{n}{i}\\right)^3\\left(1 - \\frac{i}{n} \\right)^2 \\]",
"role": "user"
}
] | O(n^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... | 565 |
[
{
"content": "An object occupies the solid region in the first octant bounded by the coordinate planes and the cylinders \\(x^2 + y^2 = 4\\) and \\(y^2 + z^2 = 4\\). If the charge density at any point is given by \\(x\\), calculate the total charge within this region.",
"role": "user"
}
] | \dfrac{3\pi}{2} |
[
{
"content": "Let $a$ and $b$ be positive integers such that the range of the function \\( y = \\frac{x^2 + ax + b}{x^2 + 2x + 3} \\) as $x$ varies over all real numbers is the interval $-5 \\leq y \\leq 4$. Find $a + b$. \\( \\boxed{} \\)",
"role": "user"
}
] | 23 |
[
{
"content": "Evaluate the double integral \\(\\int_{-\\sqrt{1-y^2}}^\\sqrt{1-y^2}\\left(\\int_0^1 \\frac{6}{(1+x^2+y^2)^2} \\, dx \\right)dy\\) by converting to polar coordinates. Determine the appropriate bounds for \\(\\theta\\) and \\(r\\).",
"role": "user"
}
] | \dfrac{3\pi}{2} |
[
{
"content": "Let $G$ be a connected linear algebraic group and $H$ be a closed subgroup of $G$ with codimension one. Is $H$ necessarily a normal subgroup of $G$?",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty}n(\\sqrt[n]{n}-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... | 14.82% |
[
{
"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... | 0.064 Am along the negative x-axis direction |
[
{
"content": "The midpoint of a line segment is located at $(3, -2)$. If one of the endpoints is $(1, 6)$, what is the other endpoint? Express your answer as an ordered pair.",
"role": "user"
}
] | (5, -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... | 4.71e11 K |
[
{
"content": "Find the residue of the function \\( f(z) = \\frac{e^z}{z^2} \\) at \\( z=0 \\).",
"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... | 41.5 yr |
[
{
"content": "Evaluate the limit: \\( \\lim_{x \\to 0^+} \\sqrt{\\tan x}^{\\sqrt{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... | The child has a 50% chance of having blood type A and a 50% chance of having blood type 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... | 8.34% |
[
{
"content": "Find the measure of the Cantor set, denoted as \\( C \\), using the measure function \\( l^* : 2^{\\mathbb{R}} \\to [0,+\\infty] \\) which satisfies the following properties:\n\n- \\( l^*(\\emptyset) = 0 \\)\n- \\( l^*(S) \\leq l^*(T) \\) if \\( S \\subseteq T \\)\n- \\( l^*(\\bigcup_{i=1}^\\infty... | 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... | 5HCOOH(aq) + 2MnO_4^-(aq) + 6H^+(aq) -> 5CO_2(aq) + 2Mn^{2+}(aq) + 8H_2O(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... | 6.20% |
[
{
"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... | Payment Date | Int. Paid | Int. Exp. | Disc. Amortization | Unamortized Disc. | Bond Carrying Amount
01/13|--|-|$258,188|$1,641812
June 30.13|$47,500|$56,106|$8,606|$249,582|1,650,418
Dec. 31,13|$47,500|$56,106|$8,606|$240,976|1659024
June 30,14|$47,500|$56,106|$8,606|$232,370|1667,630
Dec. 31,14|$47,500|$56,106|$8,606|$223,764|1,676,236 |
[
{
"content": "Determine whether the following statement is true or false: If $x, y, z$ are positive integers such that $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{z}$, then $hxyz$ is a perfect cube, where $h = (x, y, z)$. Provide a justification for your answer.",
"role": "user"
}
] | False |
[
{
"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... | rate = k[Cl_2(g)][CCl_3(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... | Cash Flow Statement (Indirect method)
Cash flow from operating activities:
Net income: $33,000
Depreciation: $18,000
Gain on disposal of furniture: -$4,000
Increase in accounts receivable: -$25,000
Increase in inventory: -$48,000
Increase in accounts payable: $25,000
Net cash used by operating activities: -$1,000
Cash flow from investing activities:
Proceeds from sale of furniture: $9,000
Net cash from investing activities: $9,000
Cash flow from financing activities:
Repayment of bank loan: -$14,000
Repurchase of common shares: -$10,000
Dividends paid: -$5,000
Net cash used in financing activities: -$29,000
Net decrease in cash: -$21,000
Cash at beginning of year: $18,000
Cash at end of year: -$3,000 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\rightarrow 0} \\frac{\\ln(1-x)+\\sin(x)}{x^2 e^x} \\]",
"role": "user"
}
] | -\dfrac{1}{2} |
[
{
"content": "Determine the values of $a$ for which the following limit evaluates to $-\\infty$:\n\\[ \\lim_{n\\to\\infty} \\sqrt{n^2-n+1}-an-b \\]",
"role": "user"
}
] | a > 1 |
[
{
"content": "Solve the congruence equation $x^5 \\equiv 4 \\pmod{7}$. Determine the integer value(s) of $x$ that satisfy this equation.",
"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... | 120 N |
[
{
"content": "Find the maximum value of the expression \\( \\frac{x^{100}}{1+x+x^2+\\ldots+x^{200}} \\) for positive \\( x \\).",
"role": "user"
}
] | \dfrac{1}{201} |
[
{
"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*10^4 m/s^2 |
[
{
"content": "In a commutative integral domain with unity, consider an irreducible element $a$ such that $a = bc$, where $b$ is an associate of $a$. Does it logically follow that $c$ must be a unit? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "In a triangle with an inscribed circle of radius \\( r = 1 \\) and one of its sides \\( a = 3 \\), find the minimum possible area of the triangle.",
"role": "user"
}
] | \dfrac{27}{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... | 13.5g |
[
{
"content": "Given a distributive and complemented lattice $(L, \\leq_L, 0, 1)$ that is dense as an order, and subsets $U, V \\subset L \\setminus \\{0\\}$ such that:\n\n1. $(U, \\leq_L)$ and $(V, \\leq_L)$ are totally ordered sets.\n2. $\\inf (U \\cup V) = 0$.\n3. For all $a, b \\in U \\cup V$, $\\inf (a, b) ... | Yes |
[
{
"content": "There are 100 countries participating in an olympiad. Each country is willing to communicate in exactly \\( n \\) languages. If each set of 20 countries can communicate in at least one common language, and no language is common to all 100 countries, what is the minimum possible value of \\( n \\)?... | 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... | 7771.17 |
[
{
"content": "Evaluate the line integral \\( \\int_{C}{x\\,\\mathrm{d}z - z\\,\\mathrm{d}y + y\\,\\mathrm{d}z} \\) where the curve \\( C \\) is parameterized by \\( \\gamma(t) = (\\cos t, \\sin t, 2t) \\) for \\( t \\in [0, 2\\pi] \\).",
"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... | 5.5e-3 T |
[
{
"content": "Let O be the origin (0, 0) and C the line segment defined by \\( (x, y) : x \\in [1, 3], y = 1 \\). Let K be the curve consisting of points \\( P \\) such that for some point \\( Q \\) on \\( C \\), \\( P \\) lies on the line segment \\( OQ \\) and the distance \\( PQ = 0.01 \\). Determine whether... | Less than 2 |
[
{
"content": "Consider a square, symmetric matrix $M$ of size at least $2 \\times 2$, with diagonal entries equal to $1$ and off-diagonal entries equal to $\\pm 1$. The matrix $M$ is indefinite, meaning there exists a nonzero vector $x$ such that $x^T M x = 0$. Is it always possible to choose such a vector $x$ ... | Yes |
[
{
"content": "Determine the number of continuous functions \\( f: [0,1] \\rightarrow \\mathbb{R} \\) that satisfy the equation \\[ \\int_{0}^{1} f(x) \\, dx = \\frac{1}{3} + \\int_{0}^{1} f^2(x^2) \\, dx. \\]",
"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... | $242,500 |
[
{
"content": "A player chooses one of the numbers 1 through 4. After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered 1 through 4. If the number chosen appears on the bottom of exactly one die after it is rolled, then the player wins $\\$1.$ If ... | -\frac{1}{16} |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty} x^3 e^{-ax^2} \\, dx \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given any solution \\(u(x, y)\\) to the partial differential equation \\(au_{x} + bu_{y} = 0\\), where \\(u(1, 2) = u(3, 6)\\), determine the value of \\(\\frac{b}{a}\\) based on the fact that \\(u(x, y)\\) is constant along lines defined by \\(bx - ay = c\\), for all \\(c \\in \\mathbb{R}\\).",
... | 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... | 34.5 rad/s |
[
{
"content": "Find the limit of the ratio of consecutive terms of the series \\( \\sum _{ n=1 }^{ \\infty }{ \\frac { \\left( 2n+2 \\right) ! }{ { 2 }^{ n+1 } } } { x }^{ n }\\) as \\( n \\) approaches infinity.",
"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... | 2.49e-12 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... | 6.52e-3 m |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.