prompt list | ground_truth string |
|---|---|
[
{
"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... | 0.00045N |
[
{
"content": "Can Kolya ensure a win in the following game? There is a pile of 100 stones on the table. Players take turns dividing a pile with more than one stone into two smaller piles, with Kolya going first. The winner is the player who leaves only piles with one stone each. Determine if Kolya can guarantee... | Yes |
[
{
"content": "Find a polynomial \\( f(x) \\in \\mathbb{Q}[x] \\) of degree at most 2 such that:\n1. \\( f(x) \\equiv 2 \\pmod{x+1} \\)\n2. \\( f(x) \\equiv x+1 \\pmod{x^2+1} \\)",
"role": "user"
}
] | x^2 + x + 2 |
[
{
"content": "Evaluate the limit: $$\\lim _{x\\to \\,1} \\frac{\\left(1-x^{1/2}\\right)\\left(1-x^{1/3}\\right)\\cdots \\left(1-x^{1/n}\\right)}{(1-x)^{n-1}}$$",
"role": "user"
}
] | \dfrac{1}{n!} |
[
{
"content": "Determine the order of the pole of the product of two holomorphic functions \\( h(z) \\) and \\( j(z) \\), where \\( h(z) \\) has a zero of order \\( m > 0 \\) and \\( j(z) \\) has a zero of order \\( n > 0 \\).",
"role": "user"
}
] | m + n |
[
{
"content": "Determine whether there exists a positive constant $c$ such that for any square matrix $A$ over any field, the inequality $|\\operatorname{rank}(J \\pm A) - \\operatorname{rank}(A)| \\leq c$ holds, where $J$ is a square matrix filled with ones.",
"role": "user"
}
] | 1 |
[
{
"content": "Find the largest negative integer $x$ which satisfies the congruence $34x+6\\equiv 2\\pmod {20}$.",
"role": "user"
}
] | -6 |
[
{
"content": "Evaluate the triple integral \\( \\int_{0}^{2} \\int_{0}^{\\sqrt{2x - x^2}} \\int_{0}^{1} z \\sqrt{x^2 +y^2} \\, dz\\,dy\\,dx \\) by converting it to cylindrical coordinates.",
"role": "user"
}
] | \dfrac{8}{9} |
[
{
"content": "Given the curve \\( \\gamma \\) and the branch of the complex logarithm function \\( f(z) = \\ln z \\) defined in the domain \\( \\mathbb{C} \\setminus \\gamma \\) with \\( f(1) = 0 \\), determine the value of its analytic continuation \\( f(-3) \\) when traversing a path that avoids the negative ... | \ln 3 + \pi i |
[
{
"content": "For a field $K$ with characteristic 0 that is complete with respect to a discrete valuation, is the absolute value on $K$ always non-Archimedean?",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( f:\\mathbb{R}\\to\\mathbb{R} \\) be a continuous function such that \\( \\frac{{C_1{{\\left| {\\sin \\left( {\\lambda x} \\right)} \\right|}^a}}} {{{{\\left| x \\right|}^b}}} \\leqslant \\left| {f\\left( x \\right)} \\right| \\leqslant \\frac{{C_2{{\\left| {\\sin \\left( {\\lambda x} \\rig... | No |
[
{
"content": "Evaluate the limit: \\( \\lim_{x \\to \\infty} \\frac{x^x}{e^x x!} \\)",
"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... | 835.7 fm |
[
{
"content": "A total of \\( n \\) people, each with a unique height, randomly take their seats around a circular table with \\( n \\) chairs. Calculate the expected number of people who are shorter than both of their immediate neighbors.",
"role": "user"
}
] | \dfrac{n}{3} |
[
{
"content": "Given the input signal $x(t) = \\cos(t)$ and impulse response $h(t) = u(t)$, find the output $y(t)$ using Fourier transforms. Express the final time-domain solution $y(t)$.",
"role": "user"
}
] | \sin(t) |
[
{
"content": "Let \\(X_1, X_2, \\ldots\\) be an i.i.d. sequence of random variables with \\(E[X_1] = 1/2\\) and \\(\\operatorname{Var}[X_i] = 2\\). Compute the probability:\n\\[ P\\left(\\lim_{n \\to \\infty} \\frac{X_1 + X_2 + \\cdots + X_n}{n} > 1\\right) \\]",
"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... | $4,411.94 |
[
{
"content": "Determine whether the limit \\( \\lim_{n \\to \\infty} \\frac{n^3(\\sqrt{2}+(-1)^n)^n}{3^n} \\) exists. If it does, find its value.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\int_{S} (x + 2y - z) \\, dV \\) over the tetrahedron \\( S \\) with vertices \\((0,0,0), (1,2,3), (0,1,2), (-1,1,1)\\). You may use a suitable linear transformation as a coordinate change.",
"role": "user"
}
] | \dfrac{1}{6} |
[
{
"content": "Compute the definite integral \\( \\int_{\\mathbb{R}^2} e^{-x^2 - y^2} \\, dx \\, dy \\).",
"role": "user"
}
] | \pi |
[
{
"content": "Let $R$ be a ring and fix $a \\in R$. If $ab = 0$ for all $b \\in R$, does it necessarily follow that $a = 0$? Here, $0$ denotes the unique element such that $0 + b = b$ for all $b \\in R$. Answer yes or no and justify your answer.",
"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... | 279 |
[
{
"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 width of the box |
[
{
"content": "A piece is placed in the lower-left corner cell of a $15 \\times 15$ board. It can move to cells that are adjacent either by sides or corners of its current cell. The piece must alternate between diagonal and horizontal moves, starting with a diagonal move. What is the maximum number of moves the ... | 224 |
[
{
"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... | 58.11 rad |
[
{
"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... | int spaces_to_move(LL* R, std::string word_to_find)
{
int i = 0;
while(R->w != word_to_find){
R = R->next;
i++;
if(!R){
std::cout << "Word not found\n";
exit(0);
}
}
return i-R->n;
} |
[
{
"content": "2018 politicians gathered at a summit, where every pair planned to hold a private discussion. It was discovered that among any four politicians, there is one who has already talked with the other three. What is the greatest number of discussions yet to be held?",
"role": "user"
}
] | 2017 |
[
{
"content": "Find the limit of the sequence defined by the integral:\n\\[ u_n = \\int_0^1 \\frac{1}{1+t+t^2+\\cdots+t^n} \\, dt \\]",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Find the maximum and minimum values of the function \\( f(x,y,z) = x^4 + y^4 + z^4 \\) subject to the constraints \\( x^2 + y^2 + z^2 = 1 \\) and \\( x + y + z = 0 \\) using the method of Lagrange multipliers.",
"role": "user"
}
] | \dfrac{1}{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... | 280 A |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $3,250 |
[
{
"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... | 1.32 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... | 3.67% |
[
{
"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... | 207.9ms^-1 |
[
{
"content": "Let $A$ be a real $n \\times n$ matrix with rank $r$. Find the dimension of the vector space $V$ consisting of all real $n \\times n$ matrices $X$ such that $AX=O$, where $O$ is the $n \\times n$ zero matrix.",
"role": "user"
}
] | n(n - r) |
[
{
"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... | -$3926.80. Since the net present value is negative, I should not go forward with the project. |
[
{
"content": "A pack of 20 cards is shuffled using the \"interlacing\" method, where the original order 1, 2, 3, ..., 20 becomes 1, 11, 2, 12, ..., 10, 20 after one shuffle. Determine how many times this shuffle must be repeated for the cards to return to their original order.",
"role": "user"
}
] | 18 |
[
{
"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... | Option 1 | Option 2 | Net Income Increase (Decrease) | Sunk (S)
------- | -------- | -------- | --------
Revenues | $81,000 | $74,000 | $(7,000)
Maintenance Expenses | $6,200 | $6,200 | $0
Operating Expenses | $32,200 | $27,300 | $4,900
Equipment Upgraded | $0 | $0 | $0 (S)
Opportunity Cost | $0 | $5,000 | $5,000
**Total** | | | **$2,900** |
[
{
"content": "In the definition of a normal linear operator on a finite-dimensional inner product space, is it necessary for the ordered basis $B$ to be orthonormal? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "Determine whether it is possible to construct a homotopy between the maps $A$ and $I$ on the 3-sphere $S^3$ in the form $F: (z_1, z_2, t) \\mapsto (F_1 (z_{1}, z_2, t),F_2 (z_1, z_{2}, t) )$, where $z_1$ and $z_2$ are complex numbers satisfying $|z_1|^2 + |z_2|^2 =1$. Provide a justification for y... | Yes |
[
{
"content": "Given the quadratic function $f(x) = ax^2 + bx + c$, where $a < b$ and $f(x) \\geq 0$ for all $x \\in \\mathbb{R}$, find the minimum value of the expression $\\frac{a + b + c}{b - a}$.",
"role": "user"
}
] | 3 |
[
{
"content": "Evaluate the Riemann-Stieltjes integral: \\[ \\int_0^6 (x^2 + [x]) \\, d(|3-x|) \\] where \\([x]\\) denotes the floor function. The solution is expected to be 63.",
"role": "user"
}
] | 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... | Debt: 24.05%, Equity: 75.95% |
[
{
"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... | 19 L/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... | $0.93 |
[
{
"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... | 10.95% |
[
{
"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... | Economic feasibility |
[
{
"content": "Let $f(x)$ be a real-valued function defined on the interval $(-5, 5)$ such that \\[ e^{-x}f(x) = 2 + \\int\\limits_0^x (t^4 + 1)^{1/2} \\ dt \\] for all $x \\in (-5, 5)$. If $f^{-1}(x)$ is the inverse function of $f(x)$, find \\((f^{-1})^{'}(2)\\).",
"role": "user"
}
] | \dfrac{1}{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... | 150 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... | 3.65 times |
[
{
"content": "Determine whether there exists a linear algebraic group $G$ over an algebraically closed field $k$ such that all semisimple elements with infinite order lie in the identity component $G^0$. Consider the case where $G$ is not connected, and $1_G \\neq s \\in G$ is semisimple. The identity component... | 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... | 3.47 years |
[
{
"content": "Consider a square in chain complexes over a field $k$:\n\\[\n\\begin{array}{ccc}\nA & \\xrightarrow{f} & B \\\\\n\\downarrow{g} & & \\downarrow{h} \\\\\nC & \\xrightarrow{k} & D\n\\end{array}\n\\]\nIf $A$ is chain equivalent to a direct sum of chain complexes $\\bigoplus_i A_i$, where each summand... | 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... | 19.5 m/s |
[
{
"content": "Consider a finite abelian group $G$ and two non-empty subsets $H$ and $K$ of $G$ that are not disjoint. Does the relation $|HK| = \\frac{|H|\\cdot|K|}{|H\\cap K|}$ hold for these subsets?",
"role": "user"
}
] | No |
[
{
"content": "Given the joint probability density function of random variables \\(X\\) and \\(Y\\) as \\(f_{1,2}(x,y) = e^{-y}\\mathcal{I}_{(0<x<y<\\infty)}\\), find the numerical value of \\(P(X+Y \\le 1)\\).",
"role": "user"
}
] | 0.1548 |
[
{
"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... | 4.94e-6 W |
[
{
"content": "Given the function \\( f(x) = x^2 + 2f\\left(\\frac{x}{2}\\right) - f\\left(\\frac{x}{4}\\right) \\), find the value of \\( f(3) \\).",
"role": "user"
}
] | 16 |
[
{
"content": "Is it possible to construct a non-surjective homomorphism from a group of order 6 to a group of order 3?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the generator of the kernel of the mapping \\( \\varphi : \\mathbb{R}[x] \\rightarrow \\mathbb{C} \\) defined by \\( f(x) \\mapsto f(2+i) \\).",
"role": "user"
}
] | x^2 - 4x + 5 |
[
{
"content": "Given an affine real algebraic variety $X$ defined as the zero locus of $k$ degree-two polynomials in $\\mathbb{R}^N$, and a map $F : X \\to \\mathbb{R}$ that projects onto one of the coordinates, determine an asymptotic upper bound on the number of critical values of $F$ in terms of $k$ and $N$. ... | e^{O(N)} |
[
{
"content": "Find the radius of convergence, \\( R \\), for the power series \\( \\sum_{n = 0}^{\\infty} a_n x^n \\), where \\( a_0 = 0 \\) and \\( a_n = \\frac{\\sin(n!)}{n!} \\) for \\( n \\geq 1 \\).",
"role": "user"
}
] | \infty |
[
{
"content": "Evaluate the integral \\( \\iint_{S} \\min(x,y) \\, dx \\, dy \\) over the region \\( S = [0,1] \\times [0,1] \\subset \\mathbb{R}^2 \\).",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "In a convex pentagon where all sides are equal and four out of the five diagonals are also equal, determine if the pentagon must be regular (i.e., all sides and angles are equal).",
"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... | $16.74 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 17.5 mi/h |
[
{
"content": "Find the sum of the digits of the square root of the number formed by 2018 digits of 4, followed by 2017 digits of 8, and ending with a single digit of 9.",
"role": "user"
}
] | 12109 |
[
{
"content": "Compute the square of 9997 without a calculator.",
"role": "user"
}
] | 99940009 |
[
{
"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... | 1.78 |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to 0^-} \\left(\\frac{\\cosh x -1}{x}\\right)^x$$",
"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... | 82.278 kg |
[
{
"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... | Hoop > Hollow sphere > Solid disk = Solid cylinder > Solid sphere |
[
{
"content": "Determine the number of triples of natural numbers \\((a, b, c)\\) that satisfy the following system of equations:\n\n\\[\\begin{cases}\n\\gcd(a, b, c) = 15 \\\\\n\\text{lcm}(a, b, c) = 3^{15} \\cdot 5^{18}\n\\end{cases}\\]",
"role": "user"
}
] | 8568 |
[
{
"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... | 8.7 \mu C and -0.3 \mu C |
[
{
"content": "Determine the number of solutions to the congruence \\(x^{p+1} \\equiv 1 \\mod p^{2017}\\) in the set \\(\\{0, 1, \\ldots, p^{2017} - 1\\}\\), where \\(p\\) is a prime number greater than 2.",
"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... | 0.3 N/m |
[
{
"content": "Find the maximum value of \\( M \\) such that for all real numbers \\( a, b, c \\), there exists a point \\( x_0 \\) in the interval \\([0, 4]\\) where \\(|x_0^3 + ax_0^2 + bx_0 + c| \\geq M\\).",
"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... | -2643.125 |
[
{
"content": "The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of the possible values of $a?$\n$\\textbf{(A)}\\ 7\\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ 16\\qquad\\textbf{(D)}\\ 17\\qquad\\textbf{(E)}\\ 18$",
"role": "user"
}
] | 16 |
[
{
"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... | 295 J |
[
{
"content": "Determine the number of functions \\( g \\) from the set \\( A = \\{1, 2, 3, 4, 5, 6, 7\\} \\) to itself such that \\( g(g(g(x))) \\) is a constant function. Find the remainder when this number is divided by 1000.",
"role": "user"
}
] | 249 |
[
{
"content": "Determine the residues at all isolated singularities of the function \\( f(z) = \\tan^3(z) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the values of $a$ for which the equation $1+(10K+4)^2+(10M+8)^2=(10N+a)^2$ has no solution, and the equation $1+(10K+4)^2+(10M+8)^2=(10N+a+8)^2$ has exactly one solution.",
"role": "user"
}
] | 3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 12267.01 |
[
{
"content": "Determine if it is possible to assign values to the positive integers \\(A, B, C, x, y, z, p, q, r\\) such that the following conditions are satisfied:\n1. \\(x + y + z = p + q + r\\)\n2. \\(Ax + By + Cz = Ap + Bq + Cr\\)\n3. At least one of the following is true: \\(x \\neq p\\), \\(y \\neq q\\),... | Yes |
[
{
"content": "Determine the smallest number of colors needed to color the edges of any directed simple graph with \\( n \\) vertices such that there is no monochromatic cycle.",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty} \\frac{x^2}{(x^2+4)(x^4-1)}dx \\) using the Residue Theorem.",
"role": "user"
}
] | \dfrac{\pi}{30} |
[
{
"content": "Let \\( s(n) \\) denote the number of positive divisors of a positive integer \\( n \\). Determine the largest prime divisor of the sum of \\( (s(k))^3 \\) for all positive divisors \\( k \\) of \\( 2014^{2014} \\). \\[ \\textbf{(A)}\\ 5 \\qquad\\textbf{(B)}\\ 7 \\qquad\\textbf{(C)}\\ 11 \\qquad\\... | E |
[
{
"content": "Let $\\omega$ be a nonreal root of $z^3 = 1.$ Find the number of different possible values of\n\\[(\\omega + 1)^n,\\]where $n$ is a positive integer.",
"role": "user"
}
] | 6 |
[
{
"content": "Let \\( y(x) \\) be a function satisfying the integral equation:\n\\[\ny(x) = 1 + 2 \\int_{0}^{x} u^3 y(u) \\, du\n\\]\nDetermine if it is true that \\( y(\\sqrt{2}) < 4 \\).",
"role": "user"
}
] | False |
[
{
"content": "Evaluate the limit: $$\\lim_{a\\to0}\\int\\limits_{-x}^x\\frac{e^t\\cdot(1-\\sin^2(at))}{1+e^t}dt$$",
"role": "user"
}
] | 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... | $272,637.80 |
[
{
"content": "Determine the degree of the differential equation \\( \\left(\\frac {d^2y}{dx^2}\\right)^3-\\sqrt{\\frac {dy}{dx}+2}=0 \\) after making it rational and integral in all its derivatives.",
"role": "user"
}
] | 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... | 18700 |
[
{
"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... | $3002 |
[
{
"content": "Find the minimum value of \\( f = |5x^2 + 11xy - 5y^2| \\) for integers \\( x \\) and \\( y \\) such that \\( x \\) and \\( y \\) are not both zero.",
"role": "user"
}
] | 5 |
[
{
"content": "A cube with a side length of 10 is divided into 1000 smaller cubes, each with an edge length of 1. A number is written in each small cube such that the sum of the numbers in each column of 10 cubes (in any of the three directions) equals 0. In one of these cubes, labeled A, the number 1 is written... | -1 |
[
{
"content": "Determine the number of ways to consider \\(\\Bbb Q\\) as an \\(\\Bbb R\\)-module.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the maximum order of an element in the group $SL_2(\\mathbb{Z} / p \\mathbb{Z})$ for an odd prime $p$, given that the characteristic polynomial $P_A(x) = x^2 - \\text{tr}(A)x + 1$ has no roots in $\\mathbb{Z} / p \\mathbb{Z}$. Provide your answer as a single integer.",
"role": "user"... | p+1 |
[
{
"content": "Does the structure theorem for finitely generated abelian groups apply to the empty product of abelian groups?",
"role": "user"
}
] | Yes |
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