inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $A \in \mathbb{R}^{n \times p}$ and $B \in \mathbb{R}^{n \times d}$ be matrices such that both have a spectral norm (largest singular value) equal to 1. If we concatenate the columns of $A$ and $B$ to construct $M = [A|B]$, determine the spectral norm of $M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Determine the number of distinct partitions of the set \(\{1,2,3, \ldots , 100\}\) that satisfy the following conditions: (a) The partition consists of exactly three non-empty parts, and (b) the elements 1, 2, and 3 are each in different parts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{97}}
Find a real number \( \lambda \) such that \((A+B)^{-1} \leq (A+\lambda I)^{-1}\), where \(A\) and \(B\) are real symmetric non-singular matrices and \(A+B\) is non-singular.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda_{\min}(B)}
Determine the splitting field of the polynomial \(x^3 + x^2 + 1\) over the real numbers \(\Bbb R\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{C}}
Find the covariance of the Gaussian random variables $x[n]$ and $y[n]$ in the AR(1) model, given that $w[n]$ and $z[n]$ are independent random variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the value of the expression \( \cot^{-1}(\tan 2x) + \cot^{-1}(-\tan 3x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi + x}
Let \(\mathbb{F}_9\) be the field with nine elements. Determine the number of triples \((a, b, c)\) of elements in \(\mathbb{F}_9\) such that \(a^2 + b^2 = c^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
Determine all values of the parameter \(a\) for which the inequality \(3 - |x-a| > x^2\) is satisfied by at least one negative \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( -\frac{13}{4}, 3 \right)}
Calculate the spectral radius of the complete tripartite graph $K_{4,4,12}$. The spectral radius is the largest eigenvalue of the graph's adjacency matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the cardinality of the quotient ring \( \mathbb{Z}[x]/(x^2-3,2x+4) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Two water particles fall freely in succession from a $300 \mathrm{~m}$ high cliff. The first one has already fallen $\frac{1}{1000} \mathrm{~mm}$ when the second one starts to fall.

How far apart will the two particles be at the moment when the first particle reaches the base of the cliff? (The result should be calculated to the nearest $\frac{1}{10} \mathrm{~mm}$. Air resistance, etc., are not to be considered.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["34.6"]}
Consider four positive integers \(a_1, a_2, a_3, a_4\), each less than 45. Determine the number of different permutations \((a_1, a_2, a_3, a_4)\) such that \(a_1 + a_2 + a_3 + a_4 < 90\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1845646}
Given a matrix $A \in M_{2\times 2}(\mathbb{R})$ that satisfies the equation $A + A^{-1} = I_2$, calculate the sum $I_2 + A + A^2 + \cdots + A^{2019}$. Provide your answer as a single matrix expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2A - I_2}
Determine the cardinality of the set $X$, where $X$ is the collection of all relations $R$ over the set of natural numbers $\mathbb{N}$ such that the transitive closure of $R$, denoted by $R^{*}$, equals $\mathbb{N\times\mathbb{N}}$. Provide your answer as a cardinal number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
In an experiment where a fair coin is tossed an infinite number of times, determine the probability that the sequence of heads and tails will never repeat itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A company has a manager and $n \geq 3$ employees. The manager flips a coin $2n$ times: each time he flips heads, he gives one task to one of his employees, chosen randomly and uniformly. Let $X_1$ and $X_2$ be the number of tasks given to employee 1 and employee 2, respectively. Find the covariance $\text{Cov}(X_1, X_2)$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2n}}
In the quadrilateral pyramid \(PABCD\), the lateral face \(PAD\) is an equilateral triangle and is perpendicular to the base \(ABCD\). The base \(ABCD\) is a rhombus with side length 2, and \(\angle BAD = 60^\circ\). Let \(N\) be the midpoint of \(PB\), and the cross-section \(DAN\) intersects \(PC\) at \(M\). Find the dihedral angle between the planes \(P-AD-M\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Find the orthogonal trajectories of the family of curves given by the equation \( y^2 = x^2(1 - cx) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + 3y^2 = Cy}
Determine the degree of the field extension \( \mathbb{Q}(\xi+\xi^5+\xi^8) / \mathbb{Q} \), where \( \xi = \cos \left(\frac{2\pi}{13}\right) + i\sin \left(\frac{2\pi}{13}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Evaluate the limit: $$\lim_{\epsilon\to 0^+} \int_{\Lambda/s}^\Lambda \int_{-\infty}^\infty \frac{i}{y^2-x^2+i\epsilon} \, dy \, dx$$ where \(s\) and \(\Lambda\) are positive constants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \ln s}
Determine the number of ways to tile a $3 \times n$ rectangle using $2 \times 1$ rectangles and $1 \times 1$ squares, given that no two $2 \times 1$ rectangles share a common point and each $2 \times 1$ rectangle is placed with its longer side parallel to the shorter side of the $3 \times n$ rectangle. Assume $n > 3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2^{n+2} - (-1)^n}{3}}
Find a non-zero function $f(x)$ such that $\frac{d}{dx}f(x) = f(x-1)$. Provide the function $f(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{W(1) x}}
Determine the maximum possible order of an element in the matrix group \({\rm GL}_m(\mathbb{F}_p)\), where \(m\) and \(p\) are positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^m - 1}
Calculate the Euler characteristic of a surface formed by joining two projective planes, each with two punctures, along the punctures.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Determine the number of zeros of the equation \( ze^{3-z} - 5^2 = 0 \) inside the unit disk.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the improper integral: $$\int_{0}^\infty \int_{-\infty}^{\infty} \mathrm{exp}[-((y^2+z^2)/2)]\,dy\,dz$$ using the transformation $y=r \cos \theta$ and $z=r \sin \theta$. What is the value of the integral?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
In parallelogram \( ABCD \), point \( F \) is on side \( AB \) and point \( H \) is on the extension of side \( BC \) such that \( \frac{AB}{BF} = \frac{BC}{BH} = 5 \). Point \( G \) is chosen so that \( BFGH \) forms a parallelogram. If line \( GD \) intersects \( AC \) at point \( X \) and \( AC = 100 \), find the length of \( AX \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
A teacher has \( n \) students and divides them into groups, assigning one student as president and another as vice president in each group. Determine the exponential generating function for the number of ways \( t_n \) this can be done.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{x^2 e^x}}
Determine the denominator of the coefficient of the integral of the polynomial \((x-\alpha)^5(x-\beta)^4\) with respect to \(x\), given that \(\beta > \alpha\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2520}
Evaluate the double integral \(\displaystyle \iint_R (x^2+y^2) \, dx \, dy\), where \(R\) is the region in the first quadrant bounded by the curves \(xy=2\), \(xy=4\), \(x^2-y^2=3\), and \(x^2-y^2=5\). Use the transformation \(u=xy\) and \(v=x^2-y^2\) to express \(x^2+y^2\) in terms of \(u\) and \(v\), and evaluate the integral over the region \(1 \le u \le 3\), \(1 \le v \le 5\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A trapezoid has one angle measuring $60^{\circ}$. If a circle can be inscribed in the trapezoid and another circle can be circumscribed around it, find the ratio of the lengths of its bases.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the values of $n$ for which the parametric equations \( \begin{pmatrix} X \\ Y \end{pmatrix} = \begin{pmatrix} \sum_{i=1}^N \frac{\cos (it)}{i^n} \\ \sum_{i=1}^N \frac{\sin (it)}{i^n} \end{pmatrix} \) are divergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \leq 1}
Find the base $b$ such that for any $\epsilon > 0$, the following asymptotic behavior holds for the function \( f_n(2,2) = \sum_{k=0}^n \binom{2n}{k} 2^k \):
\[ \frac{f_n(2,2)}{(b+\epsilon)^n} \to 0 \quad \text{and} \quad \frac{f_n(2,2)}{(b-\epsilon)^n} \to \infty \quad \text{as} \quad n \to \infty. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Consider the function \(f:\mathbb{R^3} \to \mathbb{R}\) defined by \(f(x, y, z)=\left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^2-4 R^{2}\left(x^{2}+y^{2}\right)\). Let \(Z=\{ (x,y,z) \in \mathbb{R^3} \mid f(x,y,z)=0\}\). For \(R=3\) and \(r=2\), find the surface area of \(Z\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24\pi^2}
Find the minimum value of the sum of the lengths of line segments connecting the midpoints of the hypotenuses of $n$ right triangles to a common point, given that the sum of the areas of the triangles is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 - 1) = P(x)P(-x)$. Determine the maximum possible number of real roots of $ P(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4
}
Determine the value of the supremum $A = \sup \limits_{f} \{m^*(f[0, 1])\}$, where $f$ are all functions satisfying the following conditions: $f: [0, 1] \rightarrow [0, 1]$; for all $x$, $f(x) - x \in \mathbb{Q}$; and for all $x, y$, if $x - y \in \mathbb{Q}$, then $f(x) = f(y)$. Find the value of $A$, given that $0 < A \leq 1$. \(m^*\) denotes the outer measure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the system of differential equations \( \frac{dx}{dt} = -x - y \) and \( \frac{dy}{dt} = 2x - y \), suppose there is an error of magnitude \(10^{-4}\) in measuring the initial conditions \(x(0)\) and \(y(0)\). Determine the largest error that can occur when evaluating \(x(t)\) and \(y(t)\) for \(0 \leq t < \infty\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^{-4}}
Given that \(X\) and \(Y\) are jointly continuous random variables with conditional density \(f_{X|Y}(x|y) = I_{(x,x+1)}(y)\) and marginal density \(f_{X}(x) = I_{(0,1)}(x)\), find the probability \(P[X + Y < 1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Compute the value of the exterior $2$-form
$$\omega = (x_1 + x_2)e_1^* \wedge e_2^* + (x_2 + x_3)e_2^* \wedge e_3^* + \cdots + (x_{n-1} + x_n)e_{n-1}^* \wedge e_n^*$$
onumber
$$\text{on the pair of vectors } (x_1e_1 + \cdots + x_ne_n, e_1 + \cdots + e_n),$$
where $x_1, \ldots, x_n$ are variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x_1^2 - x_n^2}
Seventeen people are seated at the vertices of a large circular table, with a constant distance between each pair of neighbors. After an hour, they are rearranged randomly. Choose any two people. Let \( X \) be the positive difference between their distances before and after the rearrangement. Calculate the expected value of \( X \). Note that the maximum distance is 8.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{8}}
In a circle, there are \(a+b+c\) ducks, each choosing one of rock, paper, or scissors. Specifically, \(a\) ducks choose rock, \(b\) ducks choose paper, and \(c\) ducks choose scissors. A move occurs when:
- A rock duck is behind a scissors duck, and they switch places.
- A paper duck is behind a rock duck, and they switch places.
- A scissors duck is behind a paper duck, and they switch places.

Find the maximum number of moves that can occur, expressed in terms of \(a\), \(b\), and \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ab + bc + ca}
Evaluate the limit: $$\lim_{n\to\infty} 1+\sqrt{2+(3+(4+\dots+(n)^{\frac{1}{n}})^{\frac{1}{4}})^{\frac{1}{3}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the integral \( \iint_R \sin\left(\frac{y-x}{y+x}\right)\, dy\, dx \) using the substitution \( x = u+v, \ y = u-v \). The region \( R \) is defined by \( \{(x,y): x+y \leq \pi, \ y \geq 0, \ x \geq 0\} \). Determine the new bounds for \( u \) and \( v \) after the substitution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the surface integral \( \int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S} \) using Stokes' Theorem, where \( \vec{F}(x,y,z) = y^2\vec{i} + 2xy\vec{j} + xz^2\vec{k} \) and \( S \) is the surface of the paraboloid \( z = x^2 + y^2 \) bounded by the planes \( x=0, y=0 \), and \( z=1 \) in the first quadrant, with the surface pointing upward.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{7}}
Given the equation \(\frac{d}{114}=\left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right) \cdots\left(1-\frac{1}{c^{2}}\right)\), find the value of \(d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{58}
Suppose $a$ is a 28-digit palindromic number. Given that $a$ is a multiple of $13$ and all the digits except the 13th, 14th, 15th, and 16th are $1$. Let $A, B, C, D$ be the 13th, 14th, 15th, and 16th digits respectively. Find the minimum value of $A+B+C+D$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose that the bivariate random variable \((X,Y)\) has the joint density function:
\[
f_{X,Y}(x,y)=\left\{
 \begin{array}{ll}
  1  & \mbox{when } y > 0 , x+y<1 \text{ and } y<x+1 \\
  0 & \mbox{otherwise }
 \end{array}
\right.
\]
Find \(\operatorname{Cov}(X,Y)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the number of ways to rearrange the sequence "1234123567" such that no two identical numbers are adjacent to each other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{236880}
The area of trapezoid \(ABCD\) is \(S\), and the ratio of the bases \(\frac{AD}{BC} = 3\). On a line intersecting the extension of the base \(AD\) beyond point \(D\), there is a segment \(EF\), such that \(AE \parallel DF\), \(BE \parallel CF\), and \(\frac{AE}{DF} = \frac{CF}{BE} = 2\). Find the area of triangle \(EFD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{S}{4}}
Determine the size of the set generated by an unrestricted number of operations on the elements of the set \( F = \{c, x, \exp(x), \ln(x) \} \), where \( c \) is a constant term. The operations allowed are addition \( f+g \), multiplication \( f\cdot g \), division \( f/g \), and composition \( f\circ g \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Evaluate the cumulative distribution function (CDF) of the standard normal distribution at 0.5365, i.e., find \( N(0.5365) \), where \( N(x) \) denotes the CDF of the standard normal random variable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.7042}
Find the limit: \[ \lim_{n \rightarrow \infty} \int_0^{1} \min\left( \frac{e^{-nx^{2}}}{\sqrt{x}}, n \right) \, d\lambda \] where \( \lambda \) is the Lebesgue measure on the interval \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( B(t) \) be a standard Brownian motion. Evaluate the conditional probability \( P(B(3) \leq 3 \mid B(1) = 1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Phi(\sqrt{2})}
Let \(\alpha =2^{1/5} \) and \(\zeta=e^{2i\pi/5} \). Let \(K=\mathbb {Q}(\alpha\zeta) \). How many field automorphisms does \(K\) have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the largest number of solid $2\text{ in } \times 2\text{ in } \times 1\text{ in}$ blocks that can fit in a $3\text{ in x } 2\text{ in x } 3\text{ in}$ box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4
}
Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \left(e^{(1+\frac{k}{n})^2} - \frac{3e^{(1 + \frac{3k}{n})}}{2\sqrt{1 + \frac{3k}{n}}}\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the cohomology groups $H^q(G, M)$ for a free group $G$ with $n$ free generators and a trivial $G$-module $M$, where $q \geq 3$. Provide your answer in terms of $n$ and $M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of different ways to color a 5×5 chessboard using 5 distinct colors such that each row contains all 5 colors and no two adjacent squares have the same color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{449771520}
Evaluate the integral using contour integration: $$\int_0^{2\pi}e^{\cos\theta} \sin(n\theta-\sin\theta)~d\theta$$ where $n\in\mathbb{Z}$. Provide your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the number of unique colorings of a regular hexagon where each vertex is either Black or White and each edge is either Red, Green, or Blue. The hexagon is acted upon by the dihedral group $D_6$. Use Burnside's Lemma to determine the number of distinct colorings.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4183}
Let $f(x)$ and $g(x)$ be periodic functions with periods 5 and 10, respectively, such that \( \lim_{x\to0}\frac{f(x)}{x}=\lim_{x\to0}\frac{g(x)}{x}=k \) where \( k>0 \). Determine the value of:
\[ \lim_{n\to\infty}\frac{f(5(4+\sqrt{15})^n)}{g(10(4+\sqrt{14})^n)}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider the paraboloid $P$ defined by $P = \{(x, y, z) \in \mathbb{R}^3: z = 4 - x^2 - y^2, x^2 + y^2 < 4\}$. Find the unit normal field of $P$ and compute the integral of the vector field $F(x, y, z) = (x, y, xy + z)$ over $(P, \nu)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24\pi}
Evaluate the limit: $$\lim_{n\to\infty}n^2\left(\sin(2\pi en!)-\frac{2\pi}{n}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Points \(A_1, B_1, C_1,\) and \(D_1\) are taken on the planes of the faces of the tetrahedron \(ABCD\) such that the lines \(AA_1, BB_1, CC_1,\) and \(DD_1\) are parallel. Determine the ratio of the volumes of tetrahedrons \(ABCD\) and \(A_1B_1C_1D_1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Find the volume of the set \(\Omega = \bigcup_{t \in [0,1]} T(t)\), where \(T(t)\) is the triangle formed by the points \(A(t) = (t,t^3,t)\), \(B(t) = (t,t,t)\), and \(C(t) = (0,2t,t)\). Assume \(T(t)\) is empty if the points are collinear.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{15}}
A supermarket has four entrances. The arrivals at each entrance follow Poisson processes with rates \(\lambda_1 = 120, \lambda_2 = 100, \lambda_3 = 150,\) and \(\lambda_4 = 180\) customers per hour. 40% of all customers are children. The probability that a child customer makes a purchase is 0.2, while the probability that an adult customer makes a purchase is 0.4. Calculate the probability that the fifth child customer to make a purchase arrives within the first 20 minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9989}
In triangle ABC, segments BD, DF, and FE divide the triangle into four equal-area triangles. Points E and D are on side CA, and point F is on side BC. Given that BF = 2DE, find the ratio of AC to BC.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{9}}
Determine the values of $x$ for which the series \( \sum_{n=1}^{\infty}\frac{\sin (nx)}{n} \) converges pointwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Evaluate the integral: $$\int_{0}^{1} \left((1-x^7)^{\frac{1}{3}}-(1-x^3)^{\frac{1}{7}}\right)\,dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the input signal $x(t) = \cos(t)$ and the impulse response $h(t) = u(t)$, find the output $y(t)$ using the Fourier Transform.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin(t)}
There are 2011 positive numbers such that both their sum and the sum of their reciprocals equal 2012. Let \(x\) be one of these numbers. Determine the maximum value of \(x + \frac{1}{x}.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8045}{2012}}
Evaluate the sum \( \sum\limits_{i,j,k=0}^n{n \choose i+j}{n \choose i+k}{n \choose k+j} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4^n}
In $\triangle BAC$, $\angle BAC = 40^\circ$, $AB = 10$, and $AC = 6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE + DE + CD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14

}
Determine the value of \( c \in \mathbb{R}^+ \) for which the Banach fixed-point theorem is satisfied for the function \( g(x_1, x_2) = \frac{1}{6} \begin{pmatrix} x_1x_2+cx_2-1 \\ x_1^2-x_2+1 \end{pmatrix} \) on the set \( M = [-1,1]\times [-1,1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Two boys start moving from the same point \( \mathrm{A} \) on a circular track but in opposite directions. Their speeds are \( 5\mathrm{{ft}} \) ,per second
 and 9 ft. per second. If they start at the same time and finish when they first me at the point A again, then the number of time
 they meet, excluding the start and finish, is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
13
}
Let \( f \) be a quadratic function that satisfies the following condition: For two distinct real numbers \( a \) and \( b \), if \( f(a) = f(b) \), then \( f(a^2 - 6b - 1) = f(b^2 + 8) \). Find the value of \( \frac{f(8) - f(2)}{f(2) - f(1)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}