inclusionAI/Ring-lite-rl-data huggingface dataset:
Let \( X = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 = 1\} \cup ([-1,1] \times \{0\}) \cup (\{0\} \times [-1,1]) \). Determine the maximum number of distinct points \( n_0 \) that can be removed from \( X \) such that the remaining set \( X - \{p_1, p_2, \ldots, p_{n_0}\} \) is still connected. What is the value of \( n_0 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the infimum of the integral \( \int_{\mathbb{R}^5} u^2 - \ln(1+u^2) \, \text{dx} \) subject to the constraint \( \int_{\mathbb{R}^5} u^{14} = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose \( f:D(0,1) \longrightarrow \mathbb{C} \) is a holomorphic function, where \( D(0,1) = \{z \in \mathbb{C} \mid |z|<1\} \), and assume the maximum \(|f(z)| \le 2\). Estimate \(|f^{(3)}(i/3)|\) using Cauchy's estimate.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{81}{2}}
A group of order 15 acts on a set of order 22 with no fixed points. Determine the number of orbits in this group action.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the smallest positive integer \( n \) such that all its prime factors are greater than 18, and it can be expressed as \( n = a^3 + b^3 \) for positive integers \( a \) and \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1843}
Determine the intervals on which the series \( \sum_{n=1}^\infty ne^{-nx} \) converges uniformly.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[a, \infty)}
Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is $50\%$ more than the regular. After both consume $\tfrac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and $2$ additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
40
}
Let \( A \) be a non-singular matrix that satisfies \( A^2B = A \) for some matrix \( B \). Given the set \( S = \{-1, 0, 1\} \), and \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) where \( a, b, c, d \in S \), determine the probability that \( \det(A + B) = \det A + \det B \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Determine the interval of convergence for the series: \[ \sum_{n=1}^{\infty}\frac{((n+1)x)^n}{n^{n+1}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1)}
Evaluate the limit: \[ \lim_{n \rightarrow \infty} \left(\left(\frac{f(n)}{g(n)}\right)^q \left(\frac{1-f(n)}{1-g(n)}\right)^{(1-q)}\right)^n \] where \( n \in \mathbb{N} \), \( f(n),g(n) \in [0,1] \), \( q\in (0.5,1) \), \( f(n) \leq g(n) \), and both \( f(n) \) and \( g(n) \) converge to some real number in \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a game, three players, $A$, $B$, and $C$, compete to earn the highest score. The score of each player is represented by a Poisson distributed random variable: $A$ with parameter $\lambda_A = 2$, $B$ with $\lambda_B = 3$, and $C$ with $\lambda_C = 4$. The scores are independent. Given that the total combined score of the players is $A + B + C = 6$, with the scores being $1, 2$, and $3$, respectively, but not knowing which player got which score, calculate the conditional probability that player $C$ had the highest score, i.e., $C = 3$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{40}{81}}
Find all positive integers \( n > 2 \) such that the only functions \( f: \mathbb{Z} \to \mathbb{Z} \) satisfying the following property are linear functions of the form \( f(x) = bx + c \), where \( b \) and \( c \) are integers: If \( a_1, a_2, \dots, a_n \) form an arithmetic progression, then the numbers \( f(a_1), f(a_2), \dots, f(a_n) \) also form an arithmetic progression, possibly in a different order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the flux of the vector field \( \boldsymbol{F}(x,y,z) = \nabla \phi + \nabla \times \boldsymbol{G} \) out of the cylindrical surface of the region \( D = \{(x,y,z) : x^2 + y^2 \leq 1, 0 \leq z \leq 1\} \), where \( \phi(x,y,z) = x^2 - y^2 + \frac{3}{2\pi}z^2 \) and \( \boldsymbol{G}(x,y,z) = \frac{3}{\pi}zy\boldsymbol{i} + \arctan(x^2)\boldsymbol{j} + x^5 y \sqrt{z^2 +1}\boldsymbol{k} \). Use the divergence theorem and the identity \( \nabla \cdot(\nabla \times \boldsymbol{F}) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the value of $a$ such that the following limit holds true:

$$\lim_{x\to\frac{\pi}{6}}\left(\sqrt3 \tan x\right)^{\frac{(a+1)\tan(3x)}{\sin x}} = e^{-\frac{8\sqrt3}{3}}$$

After applying $\lim_{x\to 0}(1 + x)^{\frac{1}{x}}$, we get:

$$e^{\lim_{x\to\frac{\pi}{6}}\left(\frac{\sqrt3 \tan x - 1}{\sin x}\right) (a+1)\tan(3x)} = e^{-\frac{8\sqrt3}{3}}$$

Determine the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the value of \( \alpha \) that results in the fastest convergence of the given iteration formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the infinite series: \[ \sum_{n=1}^\infty \left(1 - \left(1+\frac{1}{n}\right)^{-n}\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given the sphere with the parametric representation \(X(\theta,\phi) = (r \sin\phi \cos\theta, r \sin\phi \sin\theta, r \cos \phi)\), find the Christoffel symbol \(\Gamma^1_{22}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let proposition $p$: $(x-2)^2 \leq 1$, and proposition $q$: $x^2 + (2a+1)x + a(a+1) \geq 0$. If $p$ is a sufficient but not necessary condition for $q$, find the range of real numbers $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -4] \cup [-1, +\infty)}
Find a non-zero solution to the equation \( K(Q,m,F_\text{x})=\frac{{F_\text{x}}^2\cdot\left(m-1\right)}{\sqrt{{\left(m\cdot{F_\text{x}}^2-1\right)}^2+{F_\text{x}}^2\cdot{\left({F_\text{x}}^2-1\right)}^2\cdot{\left(m-1\right)}^2 \cdot Q^2}} \) for any values of \( m \) and \( Q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find a basis for a generalized eigenspace of the linear operator $T$ on $P_2(\mathbb{R})$, where $T(f(x))=2f(x)-f'(x)$, consisting of a union of disjoint cycles of generalized eigenvectors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{2, -2x, x^2\}}
In a game played inside the unit disc D, Player A chooses a point p0 ∈ D. At each step n, Player B draws a line ℓn through pn−1, and Player A chooses a point pn ∈ D ∩ ℓn. Player B wins if they can force the sequence (pn) to converge. If Players A and B play randomly, what is the probability that Player B wins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $|\alpha| < \frac{\pi}{2}$. Evaluate the following definite integral.
$$\int_{0}^{\frac{\pi}{2}} \frac{dx}{\{\sin (a+x)+\cos x\}^{2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{1}{\cos a (1 + \sin a)} \]}
Evaluate the definite integral \( \int_0^1 f(x) \, dx \), where \( f(x) = \begin{cases} 2x\sin\left(\frac{1}{x}\right) - \cos\left(\frac{1}{x}\right) & : x \in (0,1] \\ 0 & : x = 0 \end{cases} \). Determine if the integral is defined, and if so, find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin 1}
Suppose 10 people consisting of 5 married couples are randomly seated around a round table. Let \( X \) be the number of wives who are seated next to their husbands. Calculate the variance \( \text{Var}(X) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{80}{81}}
Evaluate the limit: \[ \lim_{n \to \infty} \int_0^n \frac{dx}{1+n^2\cos^2x} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( z = f(x,y) \) be a differentiable function such that \( f(3,3) = 1, \) \( f_x(3,3) = -2, \) \( f_y(3,3) = 11, \) \( f(2,5) = 1, \) \( f_x(2,5) = 7, \) and \( f_y(2,5) = -3. \) Suppose \( w \) is a differentiable function of \( u \) and \( v \) satisfying the equation \( f(w,w) = f(uv, u^2+v^2) \) for all \((u,v)\). Find \( \frac{\partial w}{\partial u} \) at \((u,v,w) = (1,2,3)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{9}}
Evaluate the infinite sum: \[ \sum_{n=1}^{\infty}\frac{2H_{n}}{(n+1)^2} \] where \( H_n \) is the \( n \)th harmonic number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\zeta(3)}
Evaluate the product \( \mathop{\prod }\limits_{{k = 0}}^{{2}^{1999}}\left( {4{\sin }^{2}\frac{k\pi }{{2}^{2000}} - 3}\right) \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3

}
Evaluate the exact value of the alternating series \( \displaystyle \sum_{n=0}^\infty \frac{(-1)^{n+1}n}{(2n+1)!} \cdot \pi^{2n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Consider the transfer function matrix \(G(s)\) of a continuous-time system given by:
\[G(s) = \begin{bmatrix}\frac{1}{s^2+2s}&\frac{s+1}{s} \\ -\frac{1}{s+1} & \frac{1}{s^2+4s+3} \end{bmatrix}\]
Suppose that \((A,B,C,D)\) is a state-space model that forms a minimal realization. What is the dynamic order of this system (i.e., the dimension of the state-space)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine if there exists a natural number $n$ such that the equation $n = x^2 + y^2$ has more than three solutions in natural numbers $x$ and $y$. If such an $n$ exists, provide an example.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Determine the dimension of the quotient ring $\mathbb{R}[x]/J$, where $\mathbb{R}[x]$ is the polynomial ring over $\mathbb{R}$, and $J=\{f(x)\in \mathbb{R}[x]\mid f(0)=f'(0)=f''(0)=0\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the frequency \( \omega(x) \) of the function \( x(t) = \sin(t) + \sin(\pi t) \), where the frequency is defined by \( \omega(x) = \limsup_{T \to \infty} \frac{\pi}{T} \nu(x,T) \), and \( \nu(x,T) \) represents the number of zeros of \( x(t) \), counting with their multiplicities, on the interval \((0,T]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Compute the limit: \[ \lim \limits_{x \rightarrow 0} \dfrac{\sin(\sin(\sin(x)))}{\cos\left(\frac{\pi}{2}\cos(x)\right)}\cdot x^k \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{\pi}}
Find a function that is not bounded below and whose second derivative is bounded above.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-x^2}
Determine the maximum possible shortest distance between any two points among 8 points placed either on the boundary or inside an equilateral triangle with a side length of 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Let $k$ be a field, $f(x) \in k[x]$ be an irreducible polynomial over $k$, and $\alpha$ be a root of $f$. If $L$ is a field extension of $k$, determine what $k(\alpha) \otimes_k L$ is isomorphic to.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{L[x]/(f(x))}
Determine the values of $p$ for which the Fourier transform exists for functions in the $L^p$ space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, 2]}
设实数 $a$ 满足 $a<9 a^{3}-11 a<|a|$, 则 $a$ 的最小值的$\sqrt{3}$是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
For $ a\in\mathbb{R}$, let $ M(a)$ be the maximum value of the function $ f(x)=\int_{0}^{\pi}\sin (x-t)\sin (2t-a)\ dt$.

Evaluate $ \int_{0}^{\frac{\pi}{2}}M(a)\sin (2a)\ da$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ \frac{28}{27} $
}
Evaluate the limit: \[ \lim_{x\to\infty} \frac{\sum_{k=0}^{x/2}\binom{x}{2k}2k(x-2k)}{\sum_{k=0}^{x/2}\binom{x}{2k}(x-2k)^{2}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the smallest number of mutually visible pairs of birds among 155 birds sitting on the boundary of a circle, where two birds are mutually visible if the angle at the center between their positions is at most 10 degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{620}
Determine the order of the group of all 2x2 matrices with entries from the integers modulo a prime number \( p \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^4}
Sumith lists out the positive integer factors of $12$ in a line, writing them out in increasing order as $1$, $2$, $3$, $4$, $6$, $12$. Luke, being the mischievious person he is, writes down a permutation of those factors and lists it right under Sumith's as $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$. Luke then calculates$$gcd(a_1, 2a_2, 3a_3, 4a_4, 6a_5, 12a_6).$$Given that Luke's result is greater than $1$, how many possible permutations could he have written?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{308}
Determine the necessary and sufficient condition that the coefficients of the cubic equation \(x^{3}+a x^{2}+b x+c=0\) must satisfy if its roots form a geometric progression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^3 c = b^3}
Evaluate the limit: $$\lim_{a\to\infty}\left[\int_{t}^{t+\frac{1}{2}}\psi(x)\,dx -\frac{\psi\left(t+\frac{1}{4}\right)+\psi\left(t+\frac{1}{2}\right)}{4}\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the finite difference approximations:
- \( \frac{\partial}{\partial x} \rightarrow O(\Delta^4) \)
- \( \frac{\partial^2}{\partial x^2} \rightarrow O(\Delta^2) \)

Determine the resulting order of accuracy for the approximation of \( \frac{\partial^3 f}{\partial x^3} \) using the approach:
\[ \frac{\partial}{\partial x} \left(\frac{\partial^2 f}{\partial x^2}\right) \rightarrow O(\Delta^?) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( S \) be a nonempty closed bounded convex set in the plane. Let \( K \) be a line and \( t \) a positive number. Let \( L_1 \) and \( L_2 \) be support lines for \( S \) parallel to \( K \), and let \( \overline{L} \) be the line parallel to \( K \) and midway between \( L_1 \) and \( L_2 \). Let \( B_S(K,t) \) be the band of points whose distance from \( \overline{L} \) is at most \( \left( \frac {t}{2} \right) w \), where \( w \) is the distance between \( L_1 \) and \( L_2 \). Determine the smallest \( t \) such that \[ S \cap \bigcap_K B_S (K, t) \ne \emptyset \] for all \( S \). Here, \( K \) runs over all lines in the plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Find the maximum value of $p$, where $m$, $n$, and $p$ are natural numbers satisfying the equation $2mnp = (m + 2)(n + 2)(p + 2)$, with the condition $m \leq n \leq p$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130}
5. The sphere touches all the edges of a tetrahedron, in which there is a vertex from which the edges emanate pairwise perpendicularly. Find the radius of this sphere if the radius of the sphere circumscribed around this tetrahedron is $3 \sqrt{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6(\sqrt{2}-1)}
Given the summation \( S_n=\sum^{n-1}_{i=0}\sum^{i-1}_{j=0}\sum^{j-1}_{k=0} (i+j+k) \), there exist positive integers \( A \) and \( B \) such that \( \frac{1}{S_3}+\frac{1}{S_4}+\frac{1}{S_5}+\dots=A-\frac{2\pi^2}{B} \). Find the value of \( A+B \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the product of all quadratic nonresidues in $\mathbb{Z}_{103}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102}
Evaluate the contour integral \( \int_\gamma \frac{1}{z+1} \, \mathrm{d}z \), where \( \gamma(t) \) is a path in the complex plane such that \( \gamma(0) = -i \) and \( \gamma(1) = i \), with \( \gamma(t) \in \mathbb{C} \setminus (-\infty,-1] \) for all \( t \in [0,1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{i\pi}{2}}
Calculate the Jacobian determinant for the transformation from variables \((u, v, w)\) to \((x, y, z)\) given by the equations:

\[ u = x + y + z, \quad v = x + 2z, \quad w = x + 2y + 3z. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{3}}
In a double elimination tournament with a varying skill levels and a few byes assigned to the top players, a total of 63 matches are conducted. How many players participated in this tournament?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Given the harmonic function \(u(x,y) = x^3 + ax^2y + bxy^2 + 2y^3\) and its harmonic conjugate \(v(x,y)\), where \(v(0,0) = 1\), find the value of \(|a + b + v(1,1)|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a $101 \times 101$ square grid, each unit square contains one chocolate, and there are doorways between adjacent squares. It costs $\$1$ to pass through a doorway. Jake can start from any square and must collect all $5101$ chocolates. What is the minimum amount of money Jake needs to ensure he can collect all the chocolates?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10200}
Determine the positive integer $k$ such that the non-constant complex analytic function $f$ on $\mathbb{C}$ satisfies the functional equation:
\[ f(s) = (-1)^{k} f(2-s) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let triangle $\triangle ABC$ have side lengths $AB = 7, BC = 8,$ and $CA = 9$, and let $M$ and $D$ be the midpoint of $\overline{BC}$ and the foot of the altitude from $A$ to $\overline{BC}$, respectively. Let $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, such that $m\angle{AEM} = m\angle{AFM} = 90^{\circ}$. Let $P$ be the intersection of the angle bisectors of $\angle{AED}$ and $\angle{AFD}$. If $MP$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b,$ and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, then find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
28
}
At least how many weighings of a balanced scale are needed to order four stones with distinct weights from the lightest to the heaviest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5
}
Evaluate the integral \( I = \int_0^1 \frac{dx}{\sqrt{(1-x^2)(1-x^2k^2)}} \) for \( 0 < k < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K(k)}
Calculate the normal stress at a point on an interface with a normal unit vector \( n = \begin{bmatrix} 0.8 & -0.6 & 0 \end{bmatrix} \) and a stress tensor \( \tau_{ij} = \begin{bmatrix} 1500 & 200 & -500 \\ 200 & 2500 & 1500 \\ -500 & 1500 & -3000 \end{bmatrix} \) by rotating the stress tensor to align the rotated \( x \) axis (\( x' \)) with the normal vector \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1668}
Four entities are traveling around a 1280-meter lap. The first entity travels at 1 m/s starting at 1 meter, the second at 2 m/s starting at 2 meters, the third at 3 m/s starting at 8 meters, and the fourth at 4 m/s starting at 16 meters. Determine the number of seconds it will take for all four entities to be at the same point on the track simultaneously.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1279}
Evaluate the inverse Fourier transform given by the integral:
\[ f(x) = \int_{-\infty}^\infty \frac{dk}{2\pi} e^{-ikx} \frac{\sin^2(k/2)}{(k/2)^2} \]
Determine the function \( f(x) \) for \( x \in [-1, 1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - |x|}
Let \(N\) be the smallest positive integer such that \(f(f(f(N))) = 97\), where \(f(n) = n - g(n)\) and \(g(n)\) is the largest positive proper divisor of \(n\). Find the largest integer not exceeding \(\sqrt{N}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Let $a$, $b$, and $c$ be complex numbers. Let $a_1$ be the image of $a$ when it is rotated $90^\circ$ clockwise around $b$. Let $a_2$ be the image of $a_1$ when it is rotated $90^\circ$ counter-clockwise around $c$.

Then $a_2$ can be written in the form
$$a_2 = pa + qb + rc$$
for some complex numbers $p$, $q$, and $r$. Find $p^2 + q^2 + r^2$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - 4i}
Consider non-negative real numbers \(a, b,\) and \(c\) such that \(a \leq b + c\). For which positive real values of \(n\) does the inequality \(a^n \leq b^n + c^n\) hold?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1]}
An equilateral $12$-gon has side length $10$ and interior angle measures that alternate between $90^\circ$, $90^\circ$, and $270^\circ$. Compute the area of this $12$-gon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
500
}
Let \( \Omega \) be a circle with radius18and let \( \mathcal{S} \) be the region inside \( \Omega \) that the centroid of \( \bigtriangleup {XYZ} \) sweeps through as \( X \) varies along
 all possible points lying outside of \( \Omega \) , \( Y \) varies along all possible points lying on \( \Omega \) and \( {XZ} \) is tangent to the circle. Compute the
 greatest integer less than or equal to the area of \( \mathcal{S} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
904
}
Consider arrangements of the9numbers \( 1,2,3,\ldots ,9 \) in a \( 3 \times  3 \) array. For each such arrangement,let \( {a}_{1} \) , \( {a}_{2} \) ,and \( {a}_{3} \) be the medians of the numbers in rows 1,2,and 3 respectively,and let \( m \) be the median of \( \left\{  {{a}_{1},{a}_{2},{a}_{3}}\right\} \) . Let \( Q \) be the number of arrangements for which \( m = 5 \) . Find the remainder when \( Q \) is divided by1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
360
}
Compute the derivative of the function \( f(W) = x^TWW^Tx \) with respect to the matrix \( W \), where \( x \) is an \( n \times 1 \) vector and \( W \) is an \( n \times m \) matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2xx^TW}
Find the area of the spherical triangle \( T = \{ x \in S^2 : d_S(x,u) \le \frac{\pi}{2}, d_S(x,v) \le \frac{\pi}{2}, d_S(x,w) \le \frac{\pi}{2} \} \), where \( u = \frac{1}{\sqrt{2}}(1,-1,0) \), \( v = \frac{1}{\sqrt{2}}(1,1,0) \), and \( w = \frac{1}{\sqrt{2}}(0,1,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Evaluate the integral: $$ \frac{2}{\pi}\int_{-\pi}^\pi\frac{\sin\frac{9x}{2}}{\sin\frac{x}{2}}dx $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}