inclusionAI/Ring-lite-rl-data huggingface dataset:
Given positive numbers \( h, s_{1}, s_{2} \) and a triangle \( ABC \) in space, determine the number of ways to choose a point \( D \) such that in the tetrahedron \( ABCD \), the height from vertex \( D \) is \( h \), and the areas of faces \( ACD \) and \( BCD \) are \( s_{1} \) and \( s_{2} \) respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the value of $0^0$ such that the function $f(x) = \frac{x}{x^x}$ is continuous at $x = 0$. What is this value?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the set of all complex numbers $w$ that satisfy the inequality $|z-w|^2 \geq |1-\overline{w}z|^2$ for all complex numbers $z$ with $|z|=1$. Provide your answer as a set of complex numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{C}}
Determine the number of zeros of the function \( q(z) = z^3 + 2z^2 - 5 \) within the annulus \( A(0, 2, 3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the probability that a randomly chosen word of this sentence has exactly four letters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5/16
}
Evaluate the limit: \[ \lim _{n \to \infty} \int_{0}^1 \arcsin(\sin(nx)) \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the index of the subgroup \( H \) in \( \mathrm{GL}_2(\mathbb{Z}) \), where \( H \) is defined as:
\[
H = \left\{\begin{bmatrix} a & b \\\ c & d \end{bmatrix} \mid a\equiv 1 \pmod{3},\ b\equiv 0 \pmod{3},\ c\equiv 0 \pmod{3},\ d\equiv 1 \pmod{3} \right\}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Given that $T \sim t_7$, find the value of $t$ such that $$P(-t < T) + P(0 < T) + P(t < T) + P(2t < T) = 1.75.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.356}
Find a tight upper bound for the function \( f(m,n) \) given the following conditions: \( f(m,n) \leq \frac{f(m-1,n-1) + f(m,n-2)}{2} \), \( f(m,m-r) = 0 \) for integer \( r \geq 1 \), and \( f(0,n) = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{-m}}
Find the first three digits of \( \frac{1}{x} \) in \( \mathbb{Z}/7^3\mathbb{Z} \), where \( x = 1+2\cdot7+4\cdot7^2 +1\cdot7^3 +\ldots \in \mathbb{Z}_7 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{156}
Let $(x_{n})_{n \ge 1}$ be a sequence of real numbers such that \( \lim_{n\to\infty} x_n \sum_{k=1}^{n}x^2_{k}=1 \). Compute \( \lim_{n\to\infty} (3n)^{1/3} x_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of distinct values that \( x^9 \) can take modulo 999.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Bus tickets from a transportation company are numbered with six digits, ranging from 000000 to 999999. A ticket is considered "lucky" if the sum of the first three digits equals the sum of the last three digits. For example, ticket 721055 is lucky, whereas 003101 is not.

Determine how many consecutive tickets a person must buy to guarantee obtaining at least one lucky ticket, regardless of the starting ticket number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1001
}
Determine the maximum number of distinct sets that can be generated from a single set \( B \subseteq \mathbb{R} \) by taking a finite number of closures and interiors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The area of triangle $\triangle OFA$, where line $l$ has an inclination angle of $60^\circ$ and passes through the focus $F$ of the parabola $y^2=4x$, and intersects with the part of the parabola that lies on the x-axis at point $A$, is equal to $\frac{1}{2}\cdot OA \cdot\frac{1}{2} \cdot OF \cdot \sin \theta$. Determine the value of this expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt {3}}
Determine the values of $b$ for which the system of differential equations \( \begin{cases} x'(t)=by(t) \\ y'(t)=x(t)+y(t) \end{cases} \) has a unique non-zero solution \((x(t),y(t))\) that approaches \((0,0)\) as \(t\) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b > 0}
Three shepherds met on a large road, each driving their respective herds. Jack says to Jim:

- If I give you 6 pigs for one horse, your herd will have twice as many heads as mine.

And Dan remarks to Jack:

- If I give you 14 sheep for one horse, your herd will have three times as many heads as mine.

Jim, in turn, says to Dan:

- And if I give you 4 cows for one horse, your herd will become 6 times larger than mine.

The deals did not take place. Determine the total number of heads of livestock in the three herds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Evaluate the limit \(\lim_{x\to\infty}(x!)^{1/x}-\frac{x}{e}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given a multiset of integers, $M1 = \{6,6,4,4,4,2,2\}$, and another multiset, $M2 = \{16,8,4\}$, determine the number of distinct ways to form $M2$ by picking integers without replacement from $M1$. Each element of $M2$ must be a sum of elements from $M1$. What is the count of such distinct combinations?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Identify the set operation that generates the group \(\mathbb{Z}_4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{+_4}
Determine the range of values for \( a + b \) such that the inequality \( a \cos x + b \cos 2x \geq -1 \) holds for all \( x \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 2]}
Find the gradient of the function \( f(A) = ||Ax - y||^2 \) with respect to the matrix \( A \), where \( A \in \mathbb{R}^{n\times n} \), \( x,y \in \mathbb{R}^n \), and \( ||\cdot|| \) denotes the Euclidean norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(Ax - y)x^\top}
Find the relationship between the determinants of the matrices \( M_1 = \begin{vmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{vmatrix} \) and \( M_2 = \begin{vmatrix} x_1+ay_1 & y_1+bz_1 & z_1+cx_1 \\ x_2+ay_2 & y_2+bz_2 & z_2+cx_2 \\ x_3+ay_3 & y_3+bz_3 & z_3+cx_3 \end{vmatrix} \), where \( a \), \( b \), and \( c \) are constants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\det(M_2) = (1 + abc) \det(M_1)}
Consider a biased coin with a probability of \( p = \frac{1}{3} \) of landing heads and \( \frac{2}{3} \) of landing tails. The coin is flipped \( n \) times. Let \( X_i \) be a random variable denoting the \( i \)-th flip, where \( X_i = 1 \) means heads, and \( X_i = 0 \) means tails. Determine the smallest value of \( n \) such that the probability that more than half of the coin flips result in heads is less than 0.001, using the Chernoff bound.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{118}
Determine the maximal length of a sequence of numbers consisting of only $+1$ and $-1$ such that the sum of every 10 consecutive numbers is 0, but the sum of every 12 consecutive numbers is not 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Determine the number of consecutive composite integers that follow \( k! + 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k - 1}
Determine the number of all ordered triplets of positive integers $(a, b, c)$, which satisfy the equalities:
$$[a, b] =1000, [b, c] = 2000, [c, a] =2000.$$
([x, y]represents the least common multiple of positive integers x,y)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
70
}
Evaluate the integral \( \iint \frac{x^2}{y^3} \, dA \) over the region defined by the curves \( y=2x, \; y=x, \; y=x^2, \; y=2x^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
In how many ways can the sides of an icosahedron be numbered from 1 to 20, considering the sides indistinguishable?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{20!}{60}}
Let $F=\langle (1,0,0,2),(0,1,0,0) \rangle$ and $G=\langle (-1,1,0,0),(1,0,1,0),(0,0,0,1) \rangle$. Given that the basis of $F \cap G$ is $\{(-1,1,0,-2)\}$, find the basis of the quotient space $F/(F \cap G)$. Provide your answer as a set of vectors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{(1,0,0,2)\}}
Evaluate the definite integral \( \int_{1}^{2} \frac{e^x}{x} \, dx \) to five decimal places using the Taylor series expansion for \( \frac{e^x}{x} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.05912}
Calculate the difference between the sum of real roots of the function \( f_{10}(x) = 1 - \sum_{k=0}^{10} \frac{x(1-x)\cdots(k-x)}{(k+1)!} \) and the sum of real roots of its derivative \( f'_{10}(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the range of values of $q$ for which the series \( \sum_{n=1}^{\infty} \frac{n^2}{(n+1)^q} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, \infty)}
The persons $A$, $B$, and $C$ are playing a game where the person with the highest points wins. The points are distributed as follows:

- $A$'s points are a Poisson distributed random variable with parameter 2.
- $B$'s points are a Poisson distributed random variable with parameter 3.
- $C$'s points are a Poisson distributed random variable with parameter 4.

The points of each player are independent of each other. The outcome of the points was 1, 2, and 3, but it is unknown who received which score. What is the conditional probability that person $C$ won the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{40}{81}}
Determine the interval of convergence for the series \( \sum\limits_{n=1}^{\infty} \frac{n+3}{2n^2+2} x^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1)}
Evaluate the contour integral \( \oint_{|z|=2} \frac{dz}{z\sqrt{1-\frac{1}{z^2}}} \) using the principal branch of the square root, where \(|z|=2\) is a circle centered at the origin with radius 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
On the board, there is a function $\sin x + \cos x$. You are allowed to write the derivative of any previously written function, as well as the sum and product of any two previously written functions. This process can be repeated many times. Determine the constant $c$ such that a function equal to $c$ for all real $x$ can appear on the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the general form of the function $f(r)$ that satisfies the differential equation $$\nabla^2 f(r) = \frac{d^2f}{dr^2} + \frac{2}{r} \frac{df}{dr} = 0,$$ where $a$ and $b$ are constants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(r) = \dfrac{a}{r} + b}
Find a positive number \( \delta \) such that if \( 0 < |x-1| < \delta \), then \( |f(x)g(x) - h(x) - 1| < \frac{1}{1000} \), given that \( |f(x) - 2| < |x-1|^2 \), \( |g(x)-3| < 2|x-1| \), and \( |h(x)-5| < |x-1| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{10000}}
Compute the Krull dimension of the ring \( k[x,y,z]/(xz, yz) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \(f(x)\) and \(g(x)\) be continuous, positive functions such that \(f(-x) = g(x) - 1\), \(f(x)=\frac{g(x)}{g(-x)}\) and \(\int_{ - 20}^{20} {f\left( x \right)dx} = 2020\). Find the value of \(\int_{ - 20}^{20} {\frac{{f\left( x \right)}}{{g\left( x \right)}}dx} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let $x > 0$. Solve the inequality: $$ \Gamma(x) \le \frac{1}{x} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1]}
How many trailing zeros does the value
\[300\cdot305\cdot310\dots1090\cdot1095\cdot1100\]end with?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
161
}
Determine the cardinality of the set \( B = \{ f \in \mathbb{R}^\mathbb{N} \mid f(\mathbb{N}) \text{ is a well-ordered subset of } \mathbb{R} \text{ with respect to the standard ordering} \}. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Evaluate the integral \( \int_0^e{W(x)}\,\mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e - 1}
Calculate the contour integral \( \int_{|z|=3} \frac{z^{17}}{(z^2+2)^3(z^3+3)^4}dz \). Assume the singular points are within the circle \(|z|=3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Find the distribution of the statistic \( T = \frac{S_{n-1}^2/\sigma_{1}^2}{S_{m-1}^2/\sigma_{2}^2} \), where \( X \sim N(\mu_1, \sigma_1) \) and \( Y \sim N(\mu_2, \sigma_2) \). Given two simple random samples of sizes \( n \) and \( m \) from \( X \) and \( Y \) respectively, determine the distribution of \( T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F(n-1, m-1)}
Determine the number of graphs with 5 vertices labeled a, b, c, d, and e, such that exactly one vertex has degree 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{205}
Evaluate the integral \( \int_{0}^{\pi} \sin(A\sin(x))\sin(x) \,dx \) where \( A \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi J_1(A)}
Let $X=(X_1,X_2,X_3)$ be a continuous random vector with the joint probability density function given by \( f(x_1,x_2,x_3)= 12x_2 \) for \( 0<x_3<x_2<x_1<1 \) and \( 0 \) elsewhere. Calculate the probability of the event \( B = \{ x_3 \leq \frac{1}{3} \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{49}{81}}
Calculate the value of \( \frac{dy}{dx} \) at \( x = -1 \) for the equation:
\[
(\sin y)^{\sin(\frac{\pi x}{2})}+ \frac{\sqrt{3}}{2}\sec^{-1}(2x)+ 2^x\tan (\log_e(x+2))=0
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{\pi \sqrt{\pi^2 - 3}}}
Evaluate the limit of the following expression as \( n \to \infty \):
\[ M_{Y_n}(t) = e^{-t\sqrt{n}} \left(\frac{1}{1 - t/\sqrt{n}}\right)^n. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{t^2}{2}}}
Let \(ABC\) be a triangle with \(AB=13\), \(BC=14\), and \(CA=15\). The altitude from \(A\) intersects \(BC\) at \(D\). Let \(\omega_1\) and \(\omega_2\) be the incircles of \(ABD\) and \(ACD\), and let the common external tangent of \(\omega_1\) and \(\omega_2\) (other than \(BC\)) intersect \(AD\) at \(E\). Compute the length of \(AE\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A circle is inscribed in an equilateral triangle with side length $s$. Points $A$,$B$,$C$,$D$,$E$,$F$ lie on the triangle such that line segments $AB$, $CD$, and $EF$ are parallel to a side of the triangle, and tangent to the circle. If the area of hexagon $ABCDEF = \frac{9\sqrt3}{2}$ , find $s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ 3 \sqrt{3}$}
Let $\{X(t), t \geq 0\}$ be a Brownian motion with drift parameter $\mu = -2$ and variance parameter $\sigma^2 = 16$. If $X(0) = 5$, calculate $P(X(1) < 0)$. \(\text{Express your answer as a probability.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2266}
Find the coefficient of \(x^{10}\) in the expansion of \((1+2x+3x^2+4x^3+5x^4+6x^5)^4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13088}
In triangle \(ABC\), the equation of side \(BC\) is given by \(2x - y = 3\). The circumcenter of the triangle is at \((2, 4)\) and the orthocenter is at \((1, 2)\). Determine the value of \(\tan B \tan C\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the value of \( m \) such that the polynomial \( p(x) = (x + 1)^m - x^m - 1 \) is divisible by \( q(x) = (x^2 + x + 1)^2 \) without a remainder.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
How many ways are there to arrange the letters in the word 'INTELLIGENT' such that there are at least two consecutive pairs of identical letters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{286920}
Evaluate the limit: $$\lim_{n\to \infty}{\sqrt{(1-\cos(1/n))\sqrt{(1-\cos(1/n))\dots}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( \left\lVert A-B \right\rVert = (x_A-x_B)^2+(y_A-y_B)^2 \) for every two points \( A = (x_A, y_A) \) and \( B = (x_B, y_B) \) in the plane. Let \( S \) be the set of points \((x,y)\) in the plane for which \( x,y \in \{ 0,1,\dots,100 \} \). Find the number of functions \( f : S \to S \) such that \( \left\lVert A-B \right\rVert \equiv \left\lVert f(A)-f(B) \right\rVert \pmod{101} \) for any \( A, B \in S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2040200}
Evaluate the limit of the product as \( n \to \infty \):
\[
P = \lim_{n\to \infty} \prod_{i=1}^n \frac{1 - (2i + 1)a/(2n)}{1 - ia/n}
\]
where \( 0 < a < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{1 - a}}
Evaluate the limit of the series as \( n \to \infty \):
\[ x(n) = \sum\limits_{k=1}^n \frac{k^2}{n^3+k^2} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Let \(n\) and \(k\) be positive integers such that \(n \geq 2k > 3\) and \(A = \{1, 2, \ldots, n\}\). Determine all pairs \((n, k)\) such that the number of \(k\)-element subsets of \(A\) is \(2n - k\) times the number of 2-element subsets of \(A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(27, 4)}
Given that $2017$ is a prime number, find the remainder when the sum $S = \sum_{k=0}^{62} \binom{2014}{k}$ is divided by $2017$. Simplify the expression for $S$ using combinatorial identities and modulo arithmetic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
Determine the last digit of \(AD^{AD}\) in hexadecimal when expressed in octal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given that \(a, b, c > 0\) and \(a + b + c = abc\), find the maximum value of \((a-1)(b-1)(c-1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\sqrt{3} - 10}
Given the equation $8GX = XX^T$, where $G = \begin{pmatrix} 5 & 4 \\ 3 & 2 \end{pmatrix}$, find the determinant of the matrix $X$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-128}
Identify the units in the polynomial ring $R[x]$, where $R$ is a domain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R^\times}
Compute the gradient of the function \( f(x) = \|x\|^p \), where \( f(x) = (x_1^2 + x_2^2 + \ldots + x_N^2)^{p/2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p \|x\|^{p-2} x}
Find the maximum perimeter of a triangle that can be inscribed in a regular septagon with each side measuring 1 unit. Provide your answer rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.85}
Evaluate the double integral \( \iint_G (x^{4}-y^{4})xy\,dxdy \), where \( G \) is the region defined by \( 0 \leq x \), \( 1 \leq x^{2}-y^{2} \leq 2 \), and \( 1 \leq xy \leq 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{8}}
Determine the minimum value of the expression \( \left|7x-1\right|+\left|7y-5\right|+\left|7z-1\right| \) for non-negative real numbers \(x\), \(y\), and \(z\) such that \(x + y + z = 1\) and \(y^2 \leq 3xz\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52 - 28\sqrt{3}}
Evaluate the limit: $$ \lim_{n \to \infty} \frac{\int_{0}^{1} \frac{e^{-x^2} dx}{(1+x^{2})^{n}}}{\frac{1}{n^2}} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In a laboratory, a chemist adds sulfuric acid to flasks held by four groups, each consisting of 3 people. For the first group, he adds $a_1$, $a_2$, and $a_3$ drops to the flasks such that $a_1 + a_2 + a_3 = 4$. For the second group, he adds $b_1$, $b_2$, and $b_3$ drops such that $b_1 + b_2 + b_3 = 3$. For the third group, he adds $c_1$, $c_2$, and $c_3$ drops such that $c_1 + c_2 + c_3 = 2$. For the fourth group, he adds $d_1$, $d_2$, and $d_3$ drops such that $d_1 + d_2 + d_3 = 1$. Assuming the drops are of equal size and randomly distributed, what is the probability that the flask with the most acid belongs to the first group? Express your answer as a percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\%}
Determine the number of surjective ring homomorphisms from the ring of Gaussian integers $\mathbb{Z}[i]$ onto the finite field $\mathbb{F}_{11^2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a least-squares problem \(||Ax-b||=0\) solved using a backward stable algorithm (QR decomposition with Householder projectors), and a condition number \(\kappa(A)=10^5\), determine how many digits of accuracy the solution \(x\) should have when using double-precision floating-point arithmetic (\(10^{-16}\)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Determine all positive real numbers \( p \) such that the inequality \( \sqrt{a^2 + pb^2} + \sqrt{b^2 + pa^2} \ge a + b + (p - 1) \sqrt{ab} \) holds for any positive real numbers \( a \) and \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 3]}
We say that an integer $x \in \{1, \dots, 102\}$ is \textit{square-ish} if there exists some integer $n$ such that $x \equiv n^2 + n \pmod{103}$. Compute the product of all \textit{square-ish} integers modulo $103$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Determine the number of homomorphisms from the group \( \Bbb{Z}_{10} \times \Bbb{Z}_{25} \) to the symmetric group \( S_4 \). Here, \( \Bbb{Z}_n \) denotes the integers from 0 to \( n-1 \) with addition modulo \( n \), and \( S_4 \) is the group of permutations of the set \( \{1, 2, 3, 4\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the Krull dimension of the ring \( R = \mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Two circles \(\gamma_{1}\) and \(\gamma_{2}\) intersect at points \(P\) and \(Q\). The tangent to \(\gamma_{2}\) at \(P\) intersects \(\gamma_{1}\) at point \(A\), and the tangent to \(\gamma_{1}\) at \(P\) intersects \(\gamma_{2}\) at point \(B\), with \(A\) and \(B\) distinct from \(P\). Given that \(PQ \cdot O_{1}O_{2} = PO_{1} \cdot PO_{2}\) and \(\angle APB\) is acute, find the measure of \(\angle APB\) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}