inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, and $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2523}
In a group, given that $aba = b^2$ and the order of $a$ is 2, determine the order of the element $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given a rearrangement of the numbers from 1 to $n$, each pair of consecutive elements $a$ and $b$ of the sequence can be either increasing (if $a<b$ ) or decreasing (if $b<a$ ). How many rearrangements of the numbers from 1 to $n$ have exactly two increasing pairs of consecutive elements?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{n}-(n+1) \cdot 2^{n}+n(n+1) / 2}
The towns in one country are connected with bidirectional airlines, where at least one direction is paid. In a trip from town $A$ to town $B$, there are exactly 22 routes that are free. Find the least possible number of towns in the country.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find a solution to the congruence equation $x^2 \equiv 7 \pmod{787}$, where the solution $x$ is between 1 and 786.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
已知x∈[0,2π]，且$2 \cos x \leq \vert \sqrt { 1 + \sin 2 x } - \sqrt { 1 - \sin 2 x } \vert \leq$ $\sqrt { 2 } .$求x的取值范围.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$[ \frac { \pi } { 4 } , \frac { 7 \pi } { 4 } ]$}
Find the area of the region enclosed by the curves \( r = 2 + \sin (3 \theta) \) and \( r = 4 - \cos (3 \theta) \) using iterated integrals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\pi}
A ray starting from point $A(-4,1)$ reflects off the line $l_{1}: x-y+3=0$ and the reflected ray passes through point $B(-3,2)$. Find the slope of the line containing the reflected ray.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Given seven distinct letters A, B, C, D, E, F, and G, determine the number of ways to form seven sets of three letters each such that each letter appears with every other letter exactly once. What is the total number of such combinations?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
The difference between $a$ and $5$ is greater than twice $a$. Express this as an inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a - 5 > 2a}
Simplify the product
$$
p = (10 + 1) (10^2 + 1) ((10^2)^2 + 1) ((10^2)^4 + 1) ((10^2)^8 + 1)...((10^2)^{1024} +1)
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10^{4096}-1}{9}}
设a，b，c，d是正整数，满足$a ^ { 5 } =$ $ b ^ { A } ,$ $c ^ { 3 } = d ^ { 2 }$，且$c - a = 1 9$，求$d - b .$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{757}
Given a positive integer $n (n \geqslant 2)$. From the numbers $1, 2, \cdots, 3n$, select $m$ numbers such that there must be four distinct numbers $a, b, c, d$ among them, which satisfy $a = b + c + d$. Find the minimum value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n + 2}
A cyclist traveled from point A to point B, stayed there for 30 minutes, and then returned to A. On the way to B, he overtook a pedestrian, and met him again 2 hours later on his way back. The pedestrian arrived at point B at the same time the cyclist returned to point A. How much time did it take the pedestrian to travel from A to B if his speed is four times less than the speed of the cyclist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
如果:  
(1) $a 、 b 、 c 、 d$ 都属于 $\{1,2,3,4\}$;  
(2) $a \neq b, b \neq c, c \neq d, d \neq a$;  
(3) $a$ 是 $a 、 b 、 c 、 d$ 中的最小值.  
那么，可以组成的不同的四位数 $\overline{a b c d}$ 的个数是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
On the radius \( AO \) of a circle with center \( O \), a point \( M \) was chosen. On one side of \( AO \) on the circle, points \( B \) and \( C \) were chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \( BC \), given that the radius of the circle is \( 12 \) and \(\sin \alpha = \frac{\sqrt{35}}{6}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine for which values of \( p \geq 1 \) the functional \( F(f) = \int_{0}^{1}f(x^2)dx \) is bounded in \( L_p[0,1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > 2}
If $\forall x\geqslant 0$, the inequality $xe^{x}+ae^{x}\ln \left(x+1\right)+1\geqslant e^{x}(x+1)^{a}$ always holds, then the minimum value of the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
The Denali Star train starts from Anchorage and travels towards Fairbanks at a certain speed. After some time, another train, Glacier Discovery, starts at Fairbanks on a parallel track and moves towards Anchorage at a speed of 70 mph. Both the Denali Star and Glacier Discovery trains have a length of 1/6 miles each. After the trains meet, the faster train takes 9.999999999999998 seconds to overtake the slower one. What is the speed of the Denali Star train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given that $f(x)$ is a function with its first derivative $f'(x) = f^2(x) + x$, determine how many times $f(x)$ is differentiable, assuming it is differentiable at least once.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the value of \(\lambda\) such that the number of quadratic polynomials \(ax^2+2bx+c\) meeting the following criteria is equal to \(1000\lambda\): (i) \(a, b, c\) are distinct integers, (ii) \(a, b, c \in \{1, 2, 3, \ldots, 2001, 2002\}\), and (iii) \(x+1\) is a factor of \(ax^2+2bx+c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2002}
Determine the period of the function \( f(x) = \sin(3\pi x)\cos(3\pi[x]) \), where \([x]\) is the greatest integer function and \(\{x\}\) is the fractional part function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $P(x)$ be a polynomial with real coefficients such that $P(12)=20$ and \[ (x-1) \cdot P(16x) = (8x-1) \cdot P(8x) \] holds for all real numbers $x$. Compute the remainder when $P(2014)$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{545}
Evaluate the limit \( \lim_{n\to\infty} n\int_{1}^{2} \frac{1}{x^2(1+x^n)}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Evaluate the limit: $$\lim\limits_{x\to{-2}}(2\sqrt{-1-x}-1)^{\frac{1}{4^{-x}-16}}$$ without using l'Hôpital's rule, derivatives, or Taylor series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{1}{32 \ln 2}}}
Evaluate the integral \( \int dz^{*}dz \exp\bigg[-(z^{*}z-cz^{*}-zd^{*})\bigg] \) where \( z,z^{*},c,d^{*} \in \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi e^{c d^{*}}}
In triangle $ABC$, the angle bisectors $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ intersect at the incenter $I$. Given that $DI = 3$, $BD = 4$, and $CI = 5$, find the area of triangle $ABC$. \( \text{Express your answer in square units.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{384}{7}}
John decides to buy utensils. They come in 30 packs with an equal number of knives, forks, and spoons. However, for an upcoming event, John requires a specific ratio of knives, forks, and spoons which is 2:3:5, and he wants to have at least 50 spoons at the event. How many packs does he need to buy to satisfy both the ratio and spoon requirements?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given $y = \frac{(m^2 - m)x}{m^2 + 1}$ is a linear function of $x$, then $m$ equals to __.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
The integers from $1$ to $25$, inclusive, are randomly placed into a $5 \times 5$ grid such that in each row, the numbers are increasing from left to right. If the columns from left to right are numbered $1, 2, 3, 4,$ and $5$, then the expected column number of the entry $23$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Evaluate the limit: \[ \lim_{n\to\infty} \left( \frac{\Gamma (n+1,n)}{n \Gamma (n)} - e^{-n} \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Two concentric circles have radii \(1\) and \(2\). Two points on the outer circle are chosen independently and uniformly at random. Find the probability that the chord joining the two points intersects the inner circle. Express your answer in the form \(\frac{k}{m}\) where \(\frac{k}{m}\) is a simplified fraction, and give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the remainder when \(20!\) is divided by 487.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{406}
Calculate the expected number of coin tosses required to obtain the sequence H-T-H-T-T-H with a fair coin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Let $n \geqslant 3$ be an integer and let

$$
M=\left\{\frac{a_{1}+a_{2}+\cdots+a_{k}}{k}: 1 \leqslant k \leqslant n \text { and } 1 \leqslant a_{1}<\cdots<a_{k} \leqslant n\right\}
$$

be the set of the arithmetic means of the elements of all non-empty subsets of $\{1,2, \ldots, n\}$. Find $\min \{|a-b|: a, b \in M$ with $a \neq b\}$.

## Proposed by Romania
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{(n-1)(n-2)}}
A rectangle inscribed in a triangle has its base coinciding with the base $b$ of the triangle. If the altitude of the triangle is $h$, and the altitude $x$ of the rectangle is half the base of the rectangle, then:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$x=\dfrac{bh}{2h+b}$}
A quadrilateral \(ABCD\) is inscribed in a circle. At point \(C\), a tangent line \(\ell\) to this circle is drawn. A circle \(\omega\) passes through points \(A\) and \(B\) and is tangent to the line \(\ell\) at point \(P\). The line \(PB\) intersects segment \(CD\) at point \(Q\). Find the ratio \(\frac{BC}{CQ}\), given that \(B\) is the point of tangency to the circle \(\omega\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of real solutions to the equation \(2^{\sin x} + 2^{\cos x} = 2^{3x^2 - 2x^3 - 1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Sergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$, and $b+2$. He then writes all six pairwise products of the numbers in the notebook on the blackboard. Let $S$ be the number of perfect squares on the blackboard. Find the maximum value of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For a real number $y$, find the maximum value of
\[
\frac{y^6}{y^{12} + 3y^9 - 9y^6 + 27y^3 + 81}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{27}}
Josh gets together with 7 friends. Everyone including him puts 5 dollars into a pot. First place gets a certain percentage of the money. Second and third place split the rest. Third place gets $4. What percentage of the money does first place get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80\%}
A small class of nine boys are to change their seating arrangement by drawing their new seat numbers from a box. After the seat change, what is the probability that there is only one pair of boys who have switched seats with each other and only three boys who have unchanged seats?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/32}
Three identical cones with vertex \( A \) and an apex angle of \(\frac{\pi}{3}\) touch each other externally. Each of them touches a fourth cone with vertex at point \( A \) internally. Find the apex angle of the fourth cone. (The apex angle of a cone is defined as the angle between its generatrices in an axial section.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3} + 2 \arcsin \frac{1}{\sqrt{3}}}
Evaluate the integral \( \iint _R \frac{y^2}{x} \, dx \, dy \) over the region \( R \) in the \( xy \)-plane bounded by \( y=x^2 \), \( y=4x^2 \), \( xy=1 \), and \( xy=5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
When $m$ is such that the function $y = (m+2)x + 4x - 5$ is a linear function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \neq -6}
Evaluate the limit \( \lim_{x\to 0^+} \frac{1}{x}\int_0^x \frac{f(t)}{t}\, dt \), where \( f \) is integrable on \([0,1]\) and \( \lim_{x\to 0} f(x)=L \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{L}
Given a circle O: x^2 + y^2 = 1 and a point P(-1, sqrt(3)), where A and B are two moving points on circle O, find the maximum value of angle APB. Express your answer in radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Evaluate the definite integral: \[ \int_{0}^{\frac{\pi}{2}}{\frac{dx}{\sqrt{\cos(x)}\cdot\sin(x)}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
The slant height of a cone is \( a \), and the distance from the vertex of the cone to the center of the inscribed sphere is \( b \). Find the angle between the slant height and the base plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \arctan \left(\frac{b}{a}\right)}
8. $\frac{\tan 96^{\circ}-\tan 12^{\circ}\left(1+\frac{1}{\sin 6^{\circ}}\right)}{1+\tan 96^{\circ} \tan 12^{\circ}\left(1+\frac{1}{\sin 6^{\circ}}\right)}=$ $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
Evaluate the definite integral: $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\cos^5(x)}{1+\sin^3(x)}\,dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 3}
The polynomial  $f(x)=x^3-4\sqrt{3}x^2+13x-2\sqrt{3}$  has three real roots,  $a$ ,  $b$ , and  $c$ . Find  $$ \max\{a+b-c,a-b+c,-a+b+c\}. $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3} + 2\sqrt{2}}
How many roots does the polynomial $z^6 - 5z^4 + 8z - 1$ have in the domain $|z| < 2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Bill is laying power cable for a new neighborhood. There are going to be 18 east-west streets that are 2 miles long and 10 north-south streets that are 4 miles long. It takes 5 miles of cable to electrify 1 mile of street due to underground wiring and redundancy measures. The cable costs vary, with east-west cables costing $2500/mile and north-south cables costing $3500/mile. Additional costs include $5000 for each of the 20 intersections. If there are also 5 streets that require a higher grade cable priced at $4000/mile, 3 of these streets being east-west and 2 being north-south, what is the total cost of cable for the neighborhood?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$1,530,000}
A circle with radius \( r \) passes through the vertices \( A \) and \( B \) of triangle \( ABC \) and intersects side \( BC \) at point \( D \). Find the radius of the circle passing through points \( A \), \( D \), and \( C \), given that \( AB = c \) and \( AC = b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{br}{c}}
Determine the smallest natural number $k$ such that the series \( \sum\limits_{n=1}^\infty\frac{(n!)^{k-1}(27n)^n}{(kn)!e^n} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the ellipse $\dfrac{x^2}{1-k} + \dfrac{y^2}{2+k} = 1$, if its focus lies on the $x$-axis, determine the range of values for $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\,2, -\dfrac{1}{2})}
Determine the Lebesgue measure of the set $A$ within the interval [0,1], where $A$ consists of numbers whose continued fraction expansions contain an infinite subsequence such that each term is divisible by its predecessor.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of points within the interval $[0, 6\pi]$ where the function $f(x)$, defined as $f(x) = \tan x$ for rational $x$ and $f(x) = x^2 + 1$ for irrational $x$, is continuous.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the residue at infinity for the function \( f(z)=\frac{\sin(\pi z)}{z^2(z-1)} \) by using the substitution \( z=\frac{1}{w} \) and evaluating the limit. What is the correct residue at infinity for \( f(z) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Find the coordinates of point \( A \) that is equidistant from points \( B \) and \( C \).

\( A(0, y, 0) \)

\( B(-2, 4, -6) \)

\( C(8, 5, 1) \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 17, 0)}
Determine the time complexity of an algorithm that uses Stirling's approximation to compute the hyperfactorial function for large input values $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(1)}
Peter has 25 cards, each printed with a different integer from 1 to 25. He wishes to place N cards in a single row so that the numbers on every adjacent pair of cards have a prime factor in common. What is the largest possible value of N? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given the function $f(x)= \frac {1}{x+2}$, point $O$ is the origin, point $A_{n}(n,f(n))$ where $n \in N^{*}$, vector $ \overrightarrow{i}=(0,1)$, $\theta_{n}$ is the angle between vector $ \overrightarrow{OA_{n}}$ and $ \overrightarrow{i}$. Find the range of real number $t$ such that $\frac {\cos \theta_{1}}{\sin \theta_{1}}+ \frac {\cos \theta_{2}}{\sin \theta_{2}}+ \frac {\cos \theta_{3}}{\sin \theta_{3}}+…+ \frac {\cos \theta_{n}}{\sin \theta_{n}} < t$ always holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t \geqslant \frac {3}{4}}
Reginald is selling apples. He usually sells each apple for $1.25, but he decides to offer a promotional discount for customers who buy in bulk. The discount offer is: if a customer buys 5 apples, they get 1 additional apple for free. He plans to use his profits to help repair his bike. His bike cost $80 and the repairs cost 25% of what his bike cost. After the repairs are paid for, 1/5 of the money he'd earned remains. Given that all customers purchasing apples from Reginald took advantage of the promotional discount, how many apples did he sell?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
In a compartment, any group of at least 3 passengers has a unique common friend. If friendship is mutual and no one is their own friend, determine the maximum number of friends that any single person in the compartment can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \(ABC\) be a triangle with \(AB=2021, AC=2022\), and \(BC=2023\). Compute the minimum value of \(AP+2BP+3CP\) over all points \(P\) in the plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6068}
Evaluate the limit: \[ \lim_{n\rightarrow\infty}\left(\sum\limits_{r=0}^{n}{\left(\frac{\binom{n}{r}}{n^r}\cdot\frac{1}{r+3}\right)}\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e - 2}
Determine the cardinality of the set of all injective functions from the natural numbers to the real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Compute the volume of the solid bounded by the surfaces: \( z = x^2 \), \( z = x + 2 \), \( y + z = 5 \), and \( y = -1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{99}{5}}
Evaluate the limit \(\lim\limits_{m \to \infty}\int_0^1 |\ln^m x\ln^n(1+x)|{\rm d}x\) where \(n \in \mathbb{N^+}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Consider two numerical sequences $X_n$ and $Y_n$. The sequence $X_n$ consists of random integers from one to six for each $n$. Define $Y_1 = X_1$, and for $n \geq 2$, $$Y_n = X_n + \frac{1}{Y_{n-1}}.$$ Determine the probability $P$ that $$\frac{1+\sqrt{3}}{2} \leq Y_n \leq 1+ \sqrt{3}$$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{5}}
Estimate an upper bound for \( \lim_{N \to \infty} \text{Pr}\left(\frac{Y_N}{N} \leq \alpha\right) \) where \( Y_N = \max_{1 \leq i \leq N} \left|\sum_{j=1}^{i} X_j\right| \) and \( X_1, X_2, \ldots, X_N \) are independent random variables with \( \text{Pr}(X_i = 1) = \text{Pr}(X_i = -1) = 1/2 \), for \( 0 < \alpha < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the functional derivative of the differential entropy with respect to the probability density function, i.e., determine \( \frac{\delta F[p]}{\delta p(x)} \), where \( F[p] = \int_X p(x)\ln(p(x))dx \) and \( p(x) \) is a probability density function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln(p(x)) + 1}
Evaluate the limit \( \lim_{n \to \infty} n^2 \int_0^1 \frac{dx}{(1+x^2)^n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the number of distinct diagonalizable matrices of size $9 \times 9$ with real entries that satisfy the equation $A^2 = I$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{512}
Find a group $G$ that contains an element $x$ of order 5 and an element $y$ of order 6, such that the order of the product $xy$ is less than 30.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_5}
Find the gradient \( \nabla_W J \) of the function \( J(W) = \left\| W^T W - I \right\|_F^2 \), where \( W \) is a matrix and \( \left\| \cdot \right\|_F \) denotes the Frobenius norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4W(W^T W - I)}
Evaluate the limit as \( n \to \infty \) of the expression:
\[
\frac{e^{1/n}}{n^2} + 2\frac{e^{2/n}}{n^2} + 3\frac{e^{3/n}}{n^2} + \ldots + n \frac{e^{n/n}}{n^2}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the value of \( p \) such that the function \( f(x,y) = \frac{1}{(x^2+y^3-2)^p} \) is integrable over the region defined by \( 1 \leq x \leq 2 \) and \( 1 \leq y \leq 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p < 2}