inclusionAI/Ring-lite-rl-data huggingface dataset:
Let \( f\left( z\right)  = \frac{z + a}{z + b} \) and \( g\left( z\right)  = f\left( {f\left( z\right) }\right) \) ,where \( a \) and \( b \) are complex numbers. Suppose that \( \left| a\right|  = 1 \) and \( g\left( {g\left( z\right) }\right)  = z \) for all \( z \)
 for which \( g\left( {g\left( z\right) }\right) \) is defined. What is the difference between the largest and smallest possible values of \( \left| b\right| \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \sqrt{3} - 1 \]}
In an Abelian group, if the orders of elements $a$ and $b$ are both 6, determine the possible values for the order of the element $ab$. Provide your answer as a set of integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1, 2, 3, 6\}}
Find the equation of the locus of points where the lengths of the tangents from the circles with equations \( x^2 + y^2 - 6x - 27 = 0 \) and \( x^2 + y^2 + 2x - 8 = 0 \) are equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8x + 19 = 0}
Determine the dimension of the coboundary chain-group of a graph with $n$ vertices and $k$ connected components.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - k}
Evaluate the limit: $$\lim_{x\to2\ \ y\to2}\frac{x^{6}+ \tan (x^{2}-y^{2}) - y^{6}}{\sin(x^{6}-y^{6}) - x^{5}y +xy^{5} + \arctan(x^{2}y -xy^{2})}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{49}{17}}
Find the sum of all natural numbers \( n \) such that \((n^2+n+1)^2\) divides the expression \(n^{2195} + n^{2194} + \ldots + n^2 + n + 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Calculate the accumulated value of two investments on March 31, 2010, given the force of interest function \( F(t) = \frac{0.02 + 0.01t}{1 + 0.02t + 0.005t^2} \), where \( t \) represents years starting from March 31, 2001. An initial investment of $100 was made on March 31, 2002, and another investment of $200 was made on March 31, 2006. What is the total accumulated value of these investments on March 31, 2010?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{413.41}
The equation of the line joining the complex numbers \(-5 + 4i\) and \(7 + 2i\) can be expressed in the form \(az + b \overline{z} = 38\) for some complex numbers \(a\) and \(b\). Find the product \(ab\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Calculate \( 516489222^{22} \mod 96899 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4647}
让$f（x） = \frac{\sin（x）}{x}$查找$$ \lim_{T\to\infty}\frac{1}{T}\int_0^T\sqrt{1+f'（x）^2}dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1
}
Evaluate the integral: \[ \int_{0}^{2\pi} e^{R \cos t} \cos(R \sin t + 3t) \, dt \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of non-zero coefficients in the Zhegalkin polynomial representation of the logical expression \(x_1 \lor x_2 \lor x_3 \lor \ldots \lor x_n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n - 1}
Determine the natural number \( n \) such that for any two sets \( \{ x_1, x_2, ..., x_n \} \) and \( \{ y_1, y_2, ..., y_n \} \) of \( n \) distinct points on the boundary \( \mathbb{S}^1 \) of the unit disk \( \mathbb{D} \), there exists a unique automorphism \( f: \mathbb{D} \rightarrow \mathbb{D} \) with \( f(x_i) = y_i \) for all \( i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Solve the Euler-Lagrange equation for the functional \( L(y) = \int_{-1}^7 \sqrt{1 + (y')^2} \, dx \) with the initial conditions \( y(0) = 1 \) and \( y(1) = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = x + 1}
Determine the number of zeros of \(\sin \pi x\) within the domain \(D = \{|z - 3 - 4i| < 6\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Calculate the mass of fluid flowing through the hemisphere defined by \( x^2 + y^2 + z^2 = 1, z \geq 0 \) in unit time. The fluid flow has a flux density vector \( F(x,y,z) = \langle x, -2x-y, z \rangle \). Let \( n \) denote the unit normal vector pointing out of the sphere. Find the mass of fluid flowing through the hemisphere in the direction of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{3}}
Suppose that $a$ and $b$ are positive integers such that $\frac{a}{b-20}$ and $\frac{b+21}{a}$ are positive integers. Find the maximum possible value of $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{143}
Evaluate the limit: \[ \lim_{n\rightarrow \infty}\frac{n!\cdot e^n}{\sqrt{n}\cdot n^n} \] without using Stirling's approximation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2\pi}}
Evaluate the limit: $$\lim_{x\rightarrow0}\frac{e^x \sin x - (x+1)\tan x}{x\log\cos(x)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $Y_1, \ldots, Y_{10}$ be a random sample from a normal distribution with mean $0$ and variance $\sigma^2$. Find the distribution of the statistic $\frac{10 \bar{Y}^2}{S^2}$, where $\bar{Y}$ is the sample mean and $S^2$ is the sample variance.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_{1,9}}
Determine the number of elements of order 15 in the symmetric group \( S_{13} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79567488}
Determine all positive integers \(n\) for which it is possible to construct an arithmetic sequence with \(10^n\) terms such that the digit sums of its terms form an arithmetic sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a full binary tree, determine the difference between the sum of the depths of all leaves and the sum of the depths of all internal nodes, expressed in terms of the number of internal nodes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n}
A 3 by 2 rectangle is split into four congruent right-angled triangles. Those four triangles are rearranged to form a rhombus. Calculate the ratio of the perimeter of the rectangle to the perimeter of the rhombus. Express your answer as a simplified ratio (e.g., a:b).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:1}
A certain mixture of nuts consists of 5 parts almonds, 2 parts walnuts, 3 parts cashews, and 4 parts pistachios, by weight. You have a budget of $1500 to purchase nuts for making 800 pounds of the mixture. Given that almonds cost $6 per pound, walnuts cost $5 per pound, cashews cost $8 per pound, and pistachios cost $10 per pound, and that almonds should make up at least 30% of the mixture, what is the maximum possible number of pounds of almonds that can be included in the 800-pound mixture while staying within the budget constraints?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Given the function $f(x) = e^{1+|x|} - \frac{1}{1+x^2}$, find the range of $x$ for which $f(x) > f(2x-1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{1}{3}, 1\right)}
设$\boldsymbol{u}=\left(u_{1},u_{2},u_{3}\right)$和$\boldsymbol{v}=$$\left(v_{1},v_{2},v_{3}\right)$是空间向量，满足$u_{i}、v_{i}(i=1,2,3)$均为整数，且
$0.9999<\cos\langle\boldsymbol{u},\boldsymbol{v}\rangle<1$。
记$S=\left|u_{1}\right|+\left|u_{2}\right|+\left|u_{3}\right|+\left|v_{1}\right|+\left|v_{2}\right|+\left|v_{3}\right|$。求$[\sqrt{S}]$的最小可能值，其中，$[x]$表示不超过实数$x$的最大整数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A set of six edges of a regular octahedron is called a Hamiltonian cycle if the edges in some order constitute a single continuous loop that visits each vertex exactly once. How many ways are there to partition the twelve edges into two Hamiltonian cycles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The range of the function \( y = \arcsin[\sin x] + \arccos[\cos x] \) (for \( x \in [0, 2\pi) \) and where \([x]\) denotes the greatest integer less than or equal to \( x \)) is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\{0, \frac{\pi}{2}, \pi\right\}}
Compute the number of ordered quadruples  $(a,b,c,d)$  of distinct positive integers such that  $\displaystyle \binom{\binom{a}{b}}{\binom{c}{d}}=21$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
What is the closed form for the sum $\sum_{i=0}^n {n \choose i} D(i)$, where $D(i)$ represents the number of derangements of $i$ elements? A derangement is a permutation of a set in which no element appears in its original position. For instance, $!3 = 2$ because there are two derangements of the set $\{1,2,3\}$: $\{2,3,1\}$ and $\{3,1,2\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
Two workers started paving paths in a park simultaneously from point $A$. The first worker paves the section $A-B-C$ and the second worker paves the section $A-D-E-F-C$. They worked at constant speeds and finished at the same time, having spent 9 hours on the job. It is known that the second worker works 1.2 times faster than the first worker. How many minutes did the second worker spend paving the section $DE$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
定义在$\left( -8，a \right)$上的奇函数$f\left( x \right)$在区间$\left[ 2，7 \right]$上是增函数，在区间$\left[ 3，6 \right]$上的最大值为$a$，最小值为-1，则$2f\left( -6 \right)+f\left( -3 \right)=$__________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$-15$}
Samantha has 10 green marbles and 5 purple marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 7 marbles. What is the probability that exactly four of the marbles that she removes are green? Express your answer as a decimal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.256}
Let $q(x)$ be a quadratic polynomial such that $[q(x)]^2 - x^2$ is divisible by $(x - 2)(x + 2)(x - 5)$. Find $q(10)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{250}{7}}
已知存在正整数 $a, b, c$ 满足 $a+b+c=407,10^{n} \mid a b c$ ，则 $n$ 的最大值是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In $\triangle ABC$, given $\angle A = 45^{\circ}$ and $\angle B = 60^{\circ}$, find the ratio of the distances from the circumcenter $O$ to each of the three sides of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 : \sqrt{2} : (\sqrt{3} - 1)}
通项为$a _ { n } = b [ \sqrt { n + c } ] + d$的数列，逐次算得各项是

$1,3,3,3,5,5,5,5,5,··$.

其中每一个正奇数$m$恰好连续出现$m$次.上述$b，c，d$是待定的整数.求$b+c+d $的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The base radius of a cone is $3cm$, and the slant height is $6cm$. Find the central angle of the sector in the unfolded diagram of the cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180^{\circ}}
Let $f(n) = \sum^n_{i=1}\frac{gcd(i,n)}{n}$. Find the sum of all positive integers $ n$ for which $f(n) = 6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1192}
What is the ratio of the sides of a triangle in which the sum of the lengths of the altitudes taken two at a time corresponds to the ratio 5:7:8?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10:15:6}
10. Given the sequence $\left\{a_{n}\right\}$ defined by $a_{1}=\frac{2}{3}, a_{n+1}=a_{n}^{2}+a_{n-1}^{2}+\cdots+a_{1}^{2}\left(n \in \mathbf{N}^{*}\right)$. If for any $n$ $\in \mathbf{N}^{*}, \frac{1}{a_{1}+1}+\frac{1}{a_{2}+1}+\cdots+\frac{1}{a_{n}+1}<M$ always holds, find the minimum value of $M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{57}{20}}
A moving point P starts from vertex A of the cube ABCD-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub> and moves along the edges to vertex C<sub>1</sub> and then back to A. If the path exactly passes through 6 different edges, such a path is called the "optimal path". The number of "optimal paths" is \_\_\_\_\_\_ (answer in digits).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
已知植圆 $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ 的左右焦点分别为 $F_{1}$ 与 $F_{2}$, 点 $P$ 在直线 $l: x-\sqrt{3} y+8+2 \sqrt{3}=0$ 上. 当 $\angle F_{1} P F_{2}$取最大值时，比 $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}$ 的值为 \sqrt{m}-n 的形式，求 m - n 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots , P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
9.5. Find the maximum value of the expression $(\sqrt{36-4 \sqrt{5}} \sin x-\sqrt{2(1+\cos 2 x)}-2) \cdot(3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y) \cdot$ If the answer is not an integer, round it to the nearest integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
In quadrilateral \(ABCD\), points \(X, Y, Z\) are the midpoints of segments \(AB, AD, BC\), respectively. It is known that \(XY\) is perpendicular to \(AB\), \(YZ\) is perpendicular to \(BC\), and the measure of angle \(ABC\) is \(100^\circ\). Find the measure of angle \(ACD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Josephine receives a bill from the hospital for 18,000$. The bill consists of various components with the following percentage allocations: X% of the bill is for medication, Y% is for imaging tests, and Z% is for a surgical procedure. The remaining bill has P% earmarked for overnight stays and Q% for doctor's fees. There is also a fixed fee of 300$ for food, 450$ for consultation, and 600$ for physical therapy sessions. The rest of the bill is for the ambulance ride. Given X=35%, Y=15%, Z=25%, P=10%, and Q=5%, how much did the ambulance ride cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$450}
A rectangular piece of paper measures $4$ units by $5$ units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called basic if    
$\mathrm{(i)}$ all four sides of the rectangle are segments of drawn line segments, and  
$\mathrm{(ii)}$ no segments of drawn lines lie inside the rectangle.  
Given that the total length of all lines drawn is exactly $2007$ units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{896}
The numbers \(1, 2, 3, \ldots, 400\) are written on 400 cards. Two players, \(A\) and \(B\), play the following game:

1. In the first step, \(A\) takes 200 cards for themselves.
2. \(B\) then takes 100 cards from both the remaining 200 cards and the 200 cards that \(A\) has, totaling 200 cards for themselves, and leaves the remaining 200 cards for \(A\).
3. In the next step, \(A\) takes 100 cards from both players' cards, totaling 200 cards for themselves, leaving the remaining 200 cards for \(B\).

This process continues until \(B\) completes the 200th step. At the end, the sums of the numbers on the cards held by \(A\) and \(B\) are calculated as \(C_A\) and \(C_B\) respectively. Then, player \(A\) pays player \(B\) the difference \(C_B - C_A\).

What is the maximum difference that \(B\) can achieve if both players play optimally?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20000}
Consider a product \(a_{1} a_{2} \ldots a_{100}\) written on a board, where \(a_{1}, \ldots, a_{100}\) are natural numbers. We consider 99 expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that exactly 32 of these expressions are even. What is the maximum number of even numbers among \(a_{1}, a_{2}, \ldots, a_{100}\) that could exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Given that the internal angles of a regular triangle and a dodecagon are 60° and 150° respectively, according to the conditions for plane tiling, the sum of the internal angles at a vertex must be 360°. Determine the number of possible ways to tile a plane using regular triangles and dodecagons. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Player A and player B are playing a game of Go. If player A plays as black first, the probability of player A winning is $\frac{1}{3}$; if player B plays as black first, the probability of player B winning is $\frac{1}{2}$. Assuming each game is independent and there are no draws, the loser of the first game plays first in the second game. If player A and player B play two games, with the first game having an equal probability of player A or player B playing as black first, then the probability of each player winning one game is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{29}{72}}
If the cost price of 80 articles is equal to the selling price of 60 articles, and an additional 15% tax is levied on the cost price, while a discount of 10% is given on the selling price, then determine the effective gain or loss percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.35\%}
In an equilateral triangle $\triangle ABC$ inscribed in the unit circle $C(0,1)$, find the maximum value of $\overline{PA}\cdot\overline{PB}\cdot\overline{PC}$, where $P$ is a variable point in the closed disk $\bar{D}(0,2)$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the double integral \( \iint_{D} \frac{y^2}{x} \, dx \, dy \), where \( D \) is the region bounded by the curves \( y = x^2 \), \( y = 4x^2 \), \( xy = 1 \), and \( xy = 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Evaluate the integral \( \iint_R \left[5+\frac{1}{4}\left( \frac{x^2}{c^2}-\frac{y^2}{d^2}\right)\right] \, dx \, dy \) where \( R \) is the region defined by \( \frac{x^2}{c^2}-\frac{y^2}{d^2} \ge 1 \). Use a suitable transformation to simplify the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the number of ordered triples \((a, b, c)\) of natural numbers such that \(a^2b^3c^4 = 2^{10} \times 3^{14}\) and \(\gcd(a, b, c) = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Evaluate the line integral \( \int_0^\pi F \cdot dr \) for the vector field \( F(x,y)=\langle 2y^2+y^2\cos(xy^2),4xy+2xy\cos(xy^2)\rangle \) along the path \( r(t)=\langle e^t\sin(t),e^t\cos(t)\rangle \), where \( t \in [0,\pi] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the probability that $X_1 + X_2 < 2$ and $X_1 - X_2 > 0$, given that $(X_1, X_2)$ follows a bivariate normal distribution with mean $\mu = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and covariance matrix $\Sigma = \begin{pmatrix} 3 & -1\\ -1 & 3 \end{pmatrix}$. Note that the events are not independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1116}
Find the largest real $m$, such that for all positive real $a, b, c$ with sum $1$, the inequality $\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m$ is satisfied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
m = 1
}
Using Stokes' Theorem, evaluate the line integral of the vector field \(\vec{F} = -y^3 \hat{i} + x^3 \hat{j} - z^3 \hat{k}\) along the curve \(C\), which is the intersection of the cylinder \(x^2 + y^2 = 1\) and the plane \(x + y + z = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{2}}
Determine the values of \( p \) for which the series \( \sum\limits_{n=1}^\infty n(1+n^2)^p \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -1)}
On a chessboard, \( n \) white rooks and \( n \) black rooks are placed such that rooks of different colors do not threaten each other. Find the maximum possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\cos 40^\circ,\sin 40^\circ)$, $(\cos 60^\circ,\sin 60^\circ)$, and $(\cos t^\circ,\sin t^\circ)$ is isosceles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{380}
In the circular sector \( OAB \), the central angle of which is \( 45^{\circ} \), a rectangle \( KMRT \) is inscribed. Side \( KM \) of the rectangle lies along the radius \( OA \), vertex \( P \) is on the arc \( AB \), and vertex \( T \) is on the radius \( OB \). The side \( KT \) is 3 units longer than the side \( KM \). The area of the rectangle \( KMRT \) is 18. Find the radius of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{13}}
There are 6 positive integers, from a certain number to 100, inclusive, that are not divisible by even integers greater than 1. What is the starting number of this range?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
Two distinct squares of the $8 \times 8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common. Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$, there exist two adjacent squares whose numbers differ by at least $g$. Determine the largest $C$-gap $g$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given a linear operator $T$ on an $n$-dimensional vector space $V$ (over $\mathbb R$), I want to find an orthonormal basis for $V$ in which the matrix of $T$ is sparse (has many zeros). What is the largest number $k=k(n)$ such that every real $n\times n$ matrix is orthogonally equivalent to a matrix with at least $k$ zero entries?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{2}n(n-1)\)}
Let the side lengths of a triangle be integers \( l \), \( m \), and \( n \), with \( l > m > n \). It is given that \( \left\{\frac{3^{l}}{10^{4}}\right\} = \left\{\frac{3^{m}}{10^{4}}\right\} = \left\{\frac{3^{n}}{10^{4}}\right\} \), where \(\{x\} = x - [x]\) and \([x]\) denotes the greatest integer less than or equal to \( x \). Find the minimum perimeter of such a triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3003}
Misha noticed that the tram passed by him in 2 seconds, and it went through a tunnel 96 meters long in 10 seconds. Find the speed of the tram (in meters per second), assuming it remains constant throughout the observation period.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 \text{ m/s}}