inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $M$ be a compact orientable manifold with boundary $\partial M$. Determine whether $H_n(M;\mathbb{R})$ is always zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $MATH$ be a square with $MA = 1$. Point $B$ lies on $AT$ such that $\angle MBT = 3.5 \angle BMT$. What is the area of $\vartriangle BMT$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}-1}{2}}
What is the sum of all possible values of $x$ such that $2x(x-10)=-50$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Consider the infinite strip \(S= \{x \in \mathbb{R}^3 : a < x_1 < b \} \) in three-dimensional Euclidean space. Determine whether the only \(L^2\) harmonic function in this strip is the zero function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $ABCDA'B'C'D'$ be a rectangular prism with $|AB|=2|BC|$. $E$ is a point on the edge $[BB']$ satisfying $|EB'|=6|EB|$. Let $F$ and $F'$ be the feet of the perpendiculars from $E$ at $\triangle AEC$ and $\triangle A'EC'$, respectively. If $m(\widehat{FEF'})=60^{\circ}$, then $|BC|/|BE| = ? $

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac32\sqrt{15}$}
Calculate the length of the longest rod that can fit inside the given figure and rotate around a corner.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Four circles,no two of which are congruent,have centers at \( A,B,C \) ,and \( D \) ,and point
\( P \) and \( Q \) lie on all four circles. The radius of circle \( A \) is \( \frac{5}{8} \) times the radius of circle
and the radius of circle \( C \) is \( \frac{5}{8} \) times the radius of circle \( D \) . Furthermore,
\( {AB} = {CD} = {39} \) and \( {PQ} = {48} \) . Let \( R \) be the midpoint of \( \overline{PQ} \) . What is
\( {AR} + {BR} + {CR} + {DR} \) ?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
192
}
Determine whether the value of \( \lim_{x \to 0^-} x^x \) is equal to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Example 7 Find the largest real number $c$ such that for any $n \in \mathbf{N}^{*}$, we have $\{\sqrt{2} n\}>\frac{c}{n}$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2 \sqrt{2}}}
Determine the largest subset of the set $\{A, B, \cdots, Z\}$, where each letter represents a mathematician, such that no two mathematicians in your subset have birthdates within 20 years of each other. Calculate your score for the subset using the formula $\max (3(k-3), 0)$, where $k$ is the number of elements in your subset.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{69}
Compute the integral \[ \frac{1}{2 \pi} \int_0^{2 \pi} \frac{1 - r^2}{1 - 2r \cos(\theta) + r^2} \, d\theta \] where \(0 < r < 1\). Express the integral as a complex line integral using complex analysis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $R$ be the ring of formal power series in $n$ indeterminates over $\mathbb{C}$, and let $(I_{k})_{k\in \mathbb{N}}$ be a strictly decreasing chain of unmixed radical ideals, all having the same height $s$. Assume that $\bigcap I_{n} = \mathfrak{p}$ is prime and that $I_{1}$ is prime. Is it true that $ht(\mathfrak{p}) = s$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s}
Consider the operator norm of the following $n \times n$ matrix:
$$
\|(I + 11^* + X^*X)^{-1} 11^* (I + 11^* + X^*X)^{-1}\|,
$$
where $X$ is an $n \times n$ matrix. Is this norm bounded by $\frac{1}{n}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{n}}
Call a positive integer $\textbf{monotonous}$ if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How many monotonous positive integers are there?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1524}
What is the number of ordered triples $(a,b,c)$ of positive integers, with $a\le b\le c\le 9$, such that there exists a (non-degenerate) triangle $\triangle ABC$ with an integer inradius for which $a$, $b$, and $c$ are the lengths of the altitudes from $A$ to $\overline{BC}$, $B$ to $\overline{AC}$, and $C$ to $\overline{AB}$, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3
}
Evaluate $2911_{11}-1392_{11}$. Express your answer in base 11, using A as the digit representing 10 in base 11.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{152A_{11}}
Determine whether the sequence \( \left(\alpha^{1/(n \cdot \ln n)}-1\right)^{1/n} \) converges or diverges for \( 2<\alpha<3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the minimum number of Fourier coefficients needed to uniquely identify a sparse signal \( f = \sum_{n=1}^N c_n \delta_{t_n} \), where \( N \in \mathbb{N} \), \( c_n \in \mathbb{C} \), and \( t_n \in \mathbb{R} \) for \( n=1, \dots, N \). The \( k \)-th Fourier coefficient of \( f \) is given by
\[
\hat{f}(k) = \sum_{n=1}^N c_n e^{-2\pi i k t_n}.
\]
Find the smallest set \( K \subset \mathbb{Z} \) such that if \( \hat{f}(k) = \hat{h}(k) \) for all \( k \in K \), then \( f = h \), where \( h = \sum_{n=1}^N d_n \delta_{s_n} \) is another linear combination of Dirac delta functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2N}
A projectile is launched with an initial velocity of $60\,\frac{m}{s}$ at an angle of $45^{\circ}$. At its highest point, it explodes into two fragments of equal mass. One fragment has zero velocity immediately after the explosion and falls vertically. Calculate the horizontal distance traveled by the other fragment, assuming $g=10\,\frac{m}{s^2}$. Provide your answer in meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{540}
Evaluate the integral: $$\int_0^\pi \left(\frac{\sin{2x}\sin{3x}\sin{5x}\sin{30x}}{\sin{x}\sin{6x}\sin{10x}\sin{15x}}\right)^2 dx$$ and verify that the result is $7\pi$ without using contour integrals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7\pi}
Given a positive integer $x > 1$ with $n$ divisors, define $f(x)$ to be the product of the smallest $\lceil\tfrac{n}{2}\rceil$ divisors of $x$. Let $a$ be the least value of $x$ such that $f(x)$ is a multiple of $x$, and $b$ be the least value of $n$ such that $f(y)$ is a multiple of $y$ for some $y$ that has exactly $n$ factors. Compute $a + b$. \( \text{Note: } X = x \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Two 10-digit integers are called neighbours if they diﬀer in exactly one digit (for example, integers  $1234567890$  and  $1234507890$  are neighbours). Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbours.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9 \cdot 10^8}
Determine if the radius of convergence of the series \( \sum \frac{f(n)}{g(n)}r^n \) is equal to \( R = 1 \), given that \( f(x) \) and \( g(x) \) are polynomials and \( g(n) \neq 0 \) for each \( n \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
一个圆柱形稻谷堆，它的底面直径10米，高1.5米．如果每立方米稻谷重1100千克，这稻谷堆重多少千克？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{129525}
Calculate the directional derivative of the function \( f(x,y) = \frac{x^2y}{x^4+y^2} \) at the point \((0,0)\) in the direction making an angle of \(30^{\circ}\) with the positive \(x\)-axis. Determine if this derivative exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Determine whether the series \( \sum_{k=0}^\infty \frac{x^k}{2^k(k+1)!} \) is defined at \( x=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $P_n$ be the space of polynomials over $\mathbb{R}$ of degree at most $n$. Consider the map $T:P_n\to P_{n+1}$ defined by $T(p(x))=p'(x)-\int_{0}^{x}p(t)dt$. Determine whether $\ker T=\{0\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ker T = \{0\}}
Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1540}
We have a calculator with two buttons that displays and integer  $x$ . Pressing the first button replaces  $x$  by  $\lfloor \frac{x}{2} \rfloor$ , and pressing the second button replaces  $x$  by  $4x+1$ . Initially, the calculator displays  $0$ . How many integers less than or equal to  $2014$  can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here,  $\lfloor y \rfloor$  denotes the greatest integer less than or equal to the real number  $y$ ).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{233}
Find the minimum value of \(\int _{0}^{1}[g''(x)]^2dx\) for functions \(g\in \mathbb{B} = \{g\in R([0,1]):g(0)=g(1)=1,g'(0)=b\}\) and determine the function \(g\) for which the minimum is achieved.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3b^2}
A cross-pentomino is a shape that consists of a unit square and four other unit squares each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius $R,$ what is $100R^2$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
250
}
Define$$S(n)=\sum\limits_{k=1}^{n}(-1)^{k+1}\,\, , \,\, k=(-1)^{1+1}1+(-1)^{2+1}2+...+(-1)^{n+1}n$$Investigate whether there are positive integers $m$ and $n$ that satisfy $S(m) + S(n) + S(m + n) = 2011$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{NO
}
10.1. Find the smallest of the solutions to the inequality

$$
\frac{-\log _{2}(120-2 x \sqrt{32-2 x})^{2}+\left|\log _{2} \frac{120-2 x \sqrt{32-2 x}}{\left(x^{2}-2 x+8\right)^{3}}\right|}{5 \log _{7}(71-2 x \sqrt{32-2 x})-2 \log _{2}(120-2 x \sqrt{32-2 x})} \geqslant 0
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-13-\sqrt{57}}
nic $\kappa$ y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers  $d>0$  such that for every positive integer  $n,$  nic $\kappa$ y can label at least  $dn^2$  cells of an  $n\times n$  square.
 
*Proposed by Mihir Singhal and Michael Kural*

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ d = \frac{2}{3} }
Determine the type of singularity and the residue of the function \( f(z) = \frac{\cot(z)}{(z - \frac{\pi}{2})^2} \) at \( z = \frac{\pi}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
4. Let $A=\{1,2,3, \cdots, 1997\}$, for any 999-element subset $X$ of $A$, if there exist $x, y \in X$, such that $x<y$ and $x \mid y$, then $X$ is called a good set. Find the largest natural number $a(a \in A)$, such that any 999-element subset containing $a$ is a good set.
(《Mathematics in Middle School》1999 Issue 1 Olympiad Problems)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{665}
Let $F \subseteq E$ be a field extension and let $\alpha \in E$ be transcendental over $F$. Determine the degree of the extension $[F(\alpha) : F(\alpha^3)]$. Justify why the degree cannot be 1 or 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the integral: $$\int_{0}^{\pi/2} \frac{\sin x \cos^5 x}{(1-2\sin^2x\cos^2x)^2}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{8}}
Evaluate the integral \( \int_{0}^{2\pi} \frac{1}{\exp(\sin(x))+\exp(\cos(x))}\, dx \) and determine if an analytical solution exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
p12. In convex pentagon $PEARL$, quadrilateral $PERL$ is a trapezoid with side $PL$ parallel to side $ER$. The areas of triangle $ERA$, triangle $LAP$, and trapezoid $PERL$ are all equal. Compute the ratio $\frac{PL}{ER}$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\sqrt{5}-1}{2}$}
Find all non-negative integers $n$, $a$, and $b$ satisfying
\[2^a + 5^b + 1 = n!.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ (a, b, n) = (2, 0, 3) }
Let $ A $ be a set of natural numbers, for which for $ \forall n \in \mathbb{N} $ exactly one of the numbers $ n $, $ 2n $, and $ 3n $ is an element of $ A $. If $ 2 \in A $, show whether $ 13824 \in A $.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13824 \notin A}
Determine if the integral \( \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx \) can be solved without using complex analysis methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{4}}
For every positive integer $n$ let $\tau (n)$ denote the number of its positive factors. Determine all $n \in N$ that satisfy the equality $\tau (n) = \frac{n}{3}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9, 18, 24}
For every integer $n\ge2$, let $\text{pow}(n)$ be the largest power of the largest prime that divides $n$. For example $\text{pow}(144)=\text{pow}(2^4\cdot3^2)=3^2$. What is the largest integer $m$ such that $2010^m$ divides

$$
\prod_{n=2}^{5300}\text{pow}(n)?
$$




Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77
}
Determine if there exists a point $p$ in the unit disk $\mathbb{D}$ such that the surface $S$ defined by \( E = G = \frac{4}{(1-u^2-v^2)^4}, \ F = 0 \) is geodesically complete at $p$. Answer with 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n+1}{4n-2}}
9. Let $m$ be a positive integer, and let $T$ denote the set of all subsets of $\{1,2, \ldots, m\}$. Call a subset $S$ of $T \delta$-good if for all $s_{1}, s_{2} \in S, s_{1} \neq s_{2},\left|\Delta\left(s_{1}, s_{2}\right)\right| \geq \delta m$, where $\Delta$ denotes symmetric difference (the symmetric difference of two sets is the set of elements that is in exactly one of the two sets). Find the largest possible integer $s$ such that there exists an integer $m$ and a $\frac{1024}{2047}$-good set of size $s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2048}
$(x_{n})_{-\infty<n<\infty}$  is a sequence of real numbers which satisfies  $x_{n+1}=\frac{x_{n}^2+10}{7}$  for every  $n \in \mathbb{Z}$ . If there exist a real upperbound for this sequence, find all the values  $x_{0}$  can take.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2, 5]}
Number Example 28 (2005 National High School Mathematics Competition Question) For each positive integer $n$, define the function
interval
question
$f(n)=\left\{\begin{array}{ll}0, & \text { if } n \text { is a perfect square, } \\ {\left[\frac{1}{\{\sqrt{n}\}}\right] ; \text { if } n \text { is not a perfect square. }}\end{array}\right.$
(Here $[x]$ denotes the greatest integer not exceeding $x$, and $\{x\}=x-[x]$). Try to find: $\sum_{k=1}^{240} f(k)$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{768}
$2015$ in binary is $11111011111$, which is a palindrome. What is the last year which also had this property?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1967}
Let $V$ be an inner product space and $B$ be a closed subspace of $V$. Determine whether $(B^\bot)^\bot = B$ is true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(B^\bot)^\bot = B}
Assume integer  $m \geq 2.$  There are  $3m$  people in a meeting, any two of them either shake hands with each other once or not.We call the meeting " $n$ -interesting", only if there exists  $n(n\leq 3m-1)$  people of them, the time everyone of whom shakes hands with other  $3m-1$  people is exactly  $1,2,\cdots,n,$  respectively. If in any " $n$ -interesting" meeting, there exists  $3$  people of them who shake hands with each other, find the minimum value of  $n.$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2m+1 }
Determine the number of digits, \(\mathbb{L}\), in the sum of divisors of the number \(2^2 \cdot 3^3 \cdot 5^3 \cdot 7^5\) that are of the form \(4n+1\), where \(n\) is a natural number. What is \(\mathbb{L}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Eight politicians stranded on a desert island on January 1st, 1991, decided to establish a parliament.
They decided on the following rules of attendance:

(a) There should always be at least one person present on each day.

(b) On no two days should the same subset attend.

(c) The members present on day $N$ should include for each $K<N$, $(K\ge 1)$ at least one member who was present on day $K$.

For how many days can the parliament sit before one of the rules is broken?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
$n$  is a fixed natural number. Find the least  $k$  such that for every set  $A$  of  $k$  natural numbers, there exists a subset of  $A$  with an even number of elements which the sum of it's members is divisible by  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ k = 2n }
Find the Cramer-Rao Lower Bound for the variance of any unbiased estimator of \( \log \sigma \), where \( X_1, X_2, \ldots, X_n \) form a random sample from a normal distribution with mean 0 and unknown standard deviation \( \sigma > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2n}}
Two knights placed on distinct square of an $8\times8$ chessboard, whose squares are unit squares, are said to attack each other if the distance between the centers of the squares on which the knights lie is $\sqrt{5}.$ In how many ways can two identical knights be placed on distinct squares of an $8\times8$ chessboard such that they do NOT attack each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1848}
Evaluate the sum \( \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{\alpha}} \) and determine its limit as \( \alpha \) approaches \( 0^+ \) by grouping successive terms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
As the order of trees tends to infinity, does the ratio of the number of trees of order \(2n\) with a perfect matching to the total number of trees of order \(2n\) converge to zero?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $P(x) = x^{2013}+4x^{2012}+9x^{2011}+16x^{2010}+\cdots + 4052169x + 4056196 = \sum\limits_{j=1}^{2014}j^2x^{2014-j}.$ If $a_1, a_2, \cdots a_{2013}$ are its roots, then compute the remainder when $a_1^{997}+a_2^{997}+\cdots + a_{2013}^{997}$ is divided by 997.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{993
}
Given that
 \( {34}! = {95232799cd96041408476186096435ab000000}{}_{\left( {10}\right) }, \)
 determine the digits \( a,b,c \) , and \( d \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
{a, b, c, d} = {2, 0, 0, 3}}
Let \( g(x) = C \) be a constant function. Determine whether the Riemann-Stieltjes integral \( \int_a^b f \, dg \) exists for any function \( f \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the convergence or divergence of the improper integral $\int^\infty_1 \frac{e^{1/x}}{x^2} dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e - 1}
A circle with radius \( r \) is inscribed in a right trapezoid. Find the sides of the trapezoid if its shorter base is equal to \( \frac{4r}{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2r, \frac{4r}{3}, \frac{10r}{3}, 4r}
In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{3}}
Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three aligned numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively.  The area of quadrilateral $ABDF$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{320}
What is the difference between the compound interest on Rs. 8000 for 1 1/2 years at a certain interest rate per annum compounded yearly and half-yearly, if the difference is Rs. 3.263999999999214?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.263999999999214}
Each cell of a  $29 \times 29$  table contains one of the integers  $1, 2, 3, \ldots , 29$ , and each of these integers appears  $29$  times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A manufacturer of airplane parts makes a certain engine that has a probability $p$ of failing on any given flight. There are two planes that can be made with this sort of engine, one that has 3 engines and one that has 5. A plane crashes if more than half its engines fail. For what values of $p$ do the two plane models have the same probability of crashing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, \frac{1}{2}, 1}
从1,2,3,4,5,7,9这7个数字中任取两个数字分别做成对数的底数和真数，则可构成不相等的对数值的数目是.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Two tourists started from $A$ to $B$ at the same time. The first tourist walked the first half of the time with a speed of $5 \mathrm{km/h}$ and the remaining time with a speed of $4 \mathrm{km/h}$. The second tourist walked the first half of the distance with a speed of $5 \mathrm{km/h}$ and the remaining distance with a speed of $4 \mathrm{km/h}$. Who will arrive at $B$ first?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{First tourist}