inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $f:\mathbb{R}^{3}\to\mathbb{R}^{2}$ be defined by $f(r,s,t)=(r^{3}s+t^{2},rst)$. Let $g:\mathbb{R}^{2}\to\mathbb{R}^{11}$ and $h:\mathbb{R}^{11}\to\mathbb{R}^{4}$ be differentiable functions. Compute the Jacobian matrix $J_{h\circ g\circ f}(0,21,0)$. What is the size of the resulting Jacobian matrix?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \times 3}
Calculate the flux of the vector field \( F(x,y,z) = (3z^2y, 2x^2y, 2y^2z) \) through the surface \( S = \{ (x,y,z) \in \mathbb{R}^3 : 2\sqrt{x^2+y^2} \leq z \leq 1+x^2+y^2 \} \). Use the divergence theorem to evaluate the flux.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{15}}
Evaluate the limit: $$\lim_{n\to \infty} \Biggl(\left(1+\dfrac{-n^2+2n-3}{n^3-n^2}\right)^{\dfrac{n^3-n^2}{-n^2+2n-3}}\Biggr)^{\frac{n^4-2n^3+3n^2}{n^3-n^2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find an element \( \alpha \) such that \( \mathbb{Q}(i,\sqrt[3]{2}) = \mathbb{Q}(\alpha) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i + \sqrt[3]{2}}
In an infinite and stationary universe, consider the intensity of light from distant objects. If the intensity falls off as $1/d^k$, where $d$ is the distance to the object, determine how the intensity changes as a function of dimensionality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{d^{n-1}}}
Let $G$ be a finite group of order $2p$, where $p$ is an odd prime. Suppose $G$ has a subgroup $H$ of order $p$ and for every element $g \in G$, $g^2 \in H$. Determine the number of elements of order 2 in the group $G$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the smallest number of subsets needed to partition the set of all integers such that none of the subsets contain an infinite arithmetic progression?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a Hilbert space $H$ with an orthonormal basis $(e_n)_{n \in \mathbb{N}}$, and a bounded linear operator $T \in \mathbf{B}(H)$ represented by the matrix $A = (\alpha_{nm})$ where $\alpha_{nm} = (Te_n|e_m)$, find the matrix representation of the adjoint operator $T^*$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\overline{\alpha_{mn}})}
The following diagram shows a square where each side has four dots that divide the side into three equal segments. The shaded region has area 105. Find the area of the original square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
135
}
Determine the values of \( n \) for which a regular hexagon can be divided into \( n \) parallelograms of equal area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3k}
In a regular dodecahedron, how many distinct planes can intersect the dodecahedron to form a regular hexagon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Real numbers $x,y,$ and $z$ are chosen from the interval $[-1,1]$ independently and uniformly at random. What is the probability that $$\vert{x}\vert+\vert{y}\vert+\vert{z}\vert+\vert{x+y+z}\vert=\vert{x+y}\vert+\vert{y+z}\vert+\vert{z+x}\vert?$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Jerry recently returned from a trip to South America where he helped two old factories reduce pollution output by installing more modern scrubber equipment. Factory A previously filtered $80$% of pollutants and Factory $B$ previously filled $72$% of pollutants. After installing the new scrubber system, both factories now filter $99.5$% of pollutants.

Jerry explains the level of pollution reduction to Michael, "Factory $A$ is the much larger factory. It's four times as large as Factory $B$. Without any filters at all, it would pollute four times as much as Factory $B$. Even with the better pollution filtration system, Factory $A$ was polluting nearly three times as much as Factory B."

Assuming the factories are the same in every way except size and previous percentage of pollution filtered, find $a+b$ where $a/b$ is the ratio in lowest terms of volume of pollutants unfiltered from both factories $\textit{after}$ installation of the new scrubber system to the volume of pollutants unfiltered from both factories $\textit{before}$ installation of the new scrubber system.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{221}
Determine the cardinality of the set of all sequences in \(\mathbb{R}\) that converge to a real number \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
If an infinite series $S$ converges to a number $N$, does the exponential function $e^S$ converge to $e^N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^N}
Find the value of \( c \) such that the sequence \((S^3_n - cn)_{n \in \mathbb{N}_0}\) is a martingale, where \( S_n \) is a random walk.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the splitting field and its degree over \( \mathbb{Q} \) for the polynomial \( f(x) = x^4 - 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
For a positive integer \( n \) we denote by \( d\left( n\right) \) the number of positive divisors of \( n \) and by \( s\left( n\right) \) the sum of these divisors. For
 example, \( d\left( {2018}\right) \) is equal to 4 since2018has four divisors(1,2,1009,2018)and
 \( s\left( {2018}\right)  = 1 + 2 + {1009} + {2018} = {3030} \) .
 Determine all positive integers \( x \) such that \( s\left( x\right)  \cdot  d\left( x\right)  = {96} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
14, 15, 47
}
One writes, initially, the numbers  $1,2,3,\dots,10$  in a board. An operation is to delete the numbers  $a, b$  and write the number  $a+b+\frac{ab}{f(a,b)}$ , where  $f(a, b)$  is the sum of all numbers in the board excluding  $a$  and  $b$ , one will make this until remain two numbers  $x, y$  with  $x\geq y$ . Find the maximum value of  $x$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1320}
Find all real numbers $x$ so that the product $(x + i)((x + 1) + i)((x + 2) + i)$ is pure imaginary.  Enter all the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3,-1,1}
Let $n\ge 2$ be a given integer. Find the greatest value of $N$, for which the following is true: there are infinitely many ways to find $N$ consecutive integers such that none of them has a divisor greater than $1$ that is a perfect $n^{\mathrm{th}}$ power.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n - 1}
Given an orthogonal $3 \times 3$ matrix $R$ represented as:

$$R = \begin{bmatrix}
r_{11}&r_{12}&r_{13} \\
r_{21}&r_{22}&r_{23} \\
r_{31}&r_{32}&r_{33} \\
\end{bmatrix}$$

where the parameters $r_{11}, r_{21}, r_{31}, r_{12}, r_{22}, r_{32}$ are known, determine how many unique solutions exist for the remaining parameters $r_{13}, r_{23}, r_{33}$. If there is a unique solution, describe the method to find it using the orthogonality condition, considering that $R$ represents a rotation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the average number of coin flips required such that the difference between the percentage of heads and tails is less than 0.1% with a probability of at least 99%.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6635776}
A father has a pie made for his two sons. Eating more than half of the pie will give indigestion to anyone. While he is away, the older son helps himself to a piece of the pie. The younger son then comes and has a piece of what is left by the brother. Assume that the size of each of the two pieces eaten by the sons is random and uniformly distributed over what is currently available. What is the expected size of the remaining piece given that no son has indigestion?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2 - \ln 2}{4 \ln 2}}
A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not by 100 (such as
 2012). The200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7,2012, a Tuesday. On what
 day of the week was Dickens born?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
 Friday 
}
Find the basis and determine the dimension of the tangent space at the identity for the symplectic group $Sp(2n, \mathbb{R})$. Provide your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(2n + 1)}
Evaluate the limit: \[ \lim_{n\to\infty}\int_0^\infty\frac{n\cos^2(x/n)}{n+x^4}dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
For each positive integer $n$, let $s(n)$ be the sum of the digits of $n$. Find the smallest positive integer $k$ such that
\[s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9999}
When two distinct digits are randomly chosen in $N=123456789$ and their places are swapped, one gets a new number $N'$ (for example, if 2 and 4 are swapped, then $N'=143256789$). The expected value of $N'$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute the remainder when $m+n$ is divided by $10^6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{962963}
What is the perimeter of the boundary of the region consisting of all points which can b
 expressed as \( \left( {{2u} - {3w},v + {4w}}\right) \) with \( 0 \leq  u \leq  1 \) , \( 0 \leq  v \leq  1 \) , and \( 0 \leq  w \leq  1 \) ?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
16
}
Let \( n \geq 2 \) be a given integer. For two permutations \((\alpha_1, \ldots, \alpha_n)\) and \((\beta_1, \ldots, \beta_n)\) of \(1, \ldots, n\), consider the \(n \times n\) matrix \(A = (a_{ij})_{1 \leq i,j \leq n}\) defined by \(a_{ij} = (1 + \alpha_i \beta_j)^{n-1}\). Determine all possible values of \(\det(A)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \pm \left( (n-1)! \right)^n }
公园里的自动旋转喷灌装置的喷灌半径是5米，它的最大喷灌面积是多少平方米．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78.5}
Evaluate the line integral \( \int_{\alpha} x \, dy - y \, dx \) where \( \alpha \) is the boundary of the parallelogram with vertices \((0,0), (a,b), (c,d), (a+c,b+d)\) oriented counterclockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(ad - bc)}
Find the minimum of $max ( |1+z|, |1+z^{2}|)$ if $z$ runs over all complex numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{3-\sqrt{5}}$}
Determine if there is a closed form solution to the equation $ae^a = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{W(1)}
The edges of $K_{2017}$ are each labeled with $1,2,$ or $3$ such that any triangle has sum of labels at least $5.$ Determine the minimum possible average of all $\dbinom{2017}{2}$ labels.
 
(Here $K_{2017}$ is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.)
 
[i]Proposed by Michael Ma[/i]

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4033}{2017}}
Determine the eigenvalues of the linear operator \( T: R[X]_n \rightarrow R[X]_n \) defined by \( T(f(x)) = f(ax+b) \), where \( R[X]_n \) is the vector space of polynomials of degree \( \leq n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, a, a^2, \ldots, a^n}
设数列 $\left\{x_{n}\right\}$ 满足 $x_{1} \in\{5,7\}$ 及当 $k \geqslant 1$时，有 $x_{k+1} \in\left\{5^{x_{k}}, 7^{x_{k}}\right\}$ 。试确定 $x_{2009}$ 的末两位数字的所有可能值，并给出这些可能值的和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
In the diagram, all rows, columns, and diagonals have the sum 12. Find the sum of the four corner numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
已知 2023 可以拆分为几个正整数之和，所有整数中的最大值和最小值相差不超过 1。有多少种可能？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2023}
Determine the number of onto functions \( f : A \to A \) such that \((f \circ f)(1) = 2\), where \( A = \{1, 2, 3, \ldots, 9\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35280}
Compute the expected value of $X_{n+1}$, given that $X_n = X_0 e^{\mu S_n}$, where $X_0 > 0$, $S_n$ is a symmetric random walk, and $\mu > 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X_n \cosh(\mu)}
Given a finite set $A$ with $n$ elements, find the smallest number of subsets required to generate the power set of $A$. Provide the general formula for this minimum size.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lceil \log_2 n \rceil}
Find all pairs of positive integers \((n, k)\) such that \((n + 1)^k - 1 = n!\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 1), (2, 1), (4, 2)}
Determine the isomorphism type of the quotient group $U_{26}/\langle 5 \rangle$, where $U_{26}$ is the group of units under multiplication modulo 26, the order of $U_{26}$ is 12, and the order of the subgroup generated by 5 is 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_3}
A department store received 10 suitcases and 10 keys separately, with the warning that each key opens only one suitcase and that each suitcase has a corresponding key. What is the expected number of attempts required to find the right key for each suitcase?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{65}{2}}
Find the supremum of the function \( \frac{F(x)(1-F(x))}{n} \) for \( x \in \mathbb{R} \), where \( F(x) \) is a cumulative distribution function and \( n \) is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4n}}
Consider the following balanced chemical reaction: 

2 K2Cr2O7 + 16 HCl -> 4 KCl + 4 CrCl3 + 8 H2O + 7 Cl2

Now, calculate the mass percentage of oxygen in K2Cr2O7 using the given balanced chemical equation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38.06\%}
A rectangle $ABCD$ is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of $ABCD$ if its area is $6750 cm^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{330}
Let $a \in \mathbb{R}$, and the function $f(x) = x^3 + ax^2 + (a-3)x$ has a derivative denoted by $f'(x)$. If $f'(x)$ is an even function, find the equation of the tangent line to the curve $y = f(x)$ at the origin. Express your answer in the form $y = c$, where $c$ is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was $84\%$ of the original price, by what percent was the price increased and decreased$?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40\%}
\(a, b, c\) are three different digits. If you add up all six two-digit numbers that can be formed using these digits, without repeating any digit in the same number, the total is 528. Find these digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7, 8, 9}
Find the infimum of the expression \[ \sqrt[5]{\frac{a}{b+c}}+\sqrt[5]{\frac{b}{c+d}}+\sqrt[5]{\frac{c}{d+e}}+\sqrt[5]{\frac{d}{e+a}}+\sqrt[5]{\frac{e}{a+b}} \] for positive real numbers $a, b, c, d, e$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
From a $6 \times 6$ grid square, gray triangles were cut out. What is the area of the remaining shape? The side length of each cell is 1 cm. Provide your answer in square centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Calculate the limit: 
$$\lim_{x \to 0^+}\frac{\tan x \cdot \sqrt{\tan x} - \sin x \cdot \sqrt{\sin x}}{x^3 \sqrt{x}}$$

without using L'Hôpital's Rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{3}{4}\)}
For a positive integer $n$, let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$. Find the smallest $n$ such that $\left|f_n^{\prime \prime}(0)\right|>2023$. Please provide the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
In the diagram, \( Z \) lies on \( XY \) and the three circles have diameters \( XZ \), \( ZY \), and \( XY \). If \( XZ = 12 \) and \( ZY = 8 \), then calculate the ratio of the area of the shaded region to the area of the unshaded region. Express your answer as a ratio in simplest form, with the two numbers separated by a colon (:).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12:13}
Given the equation $x^4 - 4x^3 + 4x^2 + ax + b = 0$, where $a$ and $b$ are real numbers, and the equation has two distinct positive roots $\alpha$ and $\beta$ such that $\alpha + \beta = 2\alpha\beta$, determine the maximum value of $a + b$. \(\text{Express your answer as a single number.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{16}{27}}
Evaluate the integral $$\int_1^\infty\frac{e^{4x}-e^{2x}}{x(e^{4x}+1)(e^{2x}+1)}\text{d}x$$ using the Frullani integral method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{2}\ln2\)}
Let the set $U = \{(x, y) | x \in \mathbb{R}, y \in \mathbb{R}\}$, $A = U \setminus \{(x, y) | x + y > m\}$, and $B = \{(x, y) | x^2 + y^2 \leq n\}$. Then, the necessary and sufficient condition for the point $(1, 2)$ to belong to $(C_U A) \cap B$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \geq 3, n \geq 5}
An  $m\times n(m,n\in \mathbb{N}^*)$  rectangle is divided into some smaller squares. The sides of each square are all parallel to the corresponding sides of the rectangle, and the length of each side is integer. Determine the minimum of the sum of the sides of these squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n - \gcd(m, n)}
Calculate the volume of the solid bounded by the surfaces $y^2 + z^2 - 2z = 0$, $x = y^2 + z^2$, and $x = 0$ using triple integrals. Determine the appropriate intervals of integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{2}}
Determine the positive values of $P$ for which the series \( \sum_{n=1}^{\infty}\frac{(n!)^3}{(Pn)!} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P \geq 3}
Consider a non-singular matrix $A$ that satisfies $A^2B = A$ with another matrix $B$. Given that $A$ is a $2 \times 2$ matrix represented as:

$$
A= \begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}
$$

where $a, b, c, d \in \{-1, 0, 1\}$. Determine the probability that the determinant of $A + B$ equals the sum of the determinants of $A$ and $B$, i.e., $\det(A + B) = \det(A) + \det(B)$. \(\boxed{?}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Determine whether the limit \( \lim_{x \to 0} \frac{1}{x} \int_{0}^{x} \left| \cos \left(\frac{1}{t}\right) \right| \, dt \) exists. If it exists, find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{\pi}}
## problem statement

Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4_{\text {and }}}$ its height dropped from vertex $A_{4 \text { to the face }} A_{1} A_{2} A_{3}$.

$A_{1}(4 ;-1 ; 3)$

$A_{2}(-2 ; 1 ; 0)$

$A_{3}(0 ;-5 ; 1)$

$A_{4}(3 ; 2 ;-6)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{\sqrt{5}}}
In three of the six circles of a diagram, the numbers 4, 14, and 6 are written. In how many ways can natural numbers be placed in the remaining three circles so that the products of the triplets of numbers along each of the three sides of the triangular diagram are the same?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the proof-theoretic ordinal of Primitive Recursive Arithmetic (PRA)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\omega^\omega}
实数$x _ { 1 } , x _ { 2 } , \cdots , x _ { 2 0 0 1 }$满足$\sum _ { k = 1 } ^ { 2 0 0 0 } \vert x _ { k } - x _ { k + 1 } \vert = 2 0 0 1 ,$，令$y _ { k } = \frac { 1 } { k } ( x _ { 1 } + x _ { 2 } + \cdots +$ $x _ { k } )$ k=1,2,⋯,2001.求$\sum _ { k = 1 } ^ { 2 0 0 0 } \vert y _ { k } - y _ { k + 1 }|$ 的最大可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
过半径为5的球面上的一点P作三条两两垂直的弦PA、PB、PC，且满足/PA|=2｜PB｜，则|PA|+|PB|+|PC|的最大值是
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ 2 \sqrt { 7 0 }$ }
Calculate the mass percentage of C in the biochemical compound adenosine triphosphate (ATP).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23.68\%}
What is the molecular weight of 8 moles of BaBr2
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2377.04}
Determine the value of $a$ such that the following integral equation holds:

$$\int \limits^{\infty }_{0} \frac{dx}{a^2 + \left(x - \frac{1}{x}\right)^2} = \frac{\pi}{5050}$$

where $a^2 \geq 0$. Provide your answer as a single value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2525}
Find four numbers that form a proportion, given that the sum of the first and last numbers is 14, the sum of the middle numbers is 11, and the sum of the squares of these four numbers is 221.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12, 8, 3, 2}
4.3. Two identical cylindrical vessels are connected at the bottom by a small cross-section pipe with a valve. While the valve was closed, water was poured into the first vessel, and oil into the second, so that the level of the liquids was the same and equal to \( h = 40 \) cm. At what level will the water in the first vessel stabilize if the valve is opened? The density of water is 1000 kg/ \(\mathrm{m}^{3}\), and the density of oil is 700 kg/ \(\mathrm{m}^{3}\). Neglect the volume of the connecting pipe. Give the answer in centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Solve the equation $\sqrt{a-\sqrt{a+ x}} = x$ for $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$x= \frac{-1+\sqrt{4a-3}}{2}$
}
A square was cut into four equal rectangles, and from these, a large letter P was assembled, as shown in the diagram, with a perimeter of 56. What is the perimeter of the original square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}