inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $P A B C$ be a tetrahedron such that $\angle A P B=\angle A P C=\angle B P C=90^{\circ}, \angle A B C=30^{\circ}$, and $A P^{2}$ equals the area of triangle $A B C$. Compute $\tan \angle A C B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8+5 \sqrt{3}}
A group \( G \) of order 15 acts on a set of order 22 with no fixed points. Determine the number of orbits in this group action.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the Galois group of the polynomial \( f(x) = x^3 + 2x + 2 \) over the field \( \mathbb{Q}(i\sqrt{19}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_3}
Consider a Markov chain \( \{X_n\} \) on \( \mathbb{N}_0 \) with transition probabilities \( p(x,x+1) = p(x,0) = \frac{1}{2} \) for all \( x \in \mathbb{N}_0 \). Find the expected time \( E(T_0 \mid X_0 = 1) \), where \( T_0 = \inf\{k \geq 0 : X_k = 0\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
若 $A B$ 为过抛物线 $y^{2}=4 x$ 的焦点 $F$ 的弦, $O$ 为坐标原点, 且 $\angle O F A=135^{\circ}$, $C$ 为抛物线的准线与 $x$ 轴的交点, 求 $\angle A C B$ 的正切值。原始答案的形式为 $k \sqrt{m}$，请给出 k + m 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the series \( \sum_{n=1}^{\infty}\frac{f(n)}{n(n+1)} \), where \( f(n) \) is the number of ones in the binary representation of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 2}
Find the analytic extension of the function $f(x) = |x|$ from the real numbers to the complex plane, excluding zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{z^2}}
In quadrilateral $ABCD$, $\angle DAB=\angle ABC=110^{\circ}$, $\angle BCD=35^{\circ}$, $\angle CDA=105^{\circ}$, and $AC$ bisects $\angle DAB$. Find $\angle ABD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{${40^\circ}$}
Let \( \varphi: \mathbb{R}^n \rightarrow \mathbb{R}^{n+k} \) be a global parametrization for the differentiable manifold \( M = \varphi(\mathbb{R}^n) \subseteq \mathbb{R}^{n+k} \). The inner product \( \langle , \rangle_{\varphi(0)} \) in \( T_{\varphi(0)}M \) is such that \( \{\frac{\partial}{\partial x_1}\vert_{\varphi(0)}, \ldots, \frac{\partial}{\partial x_n}\vert_{\varphi(0)}\} \) is an orthonormal basis. For \( a \in \mathbb{R}^n \), consider the map \( L_a: M \rightarrow M \) defined by \( L_a(\varphi(x)) = \varphi(x+a) \). Choose \( \langle , \rangle_{\varphi(a)} \) as an inner product in \( T_{\varphi(a)}M \) such that \( DL_a(\varphi(0)): T_{\varphi(0)}M \rightarrow T_{\varphi(a)}M \) is an Euclidean isometry. Find the components \( g_{ij} \) of the metric tensor \( (M, \langle , \rangle) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g_{ij} = \delta_{ij}}
Determine the value(s) of \(\alpha\) in the matrix \(A\) such that the geometric multiplicity of the eigenvalue \(\lambda = 4\) is 2, where \(A\) is a \(4 \times 4\) matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A collection of 8 cubes consists of one cube with edge-length $ k $ for each integer $ k, 1 \le k \le 8. $ A tower is to be built using all 8 cubes according to the rules: 

* Any cube may be the bottom cube in the tower.
* The cube immediately on top of a cube with edge-length $ k $ must have edge-length at most $ k+2. $ 

Let $ T $ be the number of different towers than can be constructed. What is the remainder when $ T $ is divided by 1000?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{458}
Find the smallest possible value of \( n \) such that there are exactly 77,000 ordered quadruples \((a, b, c, d)\) with \( \gcd(a, b, c, d) = 77 \) and \( \mathrm{lcm}(a, b, c, d) = n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27720}
Evaluate the integral \( \int_{-\infty}^\infty \Gamma(1+ix)\Gamma(1-ix)\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Determine the convergence rate of Newton's method when the slope is not updated in each iteration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the cardinality of the set $X$, where $X$ consists of all non-decreasing functions from $\mathbb{R}$ to $\{0,1\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
In triangle $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}$ and $\overline{AB}$, respectively. Let $P$ be the intersection of $\overline{BE}$ and $\overline{CF}$. Given that $PB = 14$, $PC = 4$, $PE = 7$, and $PF = 2$, compute the maximum possible area of $\triangle ABC$. \(\text{Proposed by Eugene Chen}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Let $A$ be an $R$-right module, $N$ be a submodule of the $R$-left module $M$, and $\pi: M \rightarrow M/N$ be the natural epimorphism. Determine $\ker(\pi \otimes 1_A)$, where $1_A$ is the identity map on $A$. Provide your answer in terms of submodules of $M \otimes_R A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N \otimes_R A}
9. [55] Let $N$ be the smallest positive integer for which
$$
x^{2}+x+1 \quad \text { divides } \quad 166-\sum_{d \mid N, d>0} x^{d} \text {. }
$$

Find the remainder when $N$ is divided by 1000 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{672}
Given the function $f(x)=\frac{1}{1+{2}^{x}}$, if the inequality $f(ae^{x})\leqslant 1-f\left(\ln a-\ln x\right)$ always holds, then the minimum value of $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{e}}
Select 3 teachers from 6 to teach in 3 different poverty-stricken areas, one in each area. Teachers A and B cannot go to the same area, and teachers A and C must either both go or both not go. How many different selection plans are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Grisha wrote 100 numbers on the board. Then he increased each number by 1 and noticed that the product of all 100 numbers did not change. He increased each number by 1 again, and again the product of all the numbers did not change, and so on. Grisha repeated this procedure $k$ times, and each of the $k$ times the product of the numbers did not change. Find the largest possible value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
已知椭圆的中心在原点 $O$, 焦点在 $x$ 轴上, 离心率为 $\frac{\sqrt{3}}{2}$, 且过点 $\left(\sqrt{2}, \frac{\sqrt{2}}{2}\right)$. 设不过原点 $O$ 的直线 $l$ 与该椭圆交于点 $P$ 和 $Q$, 且直线 $O P, P Q, O Q$ 的斜率构成等比数列, 求 $\triangle O P Q$ 面积的取值范围.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(0,1)$}
8. (5 points) The founder of a noble family received a plot of land. Each man in the family, upon dying, divided the land he inherited equally among his sons. If he had no sons, the land went to the state. No other members of the family gained or lost any land in any other way. In total, there were 200 people in the family. What is the smallest fraction of the original plot that any member of the family could have received?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4\cdot3^{65}}}
Suppose that the edges of a graph with 10 vertices can be colored with two colors such that the graph does not contain a monochromatic triangle. What is the maximum number of edges the graph can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Suppose you forgot one digit of the phone number you need and you are dialing it at random. What is the probability that you will need to make no more than two calls?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2}
Determine the number of values of $x$ for which the function $f(x) = \cos x + \cos(\sqrt{2}x)$ attains its absolute minimum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the number of ring homomorphisms from \( \mathbb{Z}_{12} \) to \( \mathbb{Z}_{18} \), excluding the zero homomorphism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the definite integral \( \int_{-2}^{2}\int_{0}^{\sqrt{1-x^2}}\sqrt[3]{x^2+y^2} \, dx \, dy \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{8}}
已知函数 $f:\{1,2, \cdots, 10\} \rightarrow\{1,2, \cdots, 5\}$ ，且对一切 $k=1,2, \cdots, 9$ ，有
$|f(k+1)-f(k)| \geqslant 3$.
则符合条件的函数 $f$ 的个数是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
For \( n = 2^k \) where \( k \ge 1 \), find a formula for the number of silver matrices of size \( n \). A silver matrix of size \( n \) is an \( n \times n \) matrix where the \( i \)-th row and \( i \)-th column contain all the numbers in \( \{1, 2, \ldots, 2n-1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2n - 1)!}
There is a board of size 7×12 cells and a cube, the side of which is equal to a cell. One face of the cube is painted with non-drying paint. The cube can be placed in a certain cell of the board and rolled over an edge to an adjacent face. The cube cannot be placed twice in the same cell. What is the maximum number of cells that the cube can visit without soiling the board with paint?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Let  $A$  be a subset of  $\{1, 2, \dots , 1000000\}$  such that for any  $x, y \in A$  with  $x\neq y$ , we have  $xy\notin A$ . Determine the maximum possible size of  $A$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999001}
Given that point $P$ moves on the parabola $y^2 = 8x$, $F$ is the focus of the parabola, and point $A$ has coordinates $(5,2)$. Find the minimum value of $PA + PF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Final answer is 7.}
Two \(4 \times 4\) squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. Find the area of the shaded region created by removing the circle from the squares. The answer is in the form \(k - m\pi\), please provide the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given that Figure 1 is the graph of an even function \( f(x) \) and Figure 2 is the graph of an odd function \( g(x) \). 

Let \(a\), \(b\), \(c\), and \(d\) be the number of real roots of the equations \( f(f(x)) = 0 \), \( f(g(x)) = 0 \), \( g(g(x)) = 0 \), and \( g(f(x)) = 0 \) respectively. 

Then, find the value of \( a + b + c + d \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
There are eight points on a circle that divide the circumference equally. Count the number of acute-angled triangles or obtuse-angled triangles that can be formed with these division points as vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice that of angle $B$, and the measures of segments $AD$ and $CD$ are $a$ and $b$ respectively. Find the measure of $AB$ in terms of $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a+b}
Calculate \(\int \int \vec{F} \cdot d\vec{S}\) over the surface defined by \(x = u^2\), \(y = uv\), \(z = \frac{1}{2} v^2\) bounded by the curves \(u = 0\), \(u = 1\), \(v = 0\), and \(v = 3\), for the vector field \(\vec{F} = y \vec{i} - x \vec{j} + xy \vec{k}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{111}{8}}
Consider a rectangular array of single digits \(d_{i,j}\) with 10 rows and 7 columns, such that \(d_{i+1, j} - d_{i, j}\) is always 1 or -9 for all \(1 \leq i \leq 9\) and all \(1 \leq j \leq 7\). For \(1 \leq i \leq 10\), let \(m_{i}\) be the median of \(d_{i,1}, \ldots, d_{i, 7}\). Determine the least and greatest possible values of the mean of \(m_{1}, m_{2}, \ldots, m_{10}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.5}
A person can be in one of three states: employed, unemployed, or early retirement. The transition rates per year are as follows: an employed person becomes unemployed with a rate of 0.2, an unemployed person becomes employed with a rate of 4, and an unemployed person retires early with a rate of 0.5. A working person never retires directly, and a retired person stays retired forever. A person starts employed, earning $280,000 per year while employed and $224,000 per year while unemployed, with no earnings in retirement. Calculate the expected total earnings before the person retires early.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13048000}
Alice and Bob are playing a coin flipping game where they flip coins until one of them wins. Alice wins when the sequence "h, h, h" appears, and Bob wins if the sequence "t, h, t" appears. Find the expected number of coin flips, $E[X]$, until one of them wins.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{35}{6}}
Given a fixed number $c$ and random variables $X_i$ and $Y_i$ for $i=1,2,...,n$, where $|P(X_i<c)-P(Y_i<c)| \leq \alpha$ for any $i$ and $\alpha$ is a constant, find an upper bound for $|P(\max_i X_i<c)-P(\max_i Y_i<c)|$. Assume that $X_i$ and $Y_i$ are not independent and identically distributed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \alpha}
Evaluate the integral \( \iint_R y\ dx\ dy \) where \( R \) is the region bounded by the \( x \)-axis and the parabolas \( y^2 = 4-4x \) and \( y^2=4+4x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $n$ be the maximum number of bishops that can be placed on the squares of a $6 \times 6$ chessboard such that no two bishops are attacking each other. Let $k$ be the number of ways to put $n$ bishops on an $6 \times 6$ chessboard such that no two bishops are attacking each other. Find $n+k$. (Two bishops are considered to be attacking each other if they lie on the same diagonal. Equivalently, if we label the squares with coordinates $(x, y)$, with $1 \leq x, y \leq 6$, then the bishops on $(a, b)$ and $(c, d)$ are attacking each other if and only if $|a-c|=|b-d|$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
A point is “bouncing” inside a unit equilateral triangle with vertices(0,0),(1,0), and \( \left( {1/2,\sqrt{3/2}}\right) \) . The point moves in straight
 lines inside the triangle and bounces elastically off an edge at an angle equal to the angle of incidence. Suppose that the point
 starts at the origin and begins motion in the direction of(1,1). After the ball has traveled a cumulative distance of \( {30}\sqrt{2} \) ,compute its distance from the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$30 - 17\sqrt{3}$}
Calculate the expected number of solutions found in an infinitely large, balanced search tree using a depth-first search algorithm, given that the probability of finding a solution at level $l$ is $p^l$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/(1-p)}
Let $S$ be the set of all positive integers between 1 and 2017, inclusive. Suppose that the least common multiple of all elements in $S$ is $L$. Find the number of elements in $S$ that do not divide $\frac{L}{2016}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
44
}
Using the systematic sampling method to select 32 people for a questionnaire survey from 960 people, determine the number of people among the 32 whose numbers fall within the interval [200, 480].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["10"]}
Let $A, B, C$ be points in that order along a line, such that $A B=20$ and $B C=18$. Let $\omega$ be a circle of nonzero radius centered at $B$, and let $\ell_{1}$ and $\ell_{2}$ be tangents to $\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\ell_{1}$ and $\ell_{2}$. Let $X$ lie on segment $\overline{K A}$ and $Y$ lie on segment $\overline{K C}$ such that $X Y \| B C$ and $X Y$ is tangent to $\omega$. What is the largest possible integer length for $X Y$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
A cube has six faces painted with six different colors: red, blue, green, yellow, purple, and orange. The cube is then cut into 125 small equal cubes. Calculate the number of small cubes with only one face colored for each of the six different colors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
In how many ways can 10 men and 4 women be seated at a round table such that no three women sit consecutively?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5356108800}
有 2022 个格排成一行，A,B 两人在玩游戏. 初始时，在每个左起奇数号格中填入 A、偶数号格中填入 B. 接下来从 A 开始，两人轮流进行以下操作：选择两个写有自己名字的格，这两个格不相邻，且它们之间的所有格都写有另一人的名字，然后将这些格都改为自己的名字. 当某人不能操作时游戏结束. 求最大的正整数 m，使得无论 B 如何操作，A 能保证游戏结束时总有至少 m 个格写有 A 的名字.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1011}
Let \(i_{1}, i_{2}, \cdots, i_{10}\) be a permutation of \(1, 2, \cdots, 10\). Define 
\[ 
S = \left|i_{1} - i_{2}\right| + \left|i_{3} - i_{4}\right| + \cdots + \left|i_{9} - i_{10}\right|, 
\]
find all possible values of \(S\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25}
Determine the number of integer solutions \((x_1, x_2, x_3, x_4, x_5)\) to the system of equations \(\sum_{i=1}^5 x_i = 10\) and \(\sum_{i=1}^5 x_i^2 = 40\), where each \(x_i\) is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130}
Determine the smallest number \( k \) of elements \( s_1, \ldots, s_k \in M \), where each \( s_i = A_i\text{SO}(3) \) for some \( A_i \in \text{SL}(3,\mathbb{R}) \), such that the simultaneous stabilizer of all \( s_i \) in \( \text{SL}(3,\mathbb{R}) \) is the identity matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the number of ways to arrange 4 Americans, 3 Russians, and 5 Chinese into a queue such that no nationality forms a single consecutive block. Use the inclusion-exclusion principle to find the solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{445167360}
Determine the range of values of $m$ for which there exists a $p > 0$ such that \( f(m,p) = \frac{2}{m^2} - \sum_{k=m}^{m+p} \frac{1}{(2k-1)^2} < 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \geq 7}
Determine the isomorphism class of the quotient ring \( \mathcal{O}_K / \langle \sqrt{6} - 1 \rangle \) for the number field \( K = \mathbb{Q}(\sqrt{2}, \sqrt{3}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_{25}}
Suppose  $a,b,c,x,y,z$  are pairwisely different real numbers. How many terms in the following can be  $1$  at most:   $$ \begin{aligned}
&ax+by+cz,&&&&ax+bz+cy,&&&&ay+bx+cz,
&ay+bz+cx,&&&&az+bx+cy,&&&&az+by+cx?
\end{aligned} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given an $\mathfrak{sl}(2,\mathbb{C})$-module $V$ with eigenvalues of the action $h$ being 6, 4, 2, 0, -2, -4, and -6, determine the number of irreducible submodules isomorphic to $V_4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Students in a class participate in a tree-planting activity. There are three different types of saplings, labeled A, B, and C, in the nursery. They are to be planted in a row of 5 pits, with the condition that the same type of sapling cannot be adjacent, and only type A saplings can be planted in the first and the fifth pits. Calculate the total number of planting methods. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
4.9. Each diagonal of the convex pentagon $A B C D E$ cuts off a triangle of unit area from it. Calculate the area of the pentagon $A B C D E$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}+5}{2}}
Let \(f(x)\) be a twice differentiable function on \([1,3]\) such that \(f(1) = f(3)\) and \(|f''(x)| < 2\) for all \(x\) in \([1,3]\). Determine the range of \(|f'(x)|\) on the interval \([1,3]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 2)}
A square with a side of 36 cm was cut into three rectangles such that the areas of all three rectangles are equal and any two rectangles have a common boundary section. What is the total length (in cm) of the cuts made?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60 \, \text{cm}}
Determine the number of quadratic subfields in the splitting field of the polynomial $x^4-20$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $m, n > 2$ be integers. One of the angles of a regular $n$-gon is dissected into $m$ angles of equal size by $(m-1)$ rays. If each of these rays intersects the polygon again at one of its vertices, we say $n$ is $m$-cut. Compute the smallest positive integer $n$ that is both 3-cut and 4-cut.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Determine the constant $d$ such that
$$\left(3x^3 - 2x^2 + x - \frac{5}{4}\right)(ex^3 + dx^2 + cx + f) = 9x^6 - 5x^5 - x^4 + 20x^3 - \frac{25}{4}x^2 + \frac{15}{4}x - \frac{5}{2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Three flower beds overlap as shown. Bed $A$ has $500$ plants, bed $B$ has $450$ plants, and bed $C$ has $350$ plants. Beds $A$ and $B$ share $50$ plants, while beds $A$ and $C$ share $100$. What is the total number of plants?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1150}
Count the number of different shapes of isosceles triangles with all sides being positive numbers and their perimeter not exceeding 10. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A student correctly added the two two-digit numbers on the left of the board and got the answer 137. What answer will she obtain if she adds the two four-digit numbers on the right of the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13837}
Given a convex hexagon $A B C D E F$ with all six side lengths equal, and internal angles $\angle A$, $\angle B$, and $\angle C$ are $134^{\circ}$, $106^{\circ}$, and $134^{\circ}$ respectively. Find the measure of the internal angle $\angle E$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{134^\circ}
Determine the value of $x$ for which all eigenvalues of the matrix \( A = \begin{pmatrix} 10 & 5+i & 4 \\ x & 20 & 2 \\ 4 & 2 & -10 \end{pmatrix} \) are real.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 - i}