inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the numbers \( x_{1}, \ldots, x_{n} \in\left(0, \frac{\pi}{2}\right) \), find the maximum value of the expression

\[ 
A=\frac{\cos ^{2} x_{1}+\ldots+\cos ^{2} x_{n}}{\sqrt{n}+\sqrt{\operatorname{ctg}^{4} x_{1}+\ldots+\operatorname{ctg}^{4} x_{n}}} 
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{n}}{4}}
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \dots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \); circle \( C_{6} \) passes through exactly 6 points in \( M \);..., circle \( C_{1} \) passes through exactly 1 point in \( M \). What is the minimum number of points in \( M \)? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given \(5^p + 5^3 = 140\), \(3^r + 21 = 48\), and \(4^s + 4^3 = 280\), find the product of \(p\), \(r\), and \(s\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \ (a>b>0)\) with left focus \(F\). Points \(P_{1}, P_{2}, \cdots, P_{n}\) are \(n\) points arranged in a clockwise order on the ellipse such that \(\angle P_{1}FP_{2} = \angle P_{2}FP_{3} = \angle P_{3}FP_{4} = \cdots = \angle P_{n}FP_{1}\). Find the sum of the reciprocals of the distances of these \(n\) points to the left directrix \(x = -\frac{a^{2}}{c} \ (c=\sqrt{a^{2}-b^{2}})\) expressed in terms of \(a, b, c, n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{nc}{b^{2}}}
A new model car travels 4.2 kilometers more per liter of gasoline than an old model car. Additionally, the fuel consumption for the new model is 2 liters less per 100 km. How many liters of gasoline per 100 km does the new car consume? Round your answer to the nearest hundredth if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.97}
Find the smallest integer M such that if a number N is divisible by all integers from 1 to M, then N must also be divisible by M + 1, M + 2, M + 3, and M + 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Three balls are lying on a table, each pair touching each other. The radii of the balls form a geometric progression with a common ratio \( q \neq 1 \). The radius of the middle ball is 2012. Calculate the ratio of the sum of the squares of the sides of the triangle formed by the points of contact of the balls with the table to the sum of the sides of the triangle formed by the centers of the balls.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4024}
Find the minimum value of the function \( f(x,y,z) = xy + 2xz + 3yz \) subject to the constraint \( xyz = 6 \) and \( x \ge 0 \), \( y \ge 0 \), \( z \ge 0 \). Use the method of Lagrange multipliers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given \( z \in \mathbf{C} \) (a complex number), \( \arg \left(z^{2}-4\right) = \frac{5 \pi}{6} \) and \( \arg \left(z^{2}+4\right) = \frac{\pi}{3} \), find the value of \( z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm(1 + \sqrt{3}i)}
Determine the dimension of the quotient space \(c_0/W\), where \(c_0\) is the space of sequences converging to zero, and \(W\) is the subspace of \(c_0\) defined by \(W = \{ (y_n) \in c_0 : y_1 + y_2 + y_3 = 0 \}.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a right triangle, the sine of the smaller angle is $\frac{1}{3}$. A line is drawn perpendicular to the hypotenuse, dividing the triangle into two equal areas. In what ratio does this line divide the hypotenuse?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
Let \( \phi : G \to H \) be a group homomorphism with kernel \( K \). If \( \phi(g) = h \), find the fiber over \( h \) in \( G \), i.e., the set \( \phi^{-1}(h) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{gK}
Find the largest positive integer \( n \) such that the system of equations
$$
(x+1)^{2}+y_{1}^{2}=(x+2)^{2}+y_{2}^{2}=\cdots=(x+k)^{2}+y_{k}^{2}=\cdots=(x+n)^{2}+y_{n}^{2}
$$
has integer solutions \(\left(x, y_{1}, y_{2}, \cdots, y_{n}\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A = \angle B = 90^{\circ}$. If the answer is $x^\circ$, what is the value of $x$ for the degree measure of $\angle E$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Given a cube with its eight vertices labeled with the numbers $1, 2, 3, \cdots, 8$ in any order, define the number on each edge as $|i-j|$, where $i$ and $j$ are the labels of the edge’s endpoints. Let $S$ be the sum of the numbers on all the edges. Find the minimum value of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
In the x-y plane, there is a right triangle ABC (∠B = 90°). If the length of AC is 225 and the slope of the line segment AC is 4/3, what is the length of AB?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Given a triangle $ABC$ with internal angles $A$, $B$, $C$ and the lengths of the sides opposite these angles are $a$, $b$, and $c$ respectively. If $a^{2}=b^{2}+c^{2}-2bc\sin A$, then the magnitude of internal angle $A$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac {\pi}{4}}
Elfs and trolls are seated at a round table, with a total of 60 creatures. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake. Everyone claims to sit between an elf and a troll, but exactly two elfs made a mistake. How many trolls are there at this table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Simplify and find the value of: $5x^2y-[3xy^2-2(3xy^2-\frac{7}{2}x^2y)]$, where $|2+y|+(x-1)^{2}=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \omega$, the area of $WXYZ$ can be expressed as the quadratic polynomial
\[ \text{Area}(WXYZ) = \gamma \omega - \delta \omega^2. \]
Then the coefficient $\delta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
In the Cartesian coordinate system $xOy$, the area of the closed figure formed by the curve $$y= \frac {1}{x} (x > 0)$$, the line $y=x$, and the line $y=3$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4-\ln3}
Given triangle ABC, vectors $\overrightarrow {m}$ = ($\sqrt {3}$sinA, sinB) and $\overrightarrow {n}$ = (cosB, $\sqrt {3}$cosA). If the dot product of $\overrightarrow {m}$ and $\overrightarrow {n}$ is equal to 1 + cos(A + B), find the value of angle C.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {2\pi}{3}}
Determine the number of permutations of the digits 1 through 9 such that the blocks 12, 34, and 567 do not appear.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{283560}
Determine whether the limit \( \lim_{n \rightarrow \infty} \sum_{k = 1}^\infty \frac{(1-\epsilon(n))^k}{k} \) exists and is finite, given that \( \epsilon(n) \rightarrow 0 \) as \( n \rightarrow \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In how many ways can we enter numbers from the set $\{1,2,3,4\}$ into a $4 \times 4$ array so that all of the following conditions hold? (a) Each row contains all four numbers. (b) Each column contains all four numbers. (c) Each "quadrant" contains all four numbers. (The quadrants are the four corner $2 \times 2$ squares.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
Matthew, Shawn, and their friend Amber played swimming-pool-basketball. Each basket was worth either 2 points or 3 points. Matthew scored 10 points, Shawn scored 8 points, and Amber scored 6 points. If Matthew scored 3 baskets that were worth 3 points each, and Amber scored 2 baskets that were worth 3 points each, how many baskets in total were made during this game, considering the combination of 2-point and 3-point baskets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given that $a > 0$, if $f(g(a)) = 8$, where $f(x) = x^2 + 8$ and $g(x) = x^2 - 4$, what is the value of $a$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
If the function $y=\sin \omega x$ is monotonically increasing in the interval $[0,2\pi]$, then the range of the real number $\omega$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(0, \frac {1}{4}\right]}
If two poles $20''$ and $80''$ high are $100''$ apart, then the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$16''$}
Given \(x_{i} \in \mathbf{R}_{+}\) for \(i=1,2,3,4,5\), find the minimum value of:

\[f = \frac{x_{1}+x_{3}}{x_{5}+2 x_{2}+3 x_{4}} + \frac{x_{2}+x_{4}}{x_{1}+2 x_{3}+3 x_{5}} + \frac{x_{3}+x_{5}}{x_{2}+2 x_{4}+3 x_{1}} + \frac{x_{4}+x_{1}}{x_{3}+2 x_{5}+3 x_{2}} + \frac{x_{5}+x_{2}}{x_{4}+2 x_{1}+3 x_{3}}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{3}}
In a swimming pool, there are 3 valves A, B, and C. With all three valves open, the pool can be filled with water in 40 minutes. Valve A alone would fill the pool in 3 hours, and valve B can fill the pool in 4 hours. If valve C emits 75 cubic meters of water more than valve A every minute, and valve B emits 60 cubic meters of water more than valve A every minute, what is the capacity (T) of the pool?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16200}
On the radius \(AO\) of a circle with center \(O\), a point \(M\) is selected. On one side of \(AO\) on the circumference, points \(B\) and \(C\) are chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \(BC\), given that the radius of the circle is \(28\), and \(\sin \alpha = \frac{\sqrt{45}}{7}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given an isosceles triangle where one of the legs and the base are the roots of the equation $x^2 - 6x + a = 0$, find the range of values for $a$ when there is only one such triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < a \leq 9}
Consider a volleyball net with sides 10 and 20, divided into $10 \times 20$ square cells, where each cell is further divided into four triangular cells by its diagonals: the vertices of the cell are connected to the center of the cell with strings. The nodes of the net are the vertices and centers of the cells. What is the maximum number of strings connecting neighboring nodes that can be cut so that the net does not fall apart into separate pieces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{800}
Determine the number of analytic functions that satisfy the condition \( f\left(\frac{1}{n\pi}\right)=1 \) for \( n\in\mathbb{N} \) within a domain \( D \), where \( D \) is a disk centered at \( z = a \) with radius \( r \), and \( 0 \notin D \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let $V$ be the vector space $M_{n \times n}(\mathbb{R})$, and let $B : V \times V \to \mathbb{R}$ be the bilinear form defined by $B(X,Y) = \text{tr}(XY^T)$. Calculate the signature of $B$. Here, $\text{tr}$ denotes the trace of a matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n^2, 0)}
Compute the differential form $d\beta$ for the two-form $\beta = z(y^2-x^2)\,dx\,dy - x^2y\,dx\,dz - xy^2\,dy\,dz$ on three-dimensional real space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: \[ \lim_{R \to \infty} \int_0^R \frac{\cos x}{x^2} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Use the Ritz method to find an approximate value of the smallest eigenvalue of the kernel \( K(x, t) = x t \) over the interval \([0, 1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
In a chess tournament, there are $100$ players. Each player plays one game with every other player. For a win, $1$ point is awarded; for a loss, $0$ points are given; and for a draw, both players receive $0.5$ points.

Ion scored more points than any other player. Mihai lost only one game but accumulated fewer points than any other player. Determine all possible values of the difference between the points accumulated by Ion and the points accumulated by Mihai.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Point \( A \) lies on the side \( LM \) of triangle \( KLM \) with a \( 120^\circ \) angle at vertex \( K \). The circles inscribed in triangles \( AKL \) and \( AKM \) have centers \( F \) and \( O \), respectively. Find the radius of the circumcircle of triangle \( FKO \) if \( AO = 2 \) and \( AF = 7 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{53}{3}}}
Given that $X_t$ is a Brownian bridge, compute the expected value $E[e^{iu(X_{4/5}-\frac{1}{2} X_{3/5})}|X_{3/5}]$, where $u$ is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-u^2/20}}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $2\sin A\sin B < -\cos(2B+C)$, what must be the relationship among $a$, $b$, and $c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^2 + b^2 < c^2}
Suppose $X_1, X_2, \ldots, X_n$ are non-constant real random variables such that $X_1 + X_2 + \cdots + X_n = 0$. Determine the largest possible independence depth of the set $\{X_1, X_2, \ldots, X_n\}$. Note that the independence depth is defined as the maximal number $k$ such that any $k$ distinct elements of the set are mutually independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the cardinality of the set of all continuous surjections from a non-empty open path-connected set $D \subseteq \mathbb{R}^n$ to a non-empty open path-connected set $E \subseteq \mathbb{R}^n$, given that $D$ and $E$ are homeomorphic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the condition under which the matrix expression \((P+Q-PQ-QP)\) is a projection matrix, given that \(P\) and \(Q\) are projection matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{PQ = QP}
Form a six-digit number without repeating digits from 1, 2, 3, 4, 5, 6, with the requirement that odd numbers are not adjacent, and 4 is not in the fourth position. Calculate the total number of such six-digit numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Find the maximum value of the expression \(A = \left(3(x_{1} + \ldots + x_{n}) - 5(y_{1} + \ldots + y_{n})\right) \cdot \left(5(x_{1} + \ldots + x_{n}) + 3(y_{1} + \ldots + y_{n})\right)\) given that \(x_{1}^{2} + \ldots + x_{n}^{2} + y_{1}^{2} + \ldots + y_{n}^{2} \leq 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17n}
From the digits $1, 2, \ldots, 9$, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows: $123456789$, $123456798$, $\ldots$, $987654321$. What is the $100000^{th}$ number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{358926471}
Let $ABCD$ be a rhombus with $\angle ADC = 46^\circ$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle BFC$? If the answer is $x^\circ$, what is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{113}
Let $\pi_1, \pi_2$ be distinct irreducible representations of a finite group $G$, both of degree $d$, with corresponding characters $\chi_1, \chi_2$. Given the orthogonality relation $$ \sum_{g \in G} \chi_1(g) \chi_2(g)^* = 0, $$ determine if the following matrix sum is the zero matrix for $\pi_1 \neq \pi_2$: $$ \sum_{g \in G} \pi_1(g) \pi_2(g^{-1})^\top. $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A submarine has three navigational devices, each with an exponential failure time. The means of these failure times are 1 year, 1.5 years, and 3 years, respectively. The submarine can remain at sea as long as at least two devices are working. Calculate the average length of time the submarine can remain at sea.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{20}}
Evaluate the integral of the function \( f(z) = \tan z \) around a rectangular contour in the complex plane. The rectangle extends from \( 0 \) to \( 2\pi \) in the real direction and from \( -0.5i \) to \( 0.5i \) in the imaginary direction. Determine the value of this integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4\pi i}
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ with an angle of 120° between them, and $|\overrightarrow {a}| = |\overrightarrow {b}| = 1$, vector $\overrightarrow {c}$ is collinear with $\overrightarrow {a} + \overrightarrow {b}$. Calculate the minimum value of $|\overrightarrow {a} + \overrightarrow {c}|$. Express your answer as an exact value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{2}}
In a hexagon described around a circle, five consecutive sides are given as $a, b, c, d, e$. Find the sixth side.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a - b + c - d + e}
If $f(x)=\frac{1}{(1-x^{2011})^{\frac{1}{2011}}}$, find the number of real solutions to the equation $f(f(f(f(x)))) = \frac{1}{\sqrt{2}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Some boys went to water trees. There were 29 trees. Each of them watered the equal amount of trees, and 2 trees were left. How many boys went to water the trees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a regular $1000$-gon, all the diagonals are drawn. What is the maximum number of diagonals that can be chosen such that among any three selected diagonals, at least two have the same length?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
求最小的正整数 m，使得对任意整数 A,B，存在多项式 f(x)，满足 f(-2)=A,f(-3)=B，且 f(x) 的各项系数都是不超过 m 的非负整数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given an integer \( n \geq 3 \), determine the largest real number \( M \) such that for any sequence of positive real numbers \( x_{1}, x_{2}, \ldots, x_{n} \), there exists a permutation \( y_{1}, y_{2}, \ldots, y_{n} \) satisfying the inequality:
\[ 
\sum_{i=1}^{n} \frac{y_{i}^{2}}{y_{i+1}^{2} - y_{i+1} y_{i+2} + y_{i+2}^{2}} \geq M,
\]
where \( y_{n+1} = y_{1} \) and \( y_{n+2} = y_{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $A_n$ denote the answer to the $n$th problem on this contest ($n=1, \dots, 30$); in particular, the answer to this problem is $A_1$. Compute $2A_1(A_1 + A_2 + \dots + A_{30})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How many ways are there to divide the set of numbers \(\left\{2^{0}, 2^{1}, 2^{2}, \cdots, 2^{2005}\right\}\) into two non-empty, disjoint subsets \(A\) and \(B\) such that the equation \(x^{2} - S(A)x + S(B) = 0\) has integer roots, where \(S(M)\) denotes the sum of all elements in the set \(M\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}
If \( 4^3 + 4^r + 4^4 \) is a perfect square and \( r \) is a positive integer, find the minimum value of \( r \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the values of $b \in \mathbb{R}$ for which the function \[ h(x,y) = \begin{cases} \dfrac{x^b y^2}{x^2 + y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} \] is continuous.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b > 0}
(My problem. :D)
Call the number of times that the digits of a number change from increasing to decreasing, or vice versa, from the left to right while ignoring consecutive digits that are equal the *flux* of the number. For example, the flux of 123 is 0 (since the digits are always increasing from left to right) and the flux of 12333332 is 1, while the flux of 9182736450 is 8. What is the average value of the flux of the positive integers from 1 to 999, inclusive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{175}{333}}
Find the integer that does not divide $n^{2225} - n^{2005}$ for every integer value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Maddie went to a beauty store and bought some beauty products. She bought 3 different makeup palettes which cost $15 each, with a 20% discount on the total cost of the palettes. She also bought 4 lipsticks which cost $2.50 each, with a "buy 3 get 1 free" promotion. Additionally, she purchased 3 boxes of hair color which cost $4 each. Maddie had a 10% off coupon for one box of hair color, and she also redeemed 100 reward points to get $5 off her total purchase. 

The beauty store had a limited-time offer where customers get a 5% discount if their subtotal is between $30 and $50, or a 10% discount if their subtotal is over $50. The sales tax applied to the purchase was 5% on the first $25 of the subtotal and 8% on the remaining amount. How much did Maddie pay in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$47.41}
Suppose that $3\triangle_4=\triangle2_{11}$, where $\triangle$ represents a digit 0-9. What is the value of the missing digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, there is a point $M(2,1)$ inside it. Two lines $l_1$ and $l_2$ passing through $M$ intersect the ellipse $E$ at points $A$, $C$ and $B$, $D$ respectively, and satisfy $\overrightarrow{AM}=\lambda \overrightarrow{MC}, \overrightarrow{BM}=\lambda \overrightarrow{MD}$ (where $\lambda > 0$, and $\lambda \neq 1$). If the slope of $AB$ is always $- \frac{1}{2}$ when $\lambda$ changes, find the eccentricity of the ellipse $E$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{2}}