inclusionAI/Ring-lite-rl-data huggingface dataset:
If $\displaystyle p=\sum^{r}_{k=0}\binom{n}{2k}\binom{n-2k}{r-k}$ and $\displaystyle q=\sum^{n}_{k=r}\binom{n}{k}\binom{2k}{2r}\bigg(\frac{3}{4}\bigg)^{n-k}\bigg(\frac{1}{2}\bigg)^{2k-2r}$ , where $n\geq 2r$. Then value of $\displaystyle \frac{p}{q}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
若 $\triangle A B C$ 的内角 $A, B, C$ 满足 $\sin A=\cos B=\tan C$,  $\cos ^{3} A+\cos ^{2} A-\cos A$ 的值为$\frac{n}{m}$，求$m+n$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \( A_{11} \) denote the answer to problem 11. Determine the smallest prime \( p \) such that the arithmetic sequence \( p, p + A_{11}, p + 2A_{11}, \ldots \) begins with the largest possible number of primes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
What is the probability that the greatest common divisor of two natural numbers, $a$ and $b$, is twice a prime number? Denote the set of prime numbers by $P$. Express your answer as a numerical probability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0687}
A rectangular array of chairs is an arrangement of the chairs in rows and columns such that each row contains the same number of chairs as every other row, and each column contains the same number of chairs as every other column. With the condition that there must be at least two chairs in every row and column, and all the chairs in the room must be included, how many arrays are possible in a classroom containing $48$ chairs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The area of the closed shape formed by the graph of the function $y=2\sin x$, where $x \in \left[\frac{\pi}{2}, \frac{5\pi}{2}\right]$, and the lines $y=\pm2$ is what?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
In a convex quadrilateral \( {ABCD} \) it is given that \( \angle {CAB} = {40}^{ \circ  },\angle {CAD} = {30}^{ \circ  },\angle {DBA} = {75}^{ \circ  } \) , and \( \angle {DBC} = {25}^{ \circ  } \) . Find
 \( \angle {BDC} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
30°
}
Calculate the limit: \( \lim_{x\to0} \sum_{n=1}^\infty \frac{\sin x}{4+n^4x^4} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{8}}
Determine the cardinality of the set of equivalence classes of well-orders on \( \mathbb{N} \) under the equivalence relation \( R \equiv R' \leftrightarrow (\mathbb{N},R) \cong (\mathbb{N},R') \), where \( R \) and \( R' \) are well-orders on \( \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
What is the minimum number of vertices that must be selected from a hexagonal grid so that there are always two vertices among them such that the midpoint of the segment connecting them is also a vertex of the grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Two students are preparing to register for the independent admission tests of Zhejiang University, Fudan University, and Shanghai Jiao Tong University, with the requirement that each person can choose at most two schools. Calculate the number of different registration results.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Consider two circles $k_1$ and $k_2$ centered at points $A$ and $B$ with radii $r_1$ and $r_2$, respectively. A third circle $k$ is tangent to both $k_1$ and $k_2$ from the inside, with radius $r_3$. Determine the limit of the ratio $r_3 / r_1$ as the radius $r_1$ approaches 0, while keeping the centers $A$ and $B$ fixed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Calculate \( \lim_{\lambda \rightarrow \infty} T_{\frac{\sin(\lambda x)}{x}} \) in the sense of distributions \( \mathcal{D}'(\mathbb{R}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \delta_0}
A fancy bathroom scale is calibrated in Newtons. This scale is put on a ramp, which is at a $40^\circ$ angle to the horizontal. A box is then put on the scale and the box-scale system is then pushed up the ramp by a horizontal force $F$. The system slides up the ramp at a constant speed. If the bathroom scale reads $R$ and the coefficient of static friction between the system and the ramp is $0.40$, what is $\frac{F}{R}$? Round to the nearest thousandth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.949
}
Ostrovsky $M$.

A number from 1 to 144 is guessed. You are allowed to select one subset of the set of numbers from 1 to 144 and ask whether the guessed number belongs to it. You have to pay 2 rubles for a "yes" answer and 1 ruble for a "no" answer. What is the minimum amount of money needed to surely guess the number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
On the lateral sides \( A B \) and \( C D \) of trapezoid \( A B C D \), points \( P \) and \( Q \) are marked such that line \( P Q \) is parallel to \( A D \), and segment \( P Q \) is divided by the diagonals of the trapezoid into three equal parts. Find the length of the base \( B C \), given that \( A D = a \), \( P Q = m \), and the intersection point of the diagonals of the trapezoid lies inside the quadrilateral \( B P C Q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{am}{3a - 2m}}
记 I=120\displaystyle\int_0^1\sqrt{\df{1-x}{1+x}}\ \dd x，则 [I]=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68}
Evaluate the improper integral \( \int_{-\infty}^\infty{\delta(t^2)} \, dt \) using the properties of the Dirac delta function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In a bag, there are $n$ red balls and $5$ white balls, where $n \in \mathbb{N}^*$. Two balls are randomly drawn from the bag each time. If the probability of drawing two balls of different colors is denoted as $p(n)$, then the maximum value of $p(n)$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{9}}
Determine the signature of the quadratic form associated with the matrix:

$$ \begin{pmatrix}
    1 & 2 & 0 \\
    2 & 0 & 1 \\
    0 & 1 & 1 \\
    \end{pmatrix} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 1)}
Compute the limit:
$$ \lim_{n\to +\infty}\frac{1}{n^2}\sum_{i=1}^{n}\log\binom{n}{i}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{2}\)}
From the set $M = \{(x,y) \mid (|x|-1)^{2} + (|y|-1)^{2} < 4\}$, where $x$ and $y$ are integers, randomly select a point $P(x,y)$. If the probability that $xy \geqslant k$ (where $k > 0$) is $\frac{6}{25}$, then the maximum value of $k$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{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}
What is the minimum number of operations required to transform 2014 into 1, where an operation is defined as adding, subtracting, multiplying, or dividing by a single-digit number (1-9)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In triangle $\triangle ABC$, the sides opposite to the angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=50\sqrt{6}$, $c=150$, and $C=60°$, find the angle $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Determine the Big-O complexity of the recurrence relation: \[ T(n) = \sqrt{n} \cdot T(\sqrt{n}) + \sqrt{n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n)}
An amount of money will become $1000 in 5 years with a rate of interest of 20%. What is the initial amount of money?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$401.88}
Provide an example of a real sequence \( \{a_n\} \) such that \( \lim\limits_{n\to\infty}(a_{n+1}-a_n)=0 \), but \( \lim\limits_{n\to\infty}a_n \) does not exist finitely.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_n = \sum_{k=1}^n \frac{1}{k}}
Let \(a, b \in \mathbb{N}\) be two coprime numbers. Determine the number of ordered pairs \((a, b)\) such that \(\frac{b}{a} + \frac{101b}{5050a} \in \mathbb{N}\), given that \(5050=2\cdot5\cdot505\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Solve the differential equation \( \frac{dy}{dx} = \frac{x^2-1}{y^2-1} \) with the initial condition \( y(0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = x}
The triangle $ABC$ is isosceles with $AB = BC$. The point $F$ on the side $[BC]$ and the point $D$ on the side $AC$ are the feet of the internal bisector drawn from $A$ and the altitude drawn from $B$, respectively, such that $AF = 2BD$. Find the measure of the angle $\angle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
For how many ordered triples $(x,y,z)$ of nonnegative integers less than $20$ are there exactly two distinct elements in the set $\{i^x, (1+i)^y, z\}$, where $i^2 = -1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
Solve in prime numbers:
$  \{\begin{array}{c}\\ \ 2a - b + 7c = 1826  \\3a + 5b + 7c = 2007 \end{array}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
a = 7, b = 29, c = 263
}
Let $a, b$ and $c$ be the distinct solutions to the equation $x^3-2x^2+3x-4=0$. Find the value of
$$\frac{1}{a(b^2+c^2-a^2)}+\frac{1}{b(c^2+a^2-b^2)}+\frac{1}{c(a^2+b^2-c^2)}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{8}}
The five tallest buildings in Los Angeles in 1985 had a mean height of 733 feet. The tallest of the five buildings has a height of 858 feet, the shortest of the five 625 feet. If a new building were constructed with a height of 885 feet, by how many feet would it increase the mean height of the five tallest buildings of the city?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Find the coefficient of \(x^{18}\) in the expansion of \((x+1)(x+2)\cdots(x+10)(2x+1)(2x+3)\cdots(2x+19)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5277440}
Determine the number of 3-Sylow subgroups in a simple group with 168 elements. The possible values for the number of 3-Sylow subgroups are 1, 4, 7, or 28. Given that the group is simple, find the correct value for the number of 3-Sylow subgroups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
What is the minimal number of cuts necessary to find the center of a cylindrical cake using a knife, given that you can only score the surface of the cake? Is it possible to determine the center with fewer than 7 cuts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine whether the statement "A given non-terminal A in a given context-free grammar (CFG) is ever used in the generation of a word" is decidable or undecidable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Decidable}
Let $A$ be a $55 \times 55$ diagonal matrix with characteristic polynomial $(x-c_1)(x-c_2)^2(x-c_3)^3 \ldots (x-c_{10})^{10}$, where $c_1, c_2, \ldots, c_{10}$ are distinct. Let $V$ be the vector space of all $55 \times 55$ matrices $B$ such that $AB = BA$. Determine the dimension of $V$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{385}
Determine the values of $a$ for which the matrix product $[R]_{\alpha}[S]_{\alpha}$ is skew-symmetric, given the matrices $[S]_{\alpha}$ and $[R]_{\alpha}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
设$a_{1},a_{2},\cdots,a_{k}\left(k\in\mathbf{Z}_{+}\right)$均为大于1的整数，且满足$$\left(a_{1}!\right)\left(a_{2}!\right)\cdots\left(a_{k}!\right)\mid2017!$$。当$k$变化时，求$\sum\limits_{i=1}^{k}a_{i}$的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5024}
Find all complex numbers \( z \) such that \( \sin(z) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{z = k\pi}
Given eight distinguishable rings, let  $n$  be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand.  The order of rings on each finger is significant, but it is not required that each finger have a ring.  Find the leftmost three nonzero digits of  $n.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{376}
Vertices $A, B, C$ of a equilateral triangle of side $1$ are in the surface of a sphere with radius $1$ and center $O$. Let $D$ be the orthogonal projection of $A$ on the plane $\alpha$ determined by points $B, C, O$. Let $N$ be one of the intersections of the line perpendicular to $\alpha$ passing through $O$ with the sphere. Find the angle $\angle DNO$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
Find the least positive integer such that the product of its digits is $8! = 8  \cdot 7  \cdot6  \cdot 5  \cdot 4  \cdot 3  \cdot 2 \cdot 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
257889
}
7. A row of 100 coins is laid out. All the coins look the same, but somewhere among them are 50 counterfeit ones (and the rest are genuine). All genuine coins weigh the same, and the counterfeit ones may weigh differently, but each counterfeit is lighter than a genuine one. Can at least 34 genuine coins be found with a single weighing on a balance scale without weights? (O. Dmitriev, R. Zhenodarov)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Determine whether the double integral \( I = \iint_{\mathbb{R}^2} \frac{x^2}{(1+x^2)(x^2 + y^2)^{3/2}}\,dx\,dy \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Converges}
Given an $L^1(\mathbb{R})$ function $f$ such that $f(x) = -f(-x)$, does it follow that $\lim_{\xi \to \infty} \hat{f}(\xi) = \hat{f}(0)$, given that $\hat{f}(\xi) = -\hat{f}(-\xi)$ and $\lim_{\xi \to \infty} \hat{f}(\xi) = 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose $G$ is a Lie group acting smoothly and freely, but not necessarily properly, on a smooth manifold $M$. If $M/G$ is known to be a topological manifold, determine whether the dimension of $M/G$ is equal to $\dim M - \dim G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dim M - \dim G}
In a \(\sigma\)-finite measure space \((X, \mathcal{M}, \mu)\) and the Lebesgue measurable space \((\mathbb{R},\mathcal{L}, m)\), consider a function \(f:X \to \mathbb{R}\) that is integrable (\(f \in L^1(X)\)). Let \(\Gamma = \{(x, f(x)): x\in X\} \subset X \times \mathbb{R}\). Determine if \(\Gamma\) is measurable with respect to the product \(\sigma\)-algebra \(\mathcal{M} \times \mathcal{L}\), and if it is, compute the measure \((\mu \times m)(\Gamma)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given integers $f(i)$ for all integers $i=1,2,\ldots,2^{n-1}$ such that $1\le f(i)\le i$, determine if there exist integers $a_{1},a_{2},\ldots,a_{n}$ such that $1=a_{1}<a_{2}<\cdots<a_{n}\le 2^{n-1}$, and $f(a_{1})\le f(a_{2})\le \cdots\le f(a_{n})$. Answer "yes" or "no."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Determine the values of $p$ for which the closure of $L^1(\mathbb{R}) \cap L^p(\mathbb{R})$ is equal to $L^p(\mathbb{R})$. Provide your answer as a set of values for $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, \infty)}
Find the value of $k \in \mathbb{R}$ such that \( \lim_{n\to\infty} \frac{\tan n}{n^k} = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions

(a) $0\leq A<B<C\leq99$,
(b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0\leq b < a < c < p$,
(c) $p$ divides $A-a$, $B-b$, and $C-c$, and
(d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences.

Find $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
272}
Determine the points $p$ in the set $L = \{ (x,y,z) \mid f(x,y,z) = c \}$, where $f(x,y,z) = \langle x^2 + y^2 + z^2, (x - \frac{1}{2})^2 + y^2 \rangle$ and $c = (1, a^2)$ for $0 < a < \frac{3}{2}$, such that the rank of the Jacobian matrix $Df(p)$ is strictly less than 2. The Jacobian matrix is given by:
\[Df(p) = \begin{bmatrix} 2x & 2y & 2z \\ 2x-1 & 2y & 0 \end{bmatrix}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 0, 0)}
Suppose $P(x)$ is a degree $n$ monic polynomial with integer coefficients such that $2013$ divides $P(r)$ for exactly $1000$ values of $r$ between $1$ and $2013$ inclusive. Find the minimum value of $n$. \( \boxed{n} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Consider a $2 \times n$ grid where each cell is either black or white, which we attempt to tile with $2 \times 1$ black or white tiles such that tiles have to match the colors of the cells they cover. We first randomly select a random positive integer $N$ where $N$ takes the value $n$ with probability $\frac{1}{2^n}$. We then take a $2 \times N$ grid and randomly color each cell black or white independently with equal probability. Compute the probability the resulting grid has a valid tiling.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{23}}
Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?



Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
27
}
Determine whether the extension \( \mathbb{K}(X) \supset \mathbb{K}(u) \) is algebraic, where \( u = \frac{X^3}{X+1} \), and find the degree \([\mathbb{K}(X):\mathbb{K}(u)]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the maximum number of elements which can be chosen from the set $ \{1,2,3,\ldots,2003\}$ such that the sum of any two chosen elements is not divisible by 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{669}
Let $X$ be an uncountable set with the cocountable topology. Determine whether $(X, T)$ is a $T_1$ space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{T_1}
Is the Ramsey number $R(0,a)$ defined, and if so, what is its value?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine whether the series \( \sum_{n=1}^{\infty}\left(\cos\frac{1}{n}\right)^{k(n)} \) converges, where \( k(n)=\frac{1}{\sin^a\left(\frac{1}{n}\right)} \) and \( a \in \mathbb{R} \). Use the approximations \( \cos\frac{1}{n} \approx 1-\frac{1}{2n^2} \) and \( \sin^a\left(\frac{1}{n}\right) \approx \frac{1}{n^a} \) to analyze the behavior of the series. Consider applying the comparison or limit comparison test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 2}
A bag contains three tiles marked $A$, $L$, and $I$. Muhammad Ali wants to pick the letters $A$, $L$, and $I$ in that order. He randomly pulls one tile from the bag. If the letter $A$ is drawn, he keeps it. If the letter pulled is not $A$, he puts it back into the bag. He repeats this process for the next tile, keeping it if it is $L$ and putting it back otherwise. Determine the probability that Muhammad draws from the bag at most 10 times. Express this probability as a fraction $\frac{x_1}{x_2}$, where $x_1$ and $x_2$ are relatively prime numbers. Compute the remainder when $x_1 + x_2$ is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{721}
12. If $\sec \alpha \sqrt{1+\tan ^{2} \alpha}+\tan \alpha \sqrt{\csc ^{2} \alpha-1}=\tan ^{2} \alpha$, determine the sign of the product $\sin (\cos \alpha) \cdot \cos (\sin \alpha)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{positive}
Determine whether the dimension of the symplectic group $Sp(2n, F)$ is $2n^2 + n$ when $F$ is the algebraic closure of a finite field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n^2 + n}
Determine the maximum number of elements in the set \( \mathcal{S} \) of all linear maps \( \mathbf{L} (\mathbb{C}^3) \) where no two maps are similar and satisfy the equation \( \mathbf{A^{10} - 3A^9 + 2A^8 = 0} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Points $E$ and $F$ are located on square $ABCD$ so that $\triangle BEF$ is equilateral. What is the ratio of the area of $\triangle DEF$ to that of $\triangle ABE$?




Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2
}
Given the in-degree and out-degree sequences of a directed graph without loops or multiple edges, determine how many of the following sequences correspond to a valid graph:

(a) \(d_{in}=(0,1,2,3), d_{out}=(2,2,1,1)\)

(b) \(d_{in}=(2,2,1), d_{out}=(2,2,1)\)

(c) \(d_{in}=(1,1,2,3,3), d_{out}=(2,2,3,1,2)\)

Find the number of sequences that can form a valid directed graph.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Three members of the Euclid Middle School girls' softball team had the following conversation. 
Ashley: I just realized that our uniform numbers are all 2-digit primes.
Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.  
Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.  
Ashley: And the sum of you two uniform numbers is today's date.

What number does Caitlin wear?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
An error of 7% in excess is made while measuring the side of a parallelogram, and an error of 3% in deficiency is made while measuring the angle. What will be the percentage error in the calculated area?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11\%}
Suppose $X$ and $Y$ are independent and identically distributed random variables with probability density function $f(x)$, $E[X]=E[Y]=m$, and $Z=(X-Y)\mathrm{sign}(X-m)$. Determine if $E[Z]>0$ is true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E[Z] > 0}
Determine the Euler-Lagrange equation for the functional \( \int_{0.5}^2 (y')^2(1 + y'^2) \,dx \) subject to the boundary conditions \( y(0.5) = 20 \) and \( y(2) = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y'' = 0}
Suppose $\epsilon$ and $X$ are random variables, and $\epsilon_i$ and $x_i$ are the $i$-th realizations of these random variables. Determine if the condition $E[\epsilon_i|x_i] = 0$ for all $i$ implies $E[\epsilon | X] = 0$. Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}