inclusionAI/Ring-lite-rl-data huggingface dataset:
About the pentagon $ABCDE$, we know that $AB = BC = CD = DE$, $\angle C = \angle D = 108^\circ$, $\angle B = 96^\circ$. Find the value in degrees of $\angle E$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102}
For real numbers $x$, let 
\[P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)\]
where $i = \sqrt{-1}$. For how many values of $x$ with $0\leq x<2\pi$ does 
\[P(x)=0?\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of unordered triples of distinct points in the $4 \times 4 \times 4$ lattice grid $\{0,1,2,3\}^{3}$ that are collinear in $\mathbb{R}^{3}$ (i.e. there exists a line passing through the three points).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{376}
Leap Day, February $29$, $2004$, occurred on a Sunday. On what day of the week will Leap Day, February $29$, $2020$, occur?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\textrm{Saturday}$}
Any positive integer \( n \) can be written in the form \( n=2^{a} q \), where \( a \geq 0 \) and \( q \) is odd. We call \( q \) the odd part of \( n \). Define the sequence \( a_0, a_1, \ldots \), as follows: \( a_0 = 2^{2011} - 1 \) and for \( m \geq 0 \), \( a_{m+1} \) is the odd part of \( 3a_m + 1 \). Find \( a_{2011} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3^{2011} - 1}{2}}
In the cube $ABCD-A_{1}B_{1}C_{1}D_{1}$, points $P, M,$ and $N$ are the midpoints of edges $AA_{1}$, $BC,$ and $CC_{1}$ respectively. What is the specific geometric shape formed by the intersection of the plane determined by these three points with the surface of the cube? Express your answer as a specific geometric shape.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Regular Hexagon}
Given the function \( f(x) = x^2 + ax + b \), where \( a \) and \( b \) are real numbers, let \( M(a, b) \) be the maximum value of \( |f(x)| \) on the interval \([-1,1]\). When \( M(a, b) \leq 2 \), find the maximum value of \( |a| + |b| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $f(x)$ be an odd function defined on $\mathbb{R}$ that satisfies $f(x+2)=-f(x)$. For $0 \leq x \leq 1$, $f(x)=\frac{1}{2}x$. Let $g(x)=f(x)+\frac{1}{2}$. Find the sum of all zeros of the function $g(x)$ in the interval $[-10,10]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
What is the residue of the function $\frac{1}{z-\sin z}$ at $z=0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/10}
On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.  
  
"Are you a truth-teller?" The principal gave a piece of candy to each of the $22$ children who answered yes.  
  
"Are you an alternater?" The principal gave a piece of candy to each of the $15$ children who answered yes.  
  
"Are you a liar?" The principal gave a piece of candy to each of the $9$ children who answered yes.  
  
How many pieces of candy in all did the principal give to the children who always tell the truth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given that $\alpha \in \left( 0, \pi \right)$ and $3\cos 2\alpha = \sin \left( \frac{\pi}{4} - \alpha \right)$, find the value of $\sin 2\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{17}{18}}
Find the number of ordered pairs of integers $(x, y)$ such that $2167$ divides $3x^2 + 27y^2 + 2021$ with $0 \le x, y \le 2166$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2352}
Write all the natural numbers from 1 to 1988 on the blackboard. Perform operations \( A \) and \( B \) on these numbers repeatedly in the following sequence: first \( A \), then \( B \); repeat with \( A \), then \( B \), and so on. Operation \( A \) subtracts the same natural number from each number on the blackboard (different instances of operation \( A \) can subtract different numbers). Operation \( B \) erases two numbers from the blackboard and replaces them with their sum. Continue this process until after one application of operation \( B \), only one non-negative number remains on the blackboard. What is this number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a family of sets \(\{A_{1}, A_{2}, \ldots, A_{n}\}\) that satisfies the following conditions:
(1) Each set \(A_{i}\) contains exactly 30 elements;
(2) For any \(1 \leq i < j \leq n\), the intersection \(A_{i} \cap A_{j}\) contains exactly 1 element;
(3) The intersection \(A_{1} \cap A_{2} \cap \ldots \cap A_{n} = \varnothing\).

Find the maximum number \(n\) of such sets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{871}
A broken line consists of  $31$  segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In the diagram, the horizontal distance between adjacent dots in the same row is 1. Also, the vertical distance between adjacent dots in the same column is 1. Find the perimeter of quadrilateral PQRS. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let $f(x)=x^{A_0}(x+1)^{A_1}(x+2)^{A_2}\cdots(x+n)^{A_n}$. If

$$\frac{f'(x)}{f(x)}=\frac{n!}{x(x+1)(x+2)(x+3)\cdots(x+n)},$$

find the value of $A_8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In a rectangular array of points, with $5$ rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered $1$ through $N,$ the second row is numbered $N + 1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so that each $P_i$ is in row $i.$ Let $x_i$ be the number associated with $P_i.$ Now renumber the array consecutively from top to bottom, beginning with the first column. Let $y_i$ be the number associated with $P_i$ after the renumbering. It is found that $x_1 = y_2,$ $x_2 = y_1,$ $x_3 = y_4,$ $x_4 = y_5,$ and $x_5 = y_3.$ Find the smallest possible value of $N.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{149}
10.5. On a plane, there are eight different points. A numbering of these points with numbers from 1 to 8 is called good if the following condition is met:

there exists a line such that all points lie on one side of it and at different distances from it, and the distances from the points to this line increase with the increase in the number. That is, the closest point is number 1, the next in distance is number 2, and so on.

What is the maximum number of different good numberings that a given set of eight points can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
James won a money prize at a charity raffle. He donated 25% of his winnings back to the charity. Then he spent 10% of his remaining winnings on a celebratory lunch and 15% of what was left after that on a gift for his friend. James then decided to invest 30% of his remaining money in a new business, and after a month, the investment yielded a 50% return. After all these transactions, he had $320 left. How many dollars did he initially win at the charity raffle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$485}
Xiaoping and Xiaoliang live in the same building, with Xiaoping residing on the fifth floor and Xiaoliang on the fourth floor. Xiaoping has to climb 80 steps to get home every day. Calculate the number of steps Xiaoliang has to climb to get home. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Tom and Jerry are running after each other on a figure-eight track. They run in the same direction at constant speeds. Initially, Jerry was exactly above Tom. After 20 minutes, Tom was exactly above Jerry, and neither of them had completed a full lap. 10 minutes later, Tom returned to the starting point. After what time from the start of running will Tom catch up with Jerry?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
对正整数 n 及实数 x(0\leq x<n)，定义f(n,x)=(1-\{x\})\cdot\text{C}_n^{[x]}+\{x\}\cdot\text{C}_n^{[x]+1},其中 [x] 表示不超过实数 x 的最大整数，\{x\}=x-[x]. 若整数 m,n\geq2 满足f\left(m,\df{1}{n}\right)+f\left(m,\df{2}{n}\right)+\cdots+f\left(m,\df{mn-1}{n}\right)=123,求 f\left(n,\df{1}{m}\right)+f\left(n,\df{2}{m}\right)+\cdots+f\left(n,\df{mn-1}{m}\right) 的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
In a park, pathways are laid as shown in the figure. Two workers started paving these paths simultaneously from point \(A\). They lay asphalt at constant speeds: the first worker on segment \(A-B-C\), the second worker on segment \(A-D-E-F-C\). They finished the job simultaneously, spending 9 hours on it. It is known that the second worker works 1.2 times faster than the first worker. How many minutes did the second worker spend laying asphalt on segment \(D-E\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Let $n$ be the smallest positive integer such that $mn$ is a perfect $k$th power of an integer for some $k \ge 2$, where $m=2^{1980} \cdot 3^{384} \cdot 5^{1694} \cdot 7^{343}$. What is $n+k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Find the volume of the solid that lies above the cone $z^2=x^2+y^2, z\geq0$, and below the sphere $x^2+y^2+z^2=4z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8π}
Define \(\phi^{!}(n)\) as the product of all positive integers less than or equal to \(n\) and relatively prime to \(n\). Compute the number of integers \(2 \leq n \leq 50\) such that \(n\) divides \(\phi^{!}(n) + 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
8. In the sequence $\left\{a_{n}\right\}_{n=1}^{+\infty}$, $x_{1}=1$, if for any positive integer $n$ there is $x_{n+1}=x_{n}+\left[\sqrt{x_{n}}\right]$, where the symbol $[x]$ denotes the greatest integer not exceeding $x$, then the smallest positive integer $n$ such that $x_{n} \geqslant 2019$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Given the parabola $y^2 = 2px$ with the axis of symmetry equation $x = -1$, find the value of $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
设 $M=\{1,2,3, \cdots, 1995\}, A$ 是 $M$ 的子集且满足条件：当 $x \in A$ 时， $15 x \notin A$ ，则 $A$ 中元素的个
数最多是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1870}
Let  $ABC$  be a triangle with  $AB=13$ ,  $BC=14$ , and  $AC=15$ .  Denote by  $\omega$  its incircle.  A line  $\ell$  tangent to  $\omega$  intersects  $\overline{AB}$  and  $\overline{AC}$  at  $X$  and  $Y$  respectively.  Suppose  $XY=5$ .  Compute the positive difference between the lengths of  $\overline{AX}$  and  $\overline{AY}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{329}}{7}}
George has 10 different colors available to paint his room and he must choose exactly 2 of them. However, George prefers to include the color blue as one of his choices if possible. In how many ways can he choose the colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Calculate the product of the base nine numbers $35_9$ and $47_9$, express it in base nine, and find the base nine sum of the digits of this product. Additionally, subtract $2_9$ from the sum of the digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22_9}
Let $z_1$, $z_2$, $z_3$, $\dots$, $z_{16}$ be the 16 zeros of the polynomial $z^{16} - 16^{4}$. For each $j$, let $w_j$ be one of $z_j$ or $iz_j$.  Find the maximum possible value of the real part of
\[\sum_{j = 1}^{16} w_j.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In 1860, someone deposited 100,000 florins at 5% interest with the goal of building and maintaining an orphanage for 100 orphans from the accumulated amount. When can the orphanage be built and opened if the construction and furnishing costs are 100,000 florins, the yearly personnel cost is 3,960 florins, and the maintenance cost for one orphan is 200 florins per year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1896}
A natural number, which does not end in zero, had one of its digits replaced with zero (if it was the leading digit, it was simply erased). As a result, the number became 9 times smaller. How many such numbers exist for which this is possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let  $ABC$  be a triangle and  $\Gamma$  the  $A$ - exscribed circle whose center is  $J$  . Let  $D$  and  $E$  be the touchpoints of  $\Gamma$   with the lines  $AB$  and  $AC$ , respectively. Let  $S$  be the area of the quadrilateral  $ADJE$ , Find the maximum value that  $\frac{S}{AJ^2}$  has and when equality holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
Darryl has a six-sided die with faces  $1, 2, 3, 4, 5, 6$ . He knows the die is weighted so that one face
comes up with probability  $1/2$  and the other five faces have equal probability of coming up. He
unfortunately does not know which side is weighted, but he knows each face is equally likely
to be the weighted one. He rolls the die  $5$  times and gets a  $1, 2, 3, 4$  and  $5$  in some unspecified
order. Compute the probability that his next roll is a  $6$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/26}
Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10004}
Compute the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_{10} \le 1007$ such that $a_i - i$ is odd for $1 \le i \le 10$. Express your answer as ${m \choose n}$ for some $m > n$ and find the remainder when $m$ is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{508}
Given a sequence $\{a_n\}$ that satisfies: $a_1=-13$, $a_6+a_8=-2$, and $a_{n-1}=2a_n-a_{n+1}$ for $n\geqslant 2$, find the sum of the first 13 terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{13}}
The ancient Chinese mathematical masterpiece "Zhou Bi Suan Jing" records the phrase "multiply the legs and the base of a right triangle by themselves, add them together, and then take the square root of the sum," which is symbolically represented as $a^{2}+b^{2}=c^{2} (a,b,c \in \mathbb{N}^{*})$. We call $a$, $b$, and $c$ Pythagorean triples. Given several sets of Pythagorean triples: $3, 4, 5$; $5, 12, 13$; $7, 24, 25$; $9, 40, 41$, and so on, we can guess that the three numbers of the fifth set of Pythagorean triples are ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11, 60, 61}
在凸四边形 $A B C D$ 中, $\overrightarrow{B C}=2 \overrightarrow{A D}$. 点 $P$ 是四边形所在平面上一点, 满足 $\overrightarrow{P A}+2020 \overrightarrow{P B}+\overrightarrow{P C}+2020 \overrightarrow{P D}=\mathbf{0}$. 设 $s, t$ 分别为四边形 $A B C D$ 与 $\triangle P A B$的面积，则 $\frac{t}{s}=$ $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2358}
The angle bisector of one of the acute angles of a right triangle, which is also the altitude dropped to the hypotenuse, is divided into segments whose ratio is $1+\sqrt{2}$, starting from the vertex. Find the acute angles of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ, 45^\circ}
In chess, the winner receives 1 point and the loser 0 points. In the event of a draw (remis), each player receives $1 / 2$ point.
Fourteen chess players, no two of whom were the same age, participated in a tournament where each played against every other. After the tournament, a ranking list was created. In the case of two players with the same number of points, the younger one received a better ranking.
After the tournament, Jan noticed that the top three players had a total number of points exactly equal to the total number of points of the last nine players. Jörg added that the number of games that ended in a draw was maximized. Determine the number of drawn games.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Find the smallest natural number $n$ such that in the two-colored complete graph $K_{n}$, there always exist two monochromatic triangles that share exactly one common vertex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A broken line consists of $31$ segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Two irreducible fractions have their denominators equal to 600 and 700. Find the minimum value for the denominator of the sum of the fractions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
A circle with a radius of 1 is randomly placed inside a $15 \times 36$ rectangle $ABCD$. Find the probability that this circle does not intersect with the diagonal $AC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{375}{442}}
A pineapple costs a certain amount and a watermelon costs Rs 5 each. If I spend Rs 38 in total and I purchased 2 pineapples, how much does each pineapple cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let $(S_n)_{n\ge0}$ be a symmetric, simple random walk. Given $a \in \mathbb{R}$, find the following $$ \lim_{n\to \infty}\mathbb{P}\left(n^{-3/2}\sum_{i=1}^n S_i>a\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-\Phi(\sqrt{3}a)}
Let \( a, b, c, d \) be 4 distinct nonzero integers such that \( a + b + c + d = 0 \) and the number \( M = (bc - ad)(ac - bd)(ab - cd) \) lies strictly between 96100 and 98000. Determine the value of \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{97344}
Come up with a statement that is true for any number except 2 and 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x-2)(x-5) \neq 0}
There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that \[\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}\]whenever $\tan 2023x$ is defined. What is $a_{2023}?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$-1$}
Given complex numbers \( z_{1} \) and \( z_{2} \) satisfying \( \left|z_{1}\right|=\left|z_{1}+z_{2}\right|=3 \) and \( \left|z_{1}-z_{2}\right|=3 \sqrt{3} \), find the value of \( \log_{2}\left| \left(z_{1} \overline{z_{2}}\right)^{2000} + \left(\overline{z_{1}} z_{2}\right)^{2000} \right| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4000}
In triangle $ABC$, if $\sin^{2}A + \sin^{2}B = \sin^{2}C - \sqrt{2}\sin A\sin B$, find the maximum value of $\sin 2A\tan^{2}B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 - 2\sqrt{2}}
Let $b_n$ be a sequence defined by the recurrence relation $b_{n+1} = \sum_{k=0}^n b_k$ with $b_0 = 1$. Find a closed form for $b_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
## Task A-3.4.

For how many natural numbers $n$ is the value of the fraction

$$
\frac{n+2022}{n-3}
$$

an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Joe is given a permutation $p = (a_1, a_2, a_3, a_4, a_5)$ of $(1, 2, 3, 4, 5)$. A swap is an ordered pair $(i, j)$ with $1 \le i < j \le 5$, allowing Joe to swap the positions $i$ and $j$ in the permutation. For example, if Joe starts with the permutation $(1, 2, 3, 4, 5)$ and uses the swaps $(1, 2)$ and $(1, 3)$, the permutation becomes:
\[(1, 2, 3, 4, 5) \rightarrow (2, 1, 3, 4, 5) \rightarrow (3, 1, 2, 4, 5). \]
Out of all $\binom{5}{2} = 10$ swaps, Joe chooses $4$ of them to be in a set of swaps $S$. Joe notices that from $p$ he could reach any permutation of $(1, 2, 3, 4, 5)$ using only the swaps in $S$. How many different sets are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125}
Given that \( a = (-1989)^{-1990} \), \( b = -1989^{1990} \), and \( c = -1989^{-1990} \), determine the relationship between \( a, b, \) and \( c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b < c < a}
Given $\sin 12^\circ = a$, then $\sin 66^\circ =$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - 2a^2}
In the 2011 Shanghai Spring College Entrance Examination, there were 8 colleges recruiting students. If exactly 3 students were admitted by 2 of these colleges, then the number of ways this could happen is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
The square root of a certain number is $a^2 + b^2$ and $4a - 6b + 13$, then this number is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{169}
Given that the graph of function $g(x)$ is symmetric with respect to the line $y=x$ to the graph of function $f(x)=x^{2}$ for $x\in [0, +\infty)$, the graph of function $g(x)$ is shifted $2$ units to the right and $2$ units down to obtain the graph of function $h(x)$. If $P$ and $Q$ are two moving points on the graphs of functions $f(x)$ and $h(x)$ respectively, then the minimum distance between these two points is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7\sqrt{2}}{4}}
\( ABCD \) is a rectangle with side lengths \( AB = CD = 1 \) and \( BC = DA = 2 \). Let \( M \) be the midpoint of \( AD \). Point \( P \) lies on the opposite side of line \( MB \) to \( A \), such that triangle \( MBP \) is equilateral. Find the value of \( \angle PCB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}