inclusionAI/Ring-lite-rl-data huggingface dataset:
A car and a bus started out from the same place, traveling in opposite directions. After 4 hours, they were 384 miles apart. At the beginning, the car was going 8 miles per hour faster than the bus. However, after 2 hours, the car's speed decreased by 10 miles per hour due to a technical problem. How fast were the car and the bus going initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54.5}
17. In $\triangle A B C, \angle C=90^{\circ}, \angle B A C<45^{\circ}$ and $A B=7$. $P$ is a point on $A B$ such that $\angle A P C$ $=2 \angle A C P$ and $C P=1$. Find $\frac{A P}{B P}$.
(2 marks)
在 $\triangle A B C$ 中, $\angle C=90^{\circ}, \angle B A C<45^{\circ}$ 且 $A B=7 \circ P$ 是 $A B$ 上的一點, 使得 $\angle A P C=2 \angle A C P$ 及 $C P=1$ 。求 $\frac{A P}{B P}$ 。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6+\sqrt{35}}
Suppose that $10$ mathematics teachers gather at a circular table with $25$ seats to discuss the upcoming mathematics competition. Each teacher is assigned a unique integer ID number between $1$ and $10$, and the teachers arrange themselves in such a way that teachers with consecutive ID numbers are not separated by any other teacher (IDs $1$ and $10$ are considered consecutive). In addition, each pair of teachers is separated by at least one empty seat. Given that seating arrangements obtained by rotation are considered identical, how many ways are there for the teachers to sit at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4004}
The slope of the tangent line to the curve $y=\frac{1}{3}{x^3}-\frac{2}{x}$ at $x=1$ is $\alpha$. Find $\frac{{sin\alpha cos2\alpha}}{{sin\alpha+cos\alpha}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3}{5}}
3.365. $\cos \frac{\alpha-\beta}{2}$, if $\sin \alpha+\sin \beta=-\frac{27}{65} ; \operatorname{tg} \frac{\alpha+\beta}{2}=\frac{7}{9} ; \frac{5}{2} \pi<\alpha<3 \pi$ and $-\frac{\pi}{2}<\beta<0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27}{7\sqrt{130}}}
Two circles \( C_{1} \) and \( C_{2} \) touch each other externally and the line \( l \) is a common tangent. The line \( m \) is parallel to \( l \) and touches the two circles \( C_{1} \) and \( C_{3} \). The three circles are mutually tangent. If the radius of \( C_{2} \) is 9 and the radius of \( C_{3} \) is 4, what is the radius of \( C_{1} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Suppose \( A, B, C \) are three angles such that \( A \geq B \geq C \geq \frac{\pi}{8} \) and \( A + B + C = \frac{\pi}{2} \). Find the largest possible value of the product \( 720 \times (\sin A) \times (\cos B) \times (\sin C) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
There are $n$ participants at a conference, each knowing at most $k$ languages. For every group of three participants, there are at least two who speak the same language. Find the smallest value of $n$ such that for any distribution of languages satisfying these properties, one can find a language spoken by at least three delegates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2k+3}
How many ordered integer pairs (x, y) are there satisfying (x^2-18)^2 ≡ y^2 (mod 31), where 0 ≤ x, y < 31? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Determine the number of quadratic polynomials $P(x)=p_{1} x^{2}+p_{2} x-p_{3}$, where $p_{1}, p_{2}, p_{3}$ are not necessarily distinct (positive) prime numbers less than 50, whose roots are distinct rational numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
After marking 12 balls with the numbers $1, 2, 3, \cdots, 12$ and placing them in a box, three people (A, B, and C) each take out 4 balls. It is known that the sum of the numbers on the balls each person took is equal. A has two balls with the numbers 6 and 11, B has two balls with the numbers 4 and 8, and C has one ball with the number 1. What are the numbers on C's other three balls?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 10, 12}
给定正整数 $n$ ，设 $a_{1}, a_{2}, \cdots, a_{n}$ 为非负整数序列，若其中连续若干项 (可以只有一项) 的算术平均值不小于 1 ，则称这些项组成一条 "龙"，其中第一项称为"龙头"，最后一项称为"龙尾"。已知 $a_{1}, a_{2}, \cdots, a_{n}$ 中每一项均为龙头或者龙尾。求 $\sum_{i=1}^{n} a_{i}$的最小值.若$n=5$，
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The café "Buratino" operates 6 days a week with Mondays off. Kolya said that from April 1 to April 20, the café was open for 17 days, and from April 10 to April 30, it was open for 18 days. It is known that he made a mistake once. What was the date of the last Tuesday in April?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Given that $a > 0$, $b > 0$, and $2 - \log_2 a = 3 - \log_3 b = \log_6 \frac{1}{a + b}$, find the value of $\frac{1}{a} + \frac{1}{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{108}}
已知数列$\left\{ {{a}_{n}} \right\}$中，${{a1}}=1$，${{a}_{n}}=n-{{a2n}}$，${{a}_{2n+1}}={{an}}+1$，则${{a}_{1}}+{{a2}}+\cdots +{{a}_{99}}$的值为 _____．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1275}
Call a day a *perfect* day if the sum of the digits of the month plus the sum of the digits of the day equals the sum of the digits of the year. For example, February $28$, $2028$ is a perfect day because $2+2+8=2+0+2+8$. Find the number of perfect days in $2018$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
In the following right-angled triangle ABC, ∠C = BC = 1 and DEF is an arc of a circle with center A. Suppose the shaded areas BDE and CEF are equal and AD = x/√π. Find the value of x. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(7, 1), (1, 7), (22,22)$}
In the fraction (x+y)/(x^2+y^2), if the values of x and y are both enlarged by a factor of 3, describe the resulting change in the fraction's value. Specify if the value is enlarged or reduced and by what factor. Express your answer in the format "enlarged/reduced by a factor of [factor]".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{reduced by a factor of 3}
In order to travel from the first to the second university building, Sasha took a car from a car-sharing service, and Zhenya rented a scooter. Sasha and Zhenya left the first building for the second one simultaneously, and at the same time, Professor Vladimir Sergeevich left the second building for the first in a car. The cars met after 3 minutes, and the scooter and the professor's car met at a distance of 1 km from the first building. Find the speeds of the cars and the scooter, given that the scooter covers a distance of 30 km in 1.25 hours longer than the car, and its speed is 4 times less than the speed of Vladimir Sergeevich's car. Provide the speeds in $\frac{\kappa M}{4}$ in ascending order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15, 40, 60}
Let $C=\{4,6,8,9,10, \ldots\}$ be the set of composite positive integers. For each $n \in C$ let $a_{n}$ be the smallest positive integer $k$ such that $k$ ! is divisible by $n$. Determine whether the following series converges: $$\sum_{n \in C}\left(\frac{a_{n}}{n}\right)^{n}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Converges}
Determine the elementary divisors and invariant factors of the abelian group: $$ \mathbb Z_4\oplus\mathbb Z_{10}\oplus\mathbb Z_{25}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{100, 10}}
Each face of a die is arranged so that the sum of the numbers on opposite faces is 7. In the arrangement shown with three dice, only seven faces are visible. What is the sum of the numbers on the faces that are not visible in the given image?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
A new website registered $2000$ people. Each of them invited $1000$ other registered people to be their friends. Two people are considered to be friends if and only if they have invited each other. What is the minimum number of pairs of friends on this website?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence
\[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \]
for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $S$ be the smallest subset of the integers with the property that $0 \in S$ and for any $x \in S$, we have $3 x \in S$ and $3 x+1 \in S$. Determine the number of non-negative integers in $S$ less than 2008.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
For a $5 \times 5$ chessboard colored as shown below, place 5 different rooks on black squares such that no two rooks can attack each other (rooks attack if they are in the same row or column). How many different ways are there to do this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1440}
某种比赛中每个队恰好与其他队各赛一场，每场比赛中胜者得2分，负者得0分，平局各得1分，比赛结束后发现，每队所得分数中恰有一半是该队同十个得分最低的队的比赛中得到的（十个得分最低的队所得分数中一半是他们彼此比赛中得到的），问共有几个队参加比赛？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
There are  $2017$  turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after  $2015$  seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form  $\tfrac{p}{q}$  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{1009}
2000 people registered on a new website. Each person invited 1000 others to be friends. Two people are considered friends if and only if both have invited each other to be friends. What is the minimum number of friend pairs that could have been formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
In $\triangle ABC$, $BC=1$, $\angle B= \frac{\pi}{3}$, and the area of $\triangle ABC$ is $S=\sqrt{3}$. Find the value of $\sin C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{39}}{13}}
Unconventional dice are to be designed such that the six faces are marked with numbers from  $1$  to  $6$  with  $1$  and  $2$  appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Find the limit: \[ \lim_{x \to \infty} \left(\sum_{n=1}^{\infty} \left(\frac{x}{n}\right)^{n}\right)^{\frac{1}{x}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{1}{e}}}
Given an ellipse with equation x^2/a^2 + y^2/b^2 = 1 (a > b > 0), if there exist three points on the ellipse such that these three points and the center of the ellipse exactly form the four vertices of a square, calculate the eccentricity of the ellipse. Express your answer as a fraction or decimal, including necessary mathematical notation (e.g., square root).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{6}}{3}}
Given the function $f(x) = mx^2 - mx - 1$. If for all $x \in \mathbb{R}$, $f(x) < 0$ always holds, then the range of the real number $m$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4 < m \leq 0}
Mary has a mystery book library. She started with 200 mystery books at the beginning of the year. She joined a book club that sent her 2 books each month of the year. After the first 3 months, she lent 10 books to her friend Jane and never got them back. In the 4th month, her other friend, Alice, returned 5 books that she had borrowed last year. She went to a bookstore in the 5th month and bought 15 more books, then bought 8 books at different yard sales throughout the next 2 months. For her birthday in the 7th month, her daughter gave her a book, and her mother gave her a series of 8 books. In the 8th month, she borrowed 4 books from a friend, Mark, and returned them after 2 months. She also received 6 books from her sister in the 9th month. In the 10th month, she got 4 books on a buy-one-get-one-free deal at a bookstore. She donated 30 books she didn't want to read again to a charity shop in the 11th month. Her neighbor borrowed 5 books from her in the last month of the year and had not returned them by the year end. Finally, she sold 7 books to a used book store in the 12th month. How many books did she have at the end of the year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{219}
Given that $f(z)$ satisfies the condition $\Im (f'(z))=6x(2y-1)$, with $f(0) = 3-2i$ and $f(1) = 6-5i$, determine the value of $f(1+i)$. \(z = x + yi\) is a complex number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 + 3i}
Given vectors $\overrightarrow{a}=(1,-1)$ and $\overrightarrow{b}=(2,-1)$, the projection of $\overrightarrow{a}+\overrightarrow{b}$ in the direction of $\overrightarrow{a}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\sqrt{2}}{2}}
Find the points on the cubic curve $y^2z = x^3 -xz^2$ where the tangent line is vertical, given that $O = (0:1:0)$ is an inflection point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0:0:1), (-1:0:1), (1:0:1)}
What is the smallest number of additional squares that must be shaded in the given figure so that it has at least one line of symmetry and rotational symmetry of order 2? Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $X_i \sim N(0,1)$ for $0 \leq i \leq n$ be i.i.d. standard normal distributions. Define $Y_n := \frac{\sqrt{n} X_0}{\sqrt{(\sum_{i=1}^n X_i^2)}}$. Determine the limiting distribution function for $Y_n$ as $n \to \infty$. Provide your answer in terms of a known distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N(0,1)}
John now has 15 marbles of different colors, including two red, two green, and one blue marble. In how many ways can he choose 5 marbles, if exactly two of the chosen marbles are red and one is green?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
Let \( t = 2016 \) and \( p = \ln 2 \). Evaluate in closed form the sum
\[
\sum_{k=1}^{\infty}\left(1 - \sum_{n=0}^{k-1} \frac{e^{-t} t^n}{n!}\right)(1-p)^{k-1} p.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - 2^{-2016}}
Let $P$ be the convex polygon in the complex plane whose vertices are the roots of
\[z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 = 0.\]The area of $P$ can be expressed in the form $\frac{a + b \sqrt{c}}{d},$ where $a,$ $b,$ $c,$ $d$ are positive integers, in simplest form.  Find $a + b + c + d.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A regular three-sided pyramid (its base is an equilateral triangle) is inscribed in a sphere, the diameter of which is nine times the height of the pyramid. What angle do two side faces enclose?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Let $N$ denote the number of ordered pairs of sets $(A, B)$ such that $A \cup B$ is a size-$999$ subset of $\{1,2,\dots,1997\}$ and $(A \cap B) \cap \{1,2\} = \{1\}$. If $m$ and $k$ are integers such that $3^m5^k$ divides $N$, find the largest possible value of $m+k$. \(\text{Proposed by Michael Tang}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1005}
Ben and Jerry are walking together inside a train tunnel when they hear a train approaching. They decide to run in opposite directions, with Ben heading towards the train and Jerry heading away from the train. As soon as Ben finishes his $1200$ meter dash to the outside, the front of the train enters the tunnel. Coincidentally, Jerry also barely survives, with the front of the train exiting the tunnel as soon as he does. Given that Ben and Jerry both run at $1/9$ of the train’s speed, how long is the tunnel in meters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1350}
Five students $A, B, C, D, E$ participate in a competition. Two people make predictions about the results of the competition:

- The first person predicts the order to be $A, B, C, D, E$. This prediction is completely wrong: not only are none of the students' positions correct, but not even any adjacent pairs of students are correctly predicted.
- The second person predicts the order to be $D, A, E, C, B$. This prediction has two students in the correct positions and two pairs of students whose positions are adjacent correctly.

What are the actual results of the competition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{EDACB}
Determine the number of distinct points in the $xy$-plane that are common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a coconut grove, (x + 2) trees yield 30 nuts per year, x trees yield a certain number of nuts per year and (x – 2) trees yield 180 nuts per year. If the average yield per year per tree is 100, and x is 10, how many nuts per year do x trees yield?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
If $A=\{1,2,3,4,5\}$, find the number of functions $f:A\to A$ such that $f(f(f(x)))=f(f(x))$ for all $x\in A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{756}
Given the function $f(x)=\sin^{2}ωx+\sqrt{3}\sin ωx\cos ωx$, where $x\in R$, $f(α)=-\frac{1}{2}$, $f(β)=\frac{1}{2}$, and the minimum value of $|α-β|$ is $\frac{3}{4}π$, calculate the value of the positive number $ω$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
There exist two distinct positive integers, both of which are divisors of  $10^{10}$ , with sum equal to  $157$ .  What are they?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32, 125}
A sequence $\left\{a_{n}\right\}_{n \geq 0}$ of real numbers satisfies the recursion $a_{n+1}=a_{n}^{3}-3 a_{n}^{2}+3$ for all positive integers $n$. For how many values of $a_{0}$ does $a_{2007}=a_{0}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{2007}}
Find the condition for the equation $|z^2| - \text{Re}(bz) + a = 0$ to have a solution, where $a \in \mathbb{R}$ and $b \in \mathbb{C}$. Determine the nature of the solution set when solutions exist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|b^2| \geq 4a}
Solve the equation \(\frac{x}{3}+\frac{x}{\sqrt{x^{2}-9}}=\frac{35}{12}\). In the answer, write the sum of all obtained solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.75}
Let the sample space be $\Omega =\{1,2,3,4,5,6,7,8\}$ with equally likely sample points, and events $A=\{1,2,3,4\}$, $B=\{1,2,3,5\}$, $C=\{1,m,n,8\}$, such that $p\left(ABC\right)=p\left(A\right)p\left(B\right)p\left(C\right)$, and satisfying that events $A$, $B$, and $C$ are not pairwise independent. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
A point is randomly thrown onto the interval \( [11; 18] \) and let \( k \) be the resulting value. Find the probability that the roots of the equation \((k^2 + 2k - 99)x^2 + (3k - 7)x + 2 = 0\) satisfy the condition \( x_1 \leq 2x_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
To build a wall of 8 m x 6 m x some height, 1600 bricks, each measuring 100 cm x 11.25 cm x 6 cm, will be needed. What is the height of the wall?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22.5}
On January 15 in the stormy town of Stormville, there is a  $50\%$  chance of rain. Every day, the probability of it raining has a  $50\%$  chance of being  $\frac{2017}{2016}$  times that of the previous day (or  $100\%$  if this new quantity is over  $100\%$ ) and a  $50\%$  chance of being  $\frac{1007}{2016}$  times that of the previous day. What is the probability that it rains on January 20?

*2018 CCA Math Bonanza Lightning Round #3.3*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243/2048}
We say that some positive integer $m$ covers the number $1998$, if $1,9,9,8$ appear in this order as digits of $m$. (For instance $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$.) Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n\ge 5$), all different from $0$. What is the remainder of $k(n)$ on division by $8$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
There are three cups of solution with equal weight, and their concentrations are $10 \%, 20 \%, 45 \%$ respectively. If $\frac{1}{2}$, $\frac{1}{4}$, and $\frac{1}{5}$ of the weights from the three cups are poured into a fourth empty cup in order, what is the concentration of the solution in the fourth cup? $\qquad$ $\%$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
在直三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, $A B=1$, $B C=C C_{1}=\sqrt{3}, \angle A B C=90^{\circ}, P$ 是平面 $A B C$ 上一动点. 则 $A_{1} P+\frac{1}{2} P C$ 的最小值是 $\qquad$.
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{7}
\[
\left(\left(\sqrt{mn} - \frac{mn}{m+\sqrt{mn}}\right) \div \frac{\sqrt[4]{mn} - \sqrt{n}}{m-n} - m \sqrt{n}\right)^{2} \div \sqrt[3]{mn \sqrt{mn}} - \left(\frac{m}{\sqrt{m^{4}-1}}\right)^{-2}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{m^2}}
A quadrilateral \(ABCD\) is inscribed in a circle with a diameter of 1, where \(\angle D\) is a right angle and \(AB = BC\).

Find the area of quadrilateral \(ABCD\) if its perimeter is \(\frac{9\sqrt{2}}{5}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{25}}
The numbers 1 to 9 are randomly placed into a 3x3 grid such that each cell contains exactly one number and each number is used exactly once. Calculate the probability that the sum of the numbers in each row and each column is odd. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{14}}
Identify a ring $R$ where the element $2$ is invertible and the equation $x^2 - 1 = 0$ has exactly four distinct solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/15\mathbb{Z}}