inclusionAI/Ring-lite-rl-data huggingface dataset:
Two identical equilateral triangles can form a rhombus. If the side length of the equilateral triangle is 10, what is the area of the largest square inside this rhombus?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Evaluate the limit $$\lim_{n \rightarrow \infty } n \sqrt[n]{n^{-n+1}+n^{-n+2}+\cdots+1} \cdot \left(1-\sin{\frac{2}{n}}\right).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $a_0 = 2,$ $b_0 = 3,$ and
\[a_{n + 1} = \frac{a_n^2}{b_n} \quad \text{and} \quad b_{n + 1} = \frac{b_n^2}{a_n}\]for all $n \ge 0.$  Then $b_8 = \frac{3^m}{2^n}$ for some integers $m$ and $n.$  Enter the ordered pair $(m,n).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3281,3280)}
$P$为边长为$a$的正三角形$ABC$所在平面外一点且$PA=PB=PC=a$，则$P$到$AB$的距离为_____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\sqrt{3}a}{2}$}
Let $p$ be a prime number such that $p \equiv 3 \pmod{4}$. Determine the remainder when the sum $\sum_{i=1}^{p-1}\left(\frac{i^4+1}{p}\right)$ is divided by $p$, where $\left(\frac{i^4+1}{p}\right)$ denotes the Legendre symbol. Use the fact that if $p-1 \nmid k$, then $\sum_{i=1}^{p-1}i^k \equiv 0 \pmod{p}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p-2}
设函数 f(x)=(x^2+ax+1)\ln x，过点 (1,0) 作其切线 y=g(x). 若对任意 x\geq 1，均有 f(x)\geq g(x)，则实数 a 的最小可能值为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Given the group $G = \mathbb{Z}_{100} \times \mathbb{Z}_{500}$, how many subgroups $H$ of $G$ are isomorphic to $\mathbb{Z}_{25} \times \mathbb{Z}_{25}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Alice and Bob are each secretly given a real number between 0 and 1 uniformly at random. Alice states, “My number is probably greater than yours.” Bob repudiates, saying, “No, my number is probably greater than yours!” Alice concedes, muttering, “Fine, your number is probably greater than mine.” If Bob and Alice are perfectly reasonable and logical, what is the probability that Bob’s number is actually greater than Alice’s?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{12}}
Find the minimum of \( \sum\limits_{k=0}^{40} \left(x+\frac{k}{2}\right)^2 \) where \( x \) is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1435}
We draw circles over the legs of an isosceles right-angled triangle, which intersect at point $D$ on the hypotenuse. What is the common area determined by the circles if each of the legs is $a$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^2}{8} (\pi - 2)}
The line segments connecting an internal point of a convex, non-equilateral $n$-gon with its vertices divide the $n$-gon into $n$ equal triangles.

What is the smallest $n$ for which this is possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a circular park 20 feet in diameter with a straight walking path 5 feet wide passing through its center, calculate the area of the park covered by grass after the path is laid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100\pi - 100}
There are two libraries, A and B, involved in an inter-library loan program. Library A has a unique collection of 150 books, all of which were in Library A at the start of the year. Library B also loans out a certain number of books to Library A. Throughout the year, books are loaned out between the libraries. By the end of the year, 60 percent of the books loaned out from Library A's unique collection are returned, and there are only 100 books from the unique collection in Library A. Additionally, it is known that 30 percent of the loaned books from Library A's unique collection were returned within the same year. During the year, Library B had loaned 20 books to Library A, and 50 percent of them were returned at the end of the year. Calculate the total number of books from Library A's unique collection that were loaned out during that year.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125}
A sum was put at simple interest at a certain rate for 10 years. Had it been put at a higher rate, it would have fetched Rs. 600 more. The sum was Rs. 1200. What was the higher rate percentage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
2016 students are lined up in a row and count off from left to right according to $1,2 \cdots, n$ $(n \geqslant 2)$. If the 2016th student counts to $n$, all students who count to $n$ in this round will receive a New Year's gift. How many students will definitely not receive a New Year's gift, regardless of what $n$ is?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{576}
9. Find the maximum value of the volume of a regular quadrilateral frustum with the upper base edge and side edge both equal to 1 (accurate to 0.001).

Note: If the upper base area of the regular quadrilateral frustum is $S_{1}$, the lower base area is $S_{2}$, and the height is $h$, then its volume
$$
V=\frac{1}{3} h\left(S_{1}+S_{2}+\sqrt{S_{1} S_{2}}\right) .
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.674}
已知$x$，$y$，$z$为正数，且满足等式$x y z ( x + y + z ) = 1$，求表达式$( x + y ) ( y + z )$的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
定义在 $\mathbf{R}$ 上的函数 $f(x)$ 满足 $f(0)=0, f(x)+f(1-x)=1, f\left(\frac{x}{5}\right)=\frac{1}{2} f(x)$ ，且当 $0 \leqslant x_{1}<x_{2} \leqslant 1$ 时, $f\left(x_{1}\right) \leqslant f\left(x_{2}\right)$. 求 $f\left(\frac{1}{2007}\right)$ 的值。答案的形式为\frac{k}{m}，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Li Ming goes home for lunch after school and cooks noodles for himself, which includes the following steps: 
① Boil the noodles for 4 minutes; 
② Wash vegetables for 5 minutes; 
③ Prepare the noodles and condiments for 2 minutes; 
④ Boil the water in the pot for 10 minutes; 
⑤ Wash the pot and add water for 2 minutes. 
Apart from step ④, only one step can be performed at a time. Calculate the minimum number of minutes Li Ming needs to cook the noodles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
12. On the highway between $\mathrm{A}$ and $\mathrm{B}$, the speed limit for vehicles traveling from $\mathrm{A}$ to $\mathrm{B}$ is 50 kilometers per hour, and for vehicles traveling from $\mathrm{B}$ to $\mathrm{A}$ is 60 kilometers per hour. Now, two cars, Car A and Car B, start simultaneously from $\mathrm{A}$ and $\mathrm{B}$ respectively, and travel back and forth between the two places. When Car A reaches $\mathrm{B}$ and is on its way back to $\mathrm{A}$, it stops due to a malfunction, and the stopping point is 30 kilometers from $\mathrm{B}$. The two cars meet for the second time at this location, which is 1 hour and 12 minutes later than the originally scheduled second meeting time. What is the distance between the two places? $\qquad$ kilometers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{198}
Find the maximum possible value obtainable by inserting a single set of parentheses into the expression $1 + 2 \times 3 + 4 \times 5 + 6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77}
Consider the sum   $$ S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|. $$  The value of  $S$  can be written as  $\tan \left( \frac{c\pi}{d} \right)$  for some relatively prime positive integers  $c, d$ , satisfying  $2c < d$ . Find the value of  $c + d$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3031}
3. A student wrote a program for recoloring a pixel into one of 128 different colors. These colors he numbered with natural numbers from 1 to 128, and the primary colors received the following numbers: white color - number 1, red - 5, orange - 13, yellow - 21, green - 45, blue - 75, dark blue - 87, purple - 91, black - 128. If the initial color of the pixel has a number \( n \leq 19 \), then the student's program recolors it to the color with the number \( n+4 \), and if the initial color of the pixel has a number \( n \geq 20 \), then the pixel is recolored to the color with the number \( |129-2n| \). Initially, the pixel was red. The student applied his program 2019 times sequentially. What color did the pixel end up as a result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Compute the flux of water through the parabolic cylinder $S: y = x^2$, from $0 \le x \le 2$ and $0 \le z \le 3$, given that the velocity vector field is $F(x,y,z) = 3z^2\mathbf{i} + 6\mathbf{j} + 6xz\mathbf{k}$. Determine the flow rate of the fluid through this surface.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
In triangle $ABC$, point $D$ divides side $AB$ in half, and point $E$ lies on side $BC$, where segment $BE$ is 3 times smaller than side $BC$. Segments $AE$ and $CD$ intersect at point $O$. Find $AB$ if it is known that $AE = 5$, $OC = 4$, and $\angle AOC = 120^\circ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{7}}
设 x_1,x_2,\cdots,x_{60}\in[-1,1]，求\ds{i=1}{60}x_i^2(x_{i+1}-x_{i-1})的最大值，其中脚标按模 60 理解.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Consider a sequence \( x_{n} \) such that \( x_{1} = x_{2} = 1 \), and \( x_{3} = \frac{2}{3} \). Suppose that \( x_{n} = \frac{x_{n-1}^{2} x_{n-2}}{2 x_{n-2}^{2} - x_{n-1} x_{n-3}} \) for all \( n \geq 4 \). Find the least \( n \) such that \( x_{n} \leq \frac{1}{10^{6}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
A customer enters a store, selects an item costing 20 rubles, and gives the seller a 100-ruble banknote. The seller notices that he doesn't have any change. He goes to a neighboring department, exchanges the 100 rubles, and gives the customer the item and the change. The customer leaves. Suddenly, the seller from the neighboring department returns with the 100-ruble note, claiming it is fake. The first seller then gives him a real 100-ruble note of his own. How much did the first seller lose in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200 \, \text{rubles}}
Please write down a rational number that is smaller than $-\frac{1}{2}$: ______. (Write down one number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let $AXYZB$ be a convex pentagon inscribed in a semicircle with diameter $AB$. Suppose that $AZ - AX = 6$, $BX - BZ = 9$, $AY = 12$, and $BY = 5$. Find the greatest integer not exceeding the perimeter of quadrilateral $OXYZ$, where $O$ is the midpoint of $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
A cube is assembled with $27$ white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.83 \times 10^{-37}}
On the surface of a sphere with center O, a point A draws three chords AB, AC, and AD of equal length, and the angle between any two of them is 60°. If the radius of the sphere is 3, then the length of chord AB is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{6}}
Today, Ivan the Confessor prefers continuous functions  $f:[0,1]\to\mathbb{R}$  satisfying  $f(x)+f(y)\geq |x-y|$  for all pairs  $x,y\in [0,1]$ . Find the minimum of  $\int_0^1 f$  over all preferred functions.

(Proposed by Fedor Petrov, St. Petersburg State University)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/4}
从 $0,1,2,3,4,5,6,7,8,9$ 这 10 个数中取 3 个数, 使其和为不小于 10 的偶数, 则不同的取法有 $\qquad$种。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Dikembe wants to teach everyone at school about the importance of flossing, so he plans to bring enough packets of floss for everyone. There are 20 students in his class. Each student needs 1.5 yards of floss. He sees in the store that each packet of floss contains a certain amount of yards. If he buys the least amount necessary, he has 5 yards of floss left over. How many yards of floss does each packet contain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Each week Carina puts 20 more seashells in a jar than she did the week before. If there are 50 seashells in the jar this week, how many will there be in a month?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130}
Given \( f(x) = \max \left| x^3 - a x^2 - b x - c \right| \) for \( 1 \leq x \leq 3 \), find the minimum value of \( f(x) \) as \( a, b, \) and \( c \) range over all real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/4}
Let the number of $2 pairs of socks be x, the number of $3 pairs of socks be y, and the number of $5 pairs of socks be z. Given that x + y + z = 15 and 2x + 3y + 5z = 35, determine the value of x.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the probability that the chord $\overline{AB}$ does not intersect with chord $\overline{CD}$ when four distinct points, $A$, $B$, $C$, and $D$, are selected from 2000 points evenly spaced around a circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
对非负实数 $x$ 令 $\pi(x)$ 为不超过 $x$ 的素数个数，如 $\pi(1)=0, \pi(2)=1$ ，再令 $f(x)$为第 $\left[\frac{x}{4}\right]+1$ 个素数, 求 $\int_{0}^{100}\left(\pi(x)+\frac{1}{4} f(x)\right) \mathrm{d} x$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2500}
已知 $x^{2}=4 y, M(2,2)$ ，过 M 点的直线交抛物线于 $A, B$ 两点，过 $A, B$ 两点作抛物线的切线交与点 P ，求 $	riangle A B P$ 的最小面积。请给出最小面积的数值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The base of the pyramid \( T A B C D \) is a rectangle \( A B C D \). The height of the pyramid coincides with the lateral edge \( T A \), and the lateral edge \( T C \) is inclined to the plane of the base at an angle of \( 30^{\circ} \). A plane passing through the edge \( T C \) and parallel to the diagonal of the base \( B D \) forms an angle of \( 45^{\circ} \) with the plane of the base, and the distance between this plane and the diagonal \( B D \) is \( a \). What is the minimum area that a cross-section of the pyramid through the diagonal of the base \( A C \) can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{4 \sqrt{3} a^2}{\sqrt{5}} }
At a bus stop near Absent-Minded Scientist's house, two bus routes stop: #152 and #251. Both go to the subway station. The interval between bus #152 is exactly 5 minutes, and the interval between bus #251 is exactly 7 minutes. The intervals are strictly observed, but these two routes are not coordinated with each other and their schedules do not depend on each other. At a completely random moment, the Absent-Minded Scientist arrives at the stop and gets on the first bus that arrives, in order to go to the subway. What is the probability that the Scientist will get on bus #251?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5/14}
A flock of 155 birds sit down on a circle \( C \). Two birds \( P_{i}, P_{j} \) are mutually visible if \( m(P_{i} P_{j}) \leq 10^\circ \). Find the smallest number of mutually visible pairs of birds. (One assumes that a position (point) on \( C \) can be occupied simultaneously by several birds.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{270}
Find the number of integer solutions to the system of equations:
$$ac+5bd=19,$$
$$ad+bc=0.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the complex number z = (x^2 - 1) + (x + 1)i, where z is purely imaginary, find the value of the real number x that satisfies this condition. Provide your answer as a single value or clearly state if multiple values satisfy the condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the function $y = \frac{kx^2}{k^2 + k - 2}$, if its graph is a hyperbola located in the second and fourth quadrants, find the value of $k$ that satisfies these conditions. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
The average age of 12 men is increased by years when two of them whose ages are 21 years and some years are replaced by two new men. The average age of the two new men is 34. What was the age of the second man who was replaced?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$.  Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively.  What is the minimum possible value of $BE+DE+CD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
The quadrilateral \(ABCD\) is inscribed in a circle. \(I\) is the incenter of triangle \(ABD\). Find the minimum value of \(BD\) given that \(AI = BC = CD = 2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
一个 $10 \times 10$ 的数表在初始状态下每个格内均填有数字1，现在对数表进行如下操作：选择某一行或某一列，并将该行（列）中所有格子内的数变为其相反数。对于非负整数 $N$ ，如果可经过有限次操作，使得最终数表中 -1 的个数恰为 $N$ ，则称 $N$ 是 "好的"，则好的 $N$ 的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Let $T$ be a $30$-$60$-$90$ triangle with a hypotenuse of length $20$. Three circles, each externally tangent to the other two, have centers at the three vertices of $T$. The area of the union of the circles intersected with $T$ is $(m + n \sqrt{3}) \pi$ for rational numbers $m$ and $n$. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
From the beginning to the end of a certain year, the price of a stock rose by a certain percent. In 2008, it dropped 25 percent. In 2009, it rose 30 percent. The price of the stock at the end of 2009 was 117 percent of the stock's starting price in that certain year. By what percent did the stock price rise in that year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In an NGO, the daily average wages of 35 illiterate employees is decreased from Rs. 40 to Rs. 15, thus the average salary of all the literate and illiterate employees is decreased by Rs. 15 per day. Find the number of educated (literate) employees working in the NGO.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Antonia received 400 emails in her mail. 1/4 were spam emails, while some of the remaining emails were promotional messages. There were 180 important emails in her inbox. What is the ratio of promotional messages to the remaining emails after removing spam emails?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2.5}
There are 1200 medical professionals at a hospital consisting of doctors, nurses, administrative staff, and medical technicians. If the ratio of doctors to nurses is 3 to 4, the ratio of nurses to administrative staff is 2 to 1, and the ratio of administrative staff to medical technicians is 3 to 5, how many nurses are at the hospital?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{389}
Given that $F\_1$ and $F\_2$ are two foci of the hyperbola $x^2-y^2=1$, and $P$ is a point on the hyperbola such that $\angle F\_1PF\_2=60^{\circ}$, determine the area of $\triangle F\_1PF\_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
设 $a \in \mathbf{R}$, 且对任意实数 $b$ 均有 $\max _{x \in[0,1]}\left|x^{2}+a x+b\right| \geq 1$, 求 $a$ 的取值范围.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(-\infty, -3] \cup [1, +\infty)$}
Through the vertices $A, B$ and $C$ of the parallelogram $A B C D$ with sides $A B=3$ and $B C=5$, a circle is drawn, intersecting the line $B D$ at point $E$, and $B E=9$. Find the diagonal $B D$.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{34}{9}}
设 P(z)=z^4-(6i+6)z^3+24iz^2-(18i-18)z-13. 求 P(z)=0 的四个根在复平面上对应的四个点所围成凸四边形的面积.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the value of $y$ that satisfies the equation:

$$\sqrt{y+\sqrt{y+\sqrt{y+\sqrt{\cdots}}}}=\frac{y+\sqrt{y^2+4y}}{2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $n=p_1^{e_1}p_2^{e_2}\dots p_k^{e_k}=\prod_{i=1}^k p_i^{e_i}$, where $p_1<p_2<\dots<p_k$ are primes and $e_1, e_2, \dots, e_k$ are positive integers. Let $f(n) = \prod_{i=1}^k e_i^{p_i}$. Find the number of integers $n$ such that $2 \le n \le 2023$ and $f(n)=128$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
设 A 是 5\times5 的方阵，A 的第一行是 (1,1,1,1,1)，第二行是 (1,-1,1,-1,1). 已知 \text{tr}(AA^T)=28，求 [\det A] 的最大可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
If $\int_{1}^{m} (2x-1)dx=6$ where $m > 1$, then the constant term of the expansion of the polynomial $(x^{2}+ \frac{1}{x^{2}}-2)^{m}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-20}
A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at the ends of each row infect the people at the ends of the rows in front of and behind them that are at the same side of the row as they are. Given that two people are chosen at random to be infected with COVID at the beginning of day 1, what is the earliest possible day that the last uninfected person will be infected with COVID?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
If a line forms a $72^{\circ}$ angle with a plane, then the maximum angle formed by this line with a line in the plane passing through the foot of the perpendicular is $\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^{\circ}}
There are 300 non-zero integers written in a circle such that each number is greater than the product of the three numbers that follow it in the clockwise direction. What is the maximum number of positive numbers that can be among these 300 written numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Anne-Marie has a deck of 16 cards, each with a distinct positive factor of 2002 written on it. She shuffles the deck and begins to draw cards from the deck without replacement. She stops when there exists a nonempty subset of the cards in her hand whose numbers multiply to a perfect square. What is the expected number of cards in her hand when she stops?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{837}{208}}
Through the point \((1,2)\), two tangent lines to the circle \(x^2 + y^2 = 1\) are drawn. What is the area of the quadrilateral formed by these two tangent lines and the \(x\)-axis and \(y\)-axis?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{13}{8}}