inclusionAI/Ring-lite-rl-data huggingface dataset:
In the set of "line segment, circle, equilateral triangle," the shapes that are both centrally symmetric and axially symmetric are ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{line segment, circle}}
Evaluate the line integral \( \int_C (x+y)^2dx - (x^2+y^2)dy \) where \( C \) is the positively oriented triangle with vertices at \((1,1)\), \((3,2)\), and \((2,5)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{140}{3}}
Given the sequence \(\left\{a_{n}\right\}\) which satisfies \(a_{0}=\frac{1}{k}\) and \(a_{n}=a_{n-1}+\frac{1}{n^{2}} a_{n-1}^{2}\), where \(k\) is a positive integer. If for all \(n \in \mathbf{N}^{*}\), we always have \(a_{n} < 1\), find the range of values for \(k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{ k \mid k \geq 3, k \in \mathbb{N}^{*} \}}
Evaluate the second derivative of the integral at $t=0$:

$$\frac{d^2}{dt^2}\left(\left.\int_0^1 e^{tf(x)} \, dx\right)\right|_{t=0}$$

where $f(x)$ is defined as follows: Fix $n$. If $x \in (0,1]$ can be written in binary in non-terminating fashion as $0.a_1a_2a_3\ldots a_n\ldots$, then $f(x)$ is the number of $1$'s among $\{a_1,\ldots,a_n\}$ minus the number of $0$'s among the same set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Determine the value of \(a\) for which the function \(f(x) = x^3 + 3x^2 + ax + 3\) is an open map from \(\mathbb{R}\) to \(\mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the set of real numbers \(s\) such that the integral \(\iint_{(a,b)^2}\frac{1}{|x-y|^s}\mathrm{d}x\mathrm{d}y\) converges, where \((a,b)\) is an arbitrary finite open interval of \(\mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1)}
Determine the possible values of the integral \( \int_{\gamma} \frac{dz}{\sqrt{1-z^2}} \) for a closed curve \( \gamma \) in \( \mathbb{C} \) that intersects the path \( \sigma \) joining \(-1\) and \(+1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi n}
求最大的正整数n，使平面内存在n个凸多边形，其中每两个都恰有一条公共边，但没有公共的内部.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A sphere with a radius of 1 unit is tangent to all the edges of a regular triangular prism. Determine the volume of this prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{4}}
现有1990堆石头，块数分别为1,2，···，1990，进行如下操作，每次可选择任意多堆，从其中每堆拿走同样多石块，问要把所有石块都拿走，最少要操作多少次？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given two triangles with side lengths $3, 4, 5$, four triangles with side lengths $4, 5, \sqrt{41}$, and six triangles with side lengths $\frac{5}{6} \sqrt{2}, 4, 5$, how many tetrahedrons can be formed using these triangles as faces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider the recurrence relation defined by:
\[ \begin{align} a_0 &= 1 \\
 c a_n &= a_{n-1} + 2 a_{n-2} + 3 a_{n-3} + \ldots + n a_0 \end{align} \]
Find the constant \( c < 0 \) that is farthest from zero such that:
\[ \begin{align} a_n &> 0, \quad \text{if } n \text{ is even} \\
 a_n &< 0, \quad \text{if } n \text{ is odd} \end{align} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{4}}
The following diagram shows an eight-sided polygon $ABCDEFGH$ with side lengths $8, 15, 8, 8, 8, 6, 8,$ and $29$ as shown. All of its angles are right angles. Transform this eight-sided polygon into a six-sided polygon by connecting $B$ to $D$ with an edge and $E$ to $G$ with an edge to form polygon $ABDEGH$. Find the perimeter of $ABDEGH$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Calculate the expected return from a game where you flip a fair 6-sided die. You win the amount of money equivalent to the number that shows up. You can play a maximum of three times but must decide whether to stop or continue after each throw. How much should you be willing to pay to play this game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{14}{3}}
Determine the number of $k$-dimensional subspaces in an $n$-dimensional vector space over the finite field $\mathbb{F}_q$. Express your answer using the Gaussian binomial coefficient.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dbinom{n}{k}_q}
From the set $\{1, 2, 3, \ldots, 20\}$, choose $k$ distinct numbers to form a set $M$, such that there are always 4 distinct numbers $a, b, c, d$ in $M$ which satisfy $a - b + c - d$ being divisible by 20. Find the smallest value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the double integral \( \iint_D xy\,\mathrm{d}x\,\mathrm{d}y \) over the region \( D \) bounded by the curves \( y = \sqrt{1 - x^2} \), \( y = \sqrt{1 - x^2/4} \), \( x = 0 \), and \( x = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
How many distinct permutations of the letters of the word REDDER are there that do not contain a palindromic substring of length at least two? (A substring is a contiguous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine whether the Cauchy product of the series $\sum_{k=0}^{\infty}\frac{1}{2^k}$ and $\sum_{k=0}^{\infty}\frac{1}{3^k}$ converges or diverges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A drink vendor has some liters of Maaza, 144 liters of Pepsi, and 368 liters of Sprite. He wants to pack them in cans, so that each can contains the same number of liters of a drink, and doesn't want to mix any two drinks in a can. The least number of cans required is 281. How many liters of Maaza does the vendor have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Let A and B be subsets of a set M, and define S_0 = {A, B}. For i ≥ 0, define S_{i+1} inductively as the collection of subsets X of M that are of the form C ∪ D, C ∩ D, or M − C, where C, D ∈ S_i. Let S = ∪_{i=0}^{∞} S_i. Determine the largest possible number of elements in the set S.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Two trains travel in opposite directions at 36 kmph and another speed, and a man sitting in the slower train passes the faster train in 4 seconds. The length of the faster train is 90.0072 meters. What is the speed of the faster train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45.00648}
In the men's team table tennis competition at the Beijing Olympics, each team consists of 3 players, and a total of five matches are to be played between the two teams, including one doubles match and four singles matches. Each player must play in two matches (the doubles match requires two players to play at the same time), and the doubles match must be played as the third match. If all five matches are played, calculate the total number of different sequences in which the three players can appear.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
In the sequence $\{a_n\}$, if $a_1=1$, $a_2=\frac{1}{2}$, and $\frac{2}{a_{n+1}}=\frac{a_n+a_{n+2}}{a_na_{n+2}}$ ($n\in\mathbb{N}^*$), find the value of $a_{20}$. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{20}}
How many pairs  $ (m,n)$  of positive integers with  $ m < n$  fulfill the equation  $ \frac {3}{2008} \equal{} \frac 1m \plus{} \frac 1n$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the complex number $z$ that satisfies the equation $\frac{1-i}{z} = -i$, where $i$ is the imaginary unit, find $\overset{ .}{z}$ = $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-i}
A counter moves in a cycle of 10. On the nth move, the counter moves n^n steps clockwise. Find the position of the counter after 1234 moves. Express your answer as a single digit (0-9).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
How many distinct arrangements of the letters in the word "basic'' are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
The set $S$ consists of 9 distinct positive integers. The average of the two smallest integers in $S$ is 5. The average of the two largest integers in $S$ is 22. What is the greatest possible average of all of the integers of $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A tourist was returning from vacation by bicycle. On the first section of the route, which was 246 km, he traveled an average of 15 km less per day than on the last section of the route, which was 276 km. He arrived home exactly on the last day of his vacation. It is also known that it took him one more day than half the number of days left from that point until the end of the vacation to complete the first section. How many days before the end of his vacation did the tourist start his journey home?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
James buys steaks for a buy two get one free offer. The price is $15 per pound, and he buys 10 pounds of steak A and 14 pounds of steak B. Additionally, there is a 12% discount on the total price of steak A and a service fee of $7.50 added to the bill. How much did he pay for the steaks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$249.90}
6. Through the midpoints of sides $A B$ and $A D$ of the base of a regular quadrilateral pyramid $S A B C D$, a plane is drawn parallel to the median of the lateral face $S D C$, drawn from vertex $D$. Find the area of the section of the pyramid by this plane, if the side of the base of the pyramid is 2, and the lateral edge is 8.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\sqrt{2}}
Let  $ a$  and  $ b$  be positive real numbers with  $ a\ge b$ . Let  $ \rho$  be the maximum possible value of  $ \frac{a}{b}$  for which the system of equations
\[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in  $ (x,y)$  satisfying  $ 0\le x<a$  and  $ 0\le y<b$ . Then  $ \rho^2$  can be expressed as a fraction  $ \frac{m}{n}$ , where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m\plus{}n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the minimum value of the function \( f(x) = x^{2} + 3x + \frac{6}{x} + \frac{4}{x^{2}} - 1 \) for \( x > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 + 6\sqrt{2}}
The equation
$$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$$
is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4032-2=4030}
In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored $23$,$14$, $11$, and $20$ points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than $18$, what is the least number of points she could have scored in the tenth game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$29$}
Some percent of the students have elected to learn from home during the pandemic. The remaining students are divided into two equal groups, only one of which is physically in school on any day. If 30% of students are present in school, what percent of students elected to learn from home?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40\%}
Given a  $3 \times 3$  symmetric real matrix  $A$ , we define  $f(A)$  as a  $3 \times 3$  matrix with the same eigenvectors of  $A$  such that if  $A$  has eigenvalues  $a$ ,  $b$ ,  $c$ , then  $f(A)$  has eigenvalues  $b+c$ ,  $c+a$ ,  $a+b$  (in that order). We define a sequence of symmetric real  $3\times3$  matrices  $A_0, A_1, A_2, \ldots$  such that  $A_{n+1} = f(A_n)$  for  $n \geq 0$ . If the matrix  $A_0$  has no zero entries, determine the maximum number of indices  $j \geq 0$  for which the matrix  $A_j$  has any null entries.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Reimu and Sanae play a game using  $4$  fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the four coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then *neither* of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5/16}
$12$  students need to form five study groups. They will form three study groups with  $2$  students each and two study groups with  $3$  students each. In how many ways can these groups be formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{138600}
Find the largest positive integer  $n$  not divisible by  $10$  which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of  $n$ . (Note:  $n$  is written in the usual base ten notation.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9999}
The product of the two $99$-digit numbers $303,030,303,...,030,303$ and $505,050,505,...,050,505$ has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$8$}
In quadrilateral $ABCD$, it is given that $AB = DB$ and $AD = BC$. If the angles are $
abla ABD = 36^{\circ}$ and $
abla BCD = 54^{\circ}$, find the measure of angle $
abla ADC$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102}
已知$x _ { i } = \frac { i } { 1 0 1 }$ i=0,,1,2,⋯,101．求下述和式的值:$S = \sum _ { i = 0 } ^ { 1 0 1 } \frac { x _ { i } ^ { 3 } } { 1 - 3 x _ { i } + 3 x _ { i } ^ { 2 } } .$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Determine the smallest set of points in a finite projective geometry $PG(d,q)$ that intersects all lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{q^d - 1}{q - 1}}
In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the diagonal $C A_{1}$, which is equal to $d$, is inclined at an angle of $60^{\circ}$ to the base plane and forms an angle of $45^{\circ}$ with the plane passing through the diagonal $A C_{1}$ and the midpoint of the side edge $B B_{1}$. Find the area of the base of the parallelepiped.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3} d^2}{8 \sqrt{5}}}
Find the derivative of the sum of the functions \( f(t) = \left(\int_0^{\sqrt{t}} e^{-x^2} \, dx\right)^2 \) and \( g(t) = \int_0^1 \frac{e^{-t(1+x^2)}}{1+x^2} \, dx \) for \( t > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Two vertical towers, \( AB \) and \( CD \), are located \( 16 \mathrm{~m} \) apart on flat ground. Tower \( AB \) is \( 18 \mathrm{~m} \) tall and tower \( CD \) is \( 30 \mathrm{~m} \) tall. Ropes are tied from \( A \) to \( C \) and from \( B \) to \( C \). Assuming the ropes are taut, calculate the total length of rope, in \(\mathrm{m}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
A furniture dealer purchased a desk for $150 and then set the selling price equal to the purchase price plus a markup that was 50% of the selling price. After keeping this desk in the showroom, the dealer decided to offer a 20% discount on the desk's marked price to attract more customers. In addition, a 5% sales tax was applied to the discounted selling price, and a 2% commission fee was given to the salesperson for the transaction. If the dealer sold the desk after offering the discount, what was the amount of the dealer's net profit from the purchase and the sale of the desk after considering the sales tax and commission fee in this scenario?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$97.20}
Find the expected value of the difference between the two greatest values in an uniform distribution in [0,1] of n random variables (x1,x2,x3,x4,...,xn).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{n+1}}
How many three-digit numbers with a hundredth digit of 3 and less than 306?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
某学校高中三个年级共有300名学生,为调查他们的课后学习时间情况,通过分层随机抽样获得了20名学生一周的课后学习时间,数据如下表(单位：小时):

|       |     |     |     |     |     |     |     |     |
|-------|-----|-----|-----|-----|-----|-----|-----|-----|
| 高一年级  | 7   | 7.5 | 8   | 8.5 | 9   |     |     |     |
| 高二年级  | 7   | 8   | 9   | 10  | 11  | 12  | 13  |     |
| 高三年级  | 6   | 6.5 | 7   | 8.5 | 11  | 13.5| 17  | 18.5|
从高一年级和高二年级抽出的学生中,各随机选取一人,高一年级选出的人记为甲,高二年级选出的人记为乙,求该周甲的课后学习时间不大于乙的课后学习时间的概率；
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{29}{35}$}
The surface area of an octagonal pyramid is $2538.34 \, \mathrm{dm}^2$, and the lateral edges form an $80^\circ$ angle with the base edges. What is the length of a base edge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x \approx 12.53 \mathrm{dm}}
Some of the towns in a country are connected with bidirectional paths, where each town can be reached by any other by going through these paths. From each town there are at least  $n \geq 3$  paths. In the country there is no such route that includes all towns exactly once. Find the least possible number of towns in this country (Answer depends from  $n$ ).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n+2}
Select 4 students to participate in a relay race from a group of 8 students, which includes 6 boys and 2 girls, using stratified sampling by gender. If girls are not to be the last runner, find the number of different arrangements. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
The given mathematical expression is $({{\lg 2})^{2}}+{\lg 5}\cdot{{\lg 20}+{(2016)}^{0}}+{0.027^{- \frac{2}{3}}}\cdot{{\left( \frac{1}{3} \right)}^{-2}} =$ \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102}
Find the center of mass of a hemisphere

$$
0 \leqslant z \leqslant \sqrt{R^{2}-x^{2}-y^{2}},
$$

where the volume density \( p \) of the material at each point is proportional to the distance to the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(0, 0, \frac{2}{5} R\right)}
In a plane, equilateral triangle $A B C$, square $B C D E$, and regular dodecagon $D E F G H I J K L M N O$ each have side length 1 and do not overlap. Find the area of the circumcircle of $\triangle A F N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2+\sqrt{3}) \pi}
The bathtub fills in 23 minutes with hot water and in 17 minutes with cold water. Masha first opened the hot water tap. After how many minutes should she open the cold water tap so that by the time the bathtub is full, the amount of hot water is 1.5 times more than the amount of cold water?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A businesswoman purchases five different pieces of real estate properties. She buys the first one for Rs. 240,000, the second one for Rs. 560,000, the third one for Rs. 850,000, the fourth one for Rs. 1,500,000, and the fifth one for Rs. 3,440,000. She sells the first property at a loss of 20%, the second at a profit of 25%, the third at a loss of 7%, the fourth at a profit of 15%, and the fifth at a loss of 30%. What is her overall profit or loss percentage from all transactions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11.75\%}
Determine the largest possible value of \(|f''(0)|\) for an analytic function \(f\) in the disk \(D = \{z \in \mathbb{C} : |z| < 2\}\) such that \(\iint_{D} |f(z)|^2 \, dx \, dy \leq 3\pi\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
A running competition on an unpredictable distance is conducted as follows. On a round track, two points \( A \) and \( B \) are chosen randomly (with the help of a spinning arrow), after which the athletes run from \( A \) to \( B \) along the shorter arc. A spectator bought a ticket to the stadium and wants the athletes to run past their seat (so they can take a good photograph). What is the probability that this will happen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Two points are chosen uniformly at random on the sides of a square with side length 1. If $p$ is the probability that the distance between them is greater than $1$, what is $\lfloor 100p \rfloor$? (Note: $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35
}
Determine the number of positive integers \( x \) such that \( x \leq 9,999,999 \) and the sum of the digits of \( x \) is 31.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{512365}
If two lines $l$ and $m$ have equations $y = -2x + 8$, and $y = -3x + 9$, what is the probability that a point randomly selected in the 1st quadrant and below $l$ will fall between $l$ and $m$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.15625}
There is a simple pendulum with a period of $T=1$ second in summer. In winter, the length of the pendulum shortens by 0.01 centimeters. How many seconds faster is this pendulum in winter over a 24-hour period compared to summer? (Round to the nearest second).

Note: The formula for the period of a simple pendulum is $T=2 \pi \sqrt{\frac{l}{g}}$, where the period $T$ is in seconds, the length $l$ is in centimeters, and the acceleration due to gravity $g$ is $980$ cm/s².
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17 \text{ seconds}}
Evaluate whether the integral \( \int_0^{2\pi}\cos(mt+\theta)\sin(nt+\phi)\,\text{d}t \) equals zero for arbitrary phases \( \theta \) and \( \phi \), where \( m \) and \( n \) are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that $X$ and $Y$ are standard normal random variables with $\operatorname{Cov}(X,Y) = \frac{1}{\sqrt{2}}$, compute the conditional probability $P(X>0 | Y<0)$. Use the relation $P(X>0 | Y<0) = 2 P(X>0 \text{ and } Y<0)$ to find your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}