inclusionAI/Ring-lite-rl-data huggingface dataset:
Suppose that \(g(x)\) is a function such that
\[g(xy) + 2x = xg(y) + g(x)\] for all real numbers \(x\) and \(y.\) If \(g(-1) = 3\) and \(g(1) = 1\), then compute \(g(-101).\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103}
One side of a rectangle measures $5 \text{ cm}$, and the other side measures $4 \text{ dm}$ and $8 \text{ cm}$. Each of the longer sides is divided into equal parts by three points. The midpoint of the bottom side is connected to the first point on the top side. Determine the area of the smaller resulting quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90 \, \text{cm}^2}
Simplify $\tan \alpha \tan 2 \alpha + \tan 2 \alpha \tan 3 \alpha + \cdots + \tan (n-1) \alpha \tan n \alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\tan n\alpha}{\tan \alpha} - n}
Solve for $x,$ where
\[\frac{x}{x - a} + \frac{x - b}{x - a - b} = \frac{x - a}{x - 2a} + \frac{x + a - b}{x - b}.\]Assume that $2a > x > b > a > 0.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2a + b}{2}}
How many zeroes of $f(z) = z^5 + 3z^4 + 6$ are within $|z| = 3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate: $({\frac{{531}}{{135}}+\frac{{579}}{{357}}+\frac{{753}}{{975}}})×({\frac{{579}}{{357}}+\frac{{753}}{{975}}+\frac{{135}}{{531}}})-({\frac{{531}}{{135}}+\frac{{579}}{{357}}+\frac{{753}}{{975}}+\frac{{135}}{{531}}})×({\frac{{579}}{{357}}+\frac{{753}}{{975}})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Alice has three daughters, each of whom has two daughters, each of Alice's six grand-daughters has one daughter. How many sets of women from the family of $16$ can be chosen such that no woman and her daughter are both in the set? (Include the empty set as a possible set.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2926}
A circle $\omega$ with center $O$ is tangent to the sides $BA$ and $BC$ of the acute angle $ABC$ at points $M$ and $N$, respectively. A line passing through point $M$ parallel to $BC$ intersects the ray $BO$ at point $K$. A point $T$ is chosen on the ray $MN$ such that $\angle MTK = \frac{1}{2} \angle ABC$. It turns out that the line $KT$ is tangent to $\omega$. Find the area of triangle $OKT$ if $BM = a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^2}{2}}
In a two-player game played on a 5×5 grid, two tokens, one red and one blue, start in random positions on the board. On each turn, a player can move either token one or two units to the right, or to the leftmost square of the above row. The last player who is able to move wins. Of the 625 possible starting positions for the two tokens, how many are winning positions for the first player if they play optimally?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{448}
A sequence of pairwise distinct positive integers is called averaging if each term after the first two is the average of the previous two terms. Let $M$ be the maximum possible number of terms in an averaging sequence in which every term is less than or equal to $2022$ and let $N$ be the number of such distinct sequences (every term less than or equal to $2022$) with exactly $M$ terms. What is $M+N?$ (Two sequences $a_1, a_2, \cdots, a_n$ and $b_1, b_2, \cdots, b_n$ are said to be distinct if $a_i \neq b_i$ for some integer $1 \leq i \leq n$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2008

}
Given the equation \( k x^2 + \frac{|x|}{x+6} = 0 \) has four distinct real solutions for \( x \). Determine the range of values for the real number \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k < -\frac{1}{9}}
find all positive integers $n$ for which $1^3+2^3+\\cdots+{16}^3+{17}^n$ is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3}
Origami. Mitya is going to fold a square sheet of paper $ABCD$. Mitya calls a fold beautiful if the side $AB$ intersects the side $CD$ and the four resulting right triangles are equal.

Before this, Vanya selects a random point $F$ on the sheet. Find the probability that Mitya can make a beautiful fold passing through the point $F$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
设数列 $\left\{a_{n}\right\}$ 是等差数列，数列 $\left\{b_{n}\right\}$ 满足 $b_{n}=a_{n+1} a_{n+2}-a_{n}^{2}, n=1,2, \cdots$ 。
（2）设数列 $\left\{a_{n}\right\},\left\{b_{n}\right\}$ 的公差均是 $d \neq 0$ ，并且存在正整数 $s, t$ ，使得 $a_{s}+b_{t}$ 是整数，求 $\left|a_{1}\right|$ 的最小值。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{19}
Among the four-digit numbers formed by the digits 0, 1, 2,..., 9 without repetition, calculate the number of cases where the absolute difference between the units digit and the hundreds digit equals 8. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Evaluate the integral \[ \frac{1}{\pi^2} \int_{0}^{\infty} \frac{(\ln{x})^2}{\sqrt{x}(1-x)^2} \, \mathrm{d}x. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the largest possible value of $a + b$ that is less than or equal to $2007$, where $a$ and $b$ are relatively prime, and there exists a positive integer $n$ such that:
\[
\frac{2^3 - 1}{2^3 + 1} \cdot \frac{3^3 - 1}{3^3 + 1} \cdot \frac{4^3 - 1}{4^3 + 1} \cdots \frac{n^3 - 1}{n^3 + 1} = \frac{a}{b}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1891}
Determine the interval of convergence for the series \(\sum_{n=1}^\infty \frac{(2n-1)(x+2)^n}{5^n n^2}.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-7, 3)}
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38$, $38$, and $34$. What is the distance between two adjacent parallel lines?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is a necessary condition for a matrix to be invertible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\det(A) \neq 0}
Find the length of the latera recta of the ellipse given by the equation \(3x^2+3y^2+2xy-12x+12y+4=0\) with one focus at \((1,-1)\) and directrix \(x-y+2=0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A $39$-tuple of real numbers $(x_1,x_2,\ldots x_{39})$ satisfies
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\] 
The ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1,x_2,\ldots x_{39})$ satisfying the condition is $\tfrac ab$ for coprime positive integers $a$, $b$ (these maxima aren't necessarily achieved using the same tuple of real numbers). Find $a + b$.

[i]Proposed by Evan Chang[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $c - a$ is equal to the height $h$ from $A$ to side $BC$, then $\left(\cos \frac{A}{2} - \sin \frac{A}{2}\right) \cdot \left(\sin \frac{C}{2} + \cos \frac{C}{2}\right) =$ _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the value of N such that the number of odd integers between 4 and N is equal to the number of even integers between 1 and 103. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
Select three different digits from $0, 1, \cdots, 9$ to form a four-digit number (one of the digits may appear twice), such as 5224. How many such four-digit numbers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3888}
Find the minimum possible value of $x+y-2xy$ under the conditions $x+y \geq 3xy$, $2(x+y) \geq 1+3xy$, and $0 \leq x \leq y \leq 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4/9}
Find the limit of the following sequence using Stolz's theorem: $$\lim _{n \rightarrow \infty}\left(\frac{2}{2^{2}-1}\right)^{\frac{1}{2^{n-1}}} \left (\frac{2^{2}}{2^{3}-1}\right)^{\frac{1}{2^{n-2}}} \cdots\left(\frac{2^{n-1}}{2^{n}-1}\right)^{\frac{1}{2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{2}\)}
The first $2007$ positive integers are each written in base $3$. How many of these base-$3$ representations are palindromes? (A palindrome is a number that reads the same forward and backward.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Find the sum of the series $\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n+1 }\left( \frac { n+2 }{ { n }^{ 5 } } \right) } $ correct to three decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.891}
求函数$\frac { 1 - R C s } { s ( 1 + R C s ) }$的拉普拉斯逆变换。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1 - 2 e ^ { - \frac { t } { R C } }$}
There are several solid and hollow circles arranged in a certain pattern as follows: ●○●●○●●●○●○●●○●●●… Among the first 2001 circles, find the number of hollow circles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{667}
Given condition $P$: The real number $x$ satisfies $(x-a)(x-3a) < 0$, where $a > 0$; condition $q$: The real number $x$ satisfies $8 < 2^{x+1} \leqslant 16$. If $q$ is a sufficient but not necessary condition for $p$, find the range of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,2]}
Negate the proposition '$\exists x\geqslant 3$, $x^{2}-2x+3 \lt 0$'. Provide your answer in the form of the negated proposition, using appropriate logical quantifiers and mathematical notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\forall x\geqslant 3$, $x^{2}-2x+3\geqslant 0$}
1. The greatest common divisor (GCD) of natural numbers $m$ and $n$ is 1. What is the greatest possible value of the GCD of the numbers $m+2000n$ and $n+2000m?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000^2-1}
In $\triangle ABC$, $\angle A = 100^\circ$, $\angle B = 50^\circ$, $\angle C = 30^\circ$, $\overline{AH}$ is an altitude, and $\overline{BM}$ is a median. If the answer is $x^\circ$, what is the value of $x$ for $\angle MHC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given: $\sqrt{1+\frac{1}{{1}^{2}}+\frac{1}{{2}^{2}}}=1\frac{1}{2}$, $\sqrt{1+\frac{1}{{2}^{2}}+\frac{1}{{3}^{2}}}=1\frac{1}{6}$, $\sqrt{1+\frac{1}{{3}^{2}}+\frac{1}{{4}^{2}}}=1\frac{1}{12}$. According to this pattern, $\sqrt{1+\frac{1}{{n}^{2}}+\frac{1}{(n+1)^{2}}}=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \dfrac{1}{n(n + 1)}}
Uncle Wang has some chickens, rabbits, and geese. Geese stand on two legs during the day and on one leg at night; chickens tuck their heads under their wings when sleeping. Careful Yue Yue discovered that the difference between the number of legs and the number of heads is always the same, regardless of whether it is day or night. If Yue Yue counts 56 legs during the day, how many heads will be counted at night?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given are  $n$  points with different abscissas in the plane. Through every pair points is drawn a parabola - a graph of a square trinomial with  leading coefficient equal to  $1$ . A parabola is called  $good$  if there are no other marked points on it, except for the two through which it is drawn, and there are no marked points above it (i.e. inside it).  What is the greatest number of  $good$  parabolas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-1}
Let \(a, b, c, d\) be positive integers such that the least common multiple (L.C.M.) of any three of them is \(3^{3} \times 7^{5}\). Determine the number of different sets of \((a, b, c, d)\) possible if the order of the numbers is taken into consideration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11457}
点 M,N 分别在 \triangle ABC 的边 AB,AC 上，且\overrightarrow{AM}=x\overrightarrow{AB},\quad \overrightarrow{AN}=y\overrightarrow{AC}.D 为线段 BC 的中点，G 为线段 MN 与 AD 的交点. 若 \overrightarrow{AG}=\df{3}{5}\overrightarrow{AD}，则 \df{1}{x^2}+\df{4}{y^2} 的最小值为__________.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{94}
Evaluate$$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\left(\left[\frac{2n}{k}\right]-2\left[\frac{n}{k}\right]-\ln4\right)$$
and express your answer in the form $\log a-b$, with $a$ and $b$ positive integers.  
Here $[x]$ is defined to be the integer such that $[x]\leqq x<[x]+1$ and $\log x$ is the logarithm of $x$ to base $e$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given an arithmetic sequence $\left\{a_n\right\}$ such that all terms are integers, the first term $a_1=2019$, and for any positive integer $n$, there always exists a positive integer $m$ such that $a_1 + a_2 + \cdots + a_n = a_n$. Determine the number of such sequences $\left\{a_n\right\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Circles with centers \(O\) and \(P\) have radii 2 and 4, respectively, and are externally tangent. Points \(A\) and \(B\) are on the circle centered at \(O\), and points \(C\) and \(D\) are on the circle centered at \(P\), such that \(\overline{AD}\) and \(\overline{BC}\) are common external tangents to the circles. Find the area of hexagon \(AOBCPD\). The original answer is in the form \(k\sqrt{m}\). Please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Compute the flux of the vector field \( F = \langle x, y, 1 \rangle \) through the surface of the paraboloid \( z = x^2 + y^2 \) bounded by the plane \( z = 1 \) using the Divergence Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given complex number $z$, define sequence $z_{0}, z_{1}, z_{2}, \ldots$ as $z_{0}=z$ and $z_{n+1}=2 z_{n}^{2}+2 z_{n}$ for $n \geq 0$. Given that $z_{10}=2017$, find the minimum possible value of $|z|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt[1024]{4035}-1}{2}}
Calculate Kendall's Tau for the random variable vector $(X_1, X_2)$, where $X_1$ follows an Exponential distribution with parameter 1 (Exp$(1)$), $X_2$ follows a standard Normal distribution (N$(0,1)$), and the dependence structure is defined by the copula:

$$ C(u_1, u_2) = \frac{1}{3}W(u_1,u_2) + \frac{2}{3}\Pi(u_1,u_2), \text{ } u \in [0,1]^2 $$

with the contra-montone copula $W(u_1,u_2) = (u_1+u_2+1)_+$ and the independence copula $\Pi(u_1,u_2) = u_1u_2$. Use the formula for Kendall's Tau:

$$ \rho_{\tau}(X_1,X_2) = 4 \int_0^1 \int_0^1 C(u_1,u_2)dC(u_1,u_2)-1 $$

Evaluate this integral to find Kendall's Tau.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{7}{27}}
Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ and $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Determine the values of the parameter \( a \) for which the equation \( 5^{x^{2}+2ax+a^{2}} = ax^{2}+2a^{2}x+a^{3}+a^{2}-6a+6 \) has exactly one solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Choose positive integers $b_1, b_2, \dotsc$ satisfying
\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]
and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$?

[i]Carl Schildkraut and Milan Haiman[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
On a plane, 10 equal line segments are drawn, and all their intersection points are marked. It turns out that each intersection point divides any segment passing through it in the ratio $3: 4$. What is the maximum possible number of marked points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A market has 100 baskets, each containing a total of 1000 kilograms of rice and 3000 eggs. A basket is considered a "Stuff Basket" if it contains exactly 10 kilograms of rice and 30 eggs. Initially, the baskets may not meet these criteria. In each step, market workers can select two baskets and transfer any amount of rice or eggs between them. What is the minimum number of steps required to ensure all 100 baskets are Stuff Baskets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8a^2}{a^2+1}}
Example 36 (1992 Shanghai High School Competition Question) Let $n$ be a given natural number, $n \geqslant 3$, and for $n$ given real numbers $a_{1}, a_{2}, \cdots, a_{n}$, denote the minimum value of $\left|a_{i}-a_{j}\right|(1 \leqslant i<j \leqslant n)$ as $m$. Find the maximum value of $m$ under the condition that $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=$ 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{12}{n(n^{2}-1)}}}
Bruce, Anne, and Carl can clean their house in 6 hours working together at their respective constant rates. If Anne's speed were doubled and Carl's speed were tripled, they could clean their house in 2 hours working at their respective rates. If Bruce and Anne together can clean the house in 4 hours, and Bruce and Carl together can clean the house in 3 hours, how many hours does it currently take Anne to clean the house on her own?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Petya considered moments in his life to be happy when his digital clock showed the number of hours to be six times the number of minutes, or vice versa. Petya fell asleep and woke up at a happy moment in his life, without missing any such moment during his sleep. What is the maximum whole number of minutes Petya's sleep could have lasted?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{361}
There are 10 people sitting around a round table, some of whom are knights and the rest are liars (knights always tell the truth, and liars always lie). It is known that among them there is at least one knight and at least one liar. What is the maximum number of people sitting at the table that can say: "Both of my neighbors are knights"? (A statement is considered false if it is even partially untrue.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
At the Paradise Park, there is a Ferris wheel with 14 seats. However, 2 of these seats are specially designed to accommodate wheelchairs, so they can only hold 4 passengers each. The remaining seats have varying capacities (in ascending order): 5, 5, 6, 6, 6, 6, 7, 7, 8, and 9 passengers per seat. Additionally, the park staff must ensure that at least 10% of the wheel's total capacity is left available for guests in the queue line who are waiting to ride. Calculate the maximum number of people that can ride the Ferris wheel at the same time, without exceeding the aforementioned constraints.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Triangle $ABC$ has $\angle C = 60^{\circ}$ and $BC = 4$. Point $D$ is the midpoint of $BC$. What is the largest possible value of $\tan{\angle BAD}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\sqrt{3}}{4\sqrt{2}-3}$}
In the first quarter, the workforce of Company X is 60% female. During the second quarter, the company hired 30 additional male workers, which reduced the female workforce percentage to 57%. In the third quarter, they decided to increase the female representation again by hiring 50 new female employees, resulting in a 62% female workforce. Finally, in the fourth quarter, the company hired 40 more male and 10 female workers, which led to a new gender ratio of 58% female employees. Calculate the total number of employees in Company X at the end of the fourth quarter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{700}
Mark and William are playing a game. Two walls are placed 1 meter apart, with Mark and William each starting an orb at one of the walls. Simultaneously, they release their orbs directly toward the other. Both orbs are enchanted such that, upon colliding with each other, they instantly reverse direction and go at double their previous speed. Furthermore, Mark has enchanted his orb so that when it collides with a wall it instantly reverses direction and goes at double its previous speed (William's reverses direction at the same speed). Initially, Mark's orb is moving at \(\frac{1}{1000}\) meters/s, and William's orb is moving at 1 meter/s. Mark wins when his orb passes the halfway point between the two walls. How fast, in meters/s, is his orb going when this first happens?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2^{17}}{125}}
Calculate the value of $\left(\left((4-1)^{-1} - 1\right)^{-1} - 1\right)^{-1} - 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{-7}{5}$}
Evaluate the double integral \( \iint_M \left|\cos(x+y)\right|\,dx\,dy \), where \( M = [0,\pi] \times [0,\pi] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Calculate: $1+i^2+i^3+\ldots+i^{10}$ (where $i$ represents the imaginary unit).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1+i}
Suppose you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\frac{p}{q}$ where $p$ and $q$ are integers, find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Four distinct integers $a$, $b$, $c$, $d$ have a product of $a \cdot b \cdot c \cdot d = 9$. Find the value of $a + b + c + d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
I have a function $f(x)$ that is continuous on the interval $[0, 1]$. I know that $f(0) < 0$ and $f(1) > 0$. I want to find a zero of $f(x)$ numerically using the bisection method. However, the time it takes to evaluate $f(x)$ numerically is different if $f(x) < 0$ or $f(x) > 0$. Let's say that for $f(x) < 0$ the time needed for evaluation is $1$ and for $f(x) > 0$ it's $C$, with $C > 1$. What is the optimal point to choose in a bisection step to minimize the expected time to find a zero of $f(x)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{1+C}\)}