inclusionAI/Ring-lite-rl-data huggingface dataset:
Assume that $z$ and $w$ are differentiable functions of $x$ and $y$, satisfying the equations $xw^3+yz^2+z^3=-1$ and $zw^3-xz^3+y^2w=1.$ Find the value of $\frac{\partial z}{\partial x}$ at the point $(x,y,z,w)=(1,-1,-1,1)$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{5}{4}}
In triangle $XYZ$, $XY=153$, $XZ=147$, and $YZ=140$. The angle bisector of angle $X$ intersects $\overline{YZ}$ at point $D$, and the angle bisector of angle $Y$ intersects $\overline{XZ}$ at point $E$. Let $P$ and $Q$ be the feet of the perpendiculars from $Z$ to $\overline{YE}$ and $\overline{XD}$, respectively. Find $PQ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Evaluate the limit: \[ \lim_{n \to \infty} n \int_1^e x^a (\log (x))^n \,dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{a+1}}
A chord of a circle is at a distance $h$ from the center. A square is inscribed in each of the segments formed by the chord such that two adjacent vertices of the square lie on the arc and the other two vertices lie on the chord. Find the difference in the side lengths of the squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8h}{5}}
Equilateral triangle $ABC$ has a side length of $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and the minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the "lune" that is inside the semicircle but outside sector $ABC$ can be expressed in the form $\sqrt{p} - \frac{q\pi}{r}$, where $p, q$, and $r$ are positive integers such that $q$ and $r$ are relatively prime. Compute $p + q + r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A group of 100 readers meet monthly to discuss their book findings in pairs. Each pair must have read at least one book that the other hasn't. What is the minimum number of distinct books needed to ensure any two random members can start a discussion?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In the game of coins, there are two bowls. The first player places 100 coins in the first bowl. The second player must decide how many coins to place in the second bowl. Each player, starting with the first, can make one of the following moves: take any number of coins from the first bowl, take any number of coins from the second bowl, or take the same number of coins from both bowls. The winner is the player who leaves no coins in either bowl after their move. How many coins should the second player place in the second bowl to guarantee a win with optimal play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{162}
Find the limit without using L'Hopital's rule or Taylor series: $\lim_{x \to 0} \left ( \frac{x}{e^x-1} -\frac{1}{\sin x}\right )$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
A student's final score on a 150-point test is directly proportional to the time spent studying multiplied by a difficulty factor for the test. The student scored 90 points on a test with a difficulty factor of 1.5 after studying for 2 hours. What score would the student receive on a second test of the same format if they studied for 5 hours and the test has a difficulty factor of 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Find the asymptotic upper and lower bounds for the recurrence relation \( T(n) = T(\sqrt{n}) + n \). Use any appropriate method, noting that the master method does not apply.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n)}
Let  $ABCD$  be a quadrilateral with  $AD = 20$  and  $BC = 13$ .  The area of  $\triangle ABC$  is  $338$  and the area of  $\triangle DBC$  is  $212$ .  Compute the smallest possible perimeter of  $ABCD$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{118}
Let $f$ be a Lebesgue integrable function on $[0,1]$ with $\lim_{x\rightarrow 1}f(x) = c$, where $c$ is a constant. Determine the value of $\int_{0}^{1} nx^n f(x) \,dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c}
There are 350 items that are members of set U. Of these items, 49 are members of set B, 59 are not members of either set A or set B, and 23 are members of both sets A and B. Additionally, 36 items are members of set C, 14 items are members of set A and set C, and 19 items are members of set B and set C. Moreover, 8 items are members of all three sets A, B, and C. How many of the members of set U are members of set A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{265}
If the negation of the proposition "There exists a real number $x_0$, such that $x + ax_0 + 1 < 0$" is true, then the range of the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2 \leq a \leq 2}
令  
$$  
S=\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{m^{2} n}{2^{m}\left(n 2^{m}+m 2^{n}\right)}  
$$  
则 $[100 S]=$ $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Evaluate the integral: \[ \int_{0}^b x\frac{2b(b^2-x^2)}{(b^2+x^2)^2} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b(1 - \ln 2)}
Find the remainder when the polynomial
\[p(x) = 1 + x^2 + x^4 + x^6 + \dots + x^{22}\]is divided by the polynomial $q(x) = 1 + x + x^2 + x^3 + \dots + x^{11}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2x^{10} + 2x^8 + 2x^6 + 2x^4 + 2x^2 + 2}
Determine all values of $b$ for which the infinite series $$\sum_{k=1}^\infty \frac{k!}{(b+1)(b+2)\cdots(b+k)}$$ converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b > 1}
Determine the number of geodesics between two distinct points on a unit cylinder, given that the points do not lie on the same circular arc.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given that \( P \) is a moving point on the parabola \( y^2 = 2x \), and points \( B \) and \( C \) are on the \( y \)-axis, the circle \((x-1)^2 + y^2 = 1\) is the incircle of \( \triangle PBC \). Then, the minimum value of \( S_{\triangle PBC} \) is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the smallest positive integer consisting only of the digit '1' repeated multiple times (e.g., 111, 1111) that is divisible by 2019.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{672}
Compute the integral \( \int_{F(S^1)} \omega \) given that the pullback of the 1-form \( \omega = dt_1 + dt_2 + dt_3 \) on \( S^1 \times S^1 \times S^1 \) by the function \( F: S^1 \rightarrow S^1 \times S^1 \times S^1 \) defined by \( F(p) = (p,p,p) \) is \( F^*\omega = 3\,dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\pi}
Evaluate the contour integral \( \int_C \frac{1}{z} \, dz \) for any contour \( C \) in the right half-plane from \( z = -3i \) to \( z = 3i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i\pi}
Find the area enclosed by the equation: \[ \left\lfloor\frac{\left|3x + 4y\right|}{5}\right\rfloor + \left\lfloor\frac{\left|4x - 3y\right|}{5}\right\rfloor = 3 \] where \(\lfloor\cdot\rfloor\) represents the greatest-integer function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A regular triangle is constructed on the diameter of a semicircle such that its sides are equal to the diameter. What is the ratio of the areas of the parts of the triangle lying outside and inside the semicircle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 \sqrt{3} - \pi}{3 \sqrt{3} + \pi}}
Evaluate the product $\frac{1}{2}\cdot\frac{4}{1}\cdot\frac{1}{8}\cdot\frac{16}{1} \dotsm \frac{1}{16384}\cdot\frac{32768}{1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Let \(ABC\) be an equilateral triangle. \(D\) and \(E\) are midpoints of \([AB]\) and \([AC]\). The ray \([DE\) cuts the circumcircle of \(\triangle ABC\) at \(F\). The original answer is in the form \(\frac{\sqrt{k} - m}{n}\). Please find the value of \(k + m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Consider the function \(f:\mathbb{R}^2\to\{1,2,...,2012\}\) that satisfies the following rule: If \(a<b<c\), then \(f(a,c)=f(b,c)=f(a,b)\) and \(f(b,a)=f(c,a)=f(c,b)\). Let \(x_1,x_2,...,x_{2010}\) be a sequence of real numbers, which are all different. Find the number of possible ordered \(2010^2\)-tuples
\[\Bigl(f(x_1,x_1),f(x_1,x_2),\ldots,f(x_1,x_{2010}),f(x_2,x_1),f(x_2,x_2),\ldots,f(x_2,x_{2010}),\ldots,f(x_{2010},x_{2010})\Bigr)\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2012^{2012}}
In a small reserve, a biologist counted a total of 300 heads comprising of two-legged birds, four-legged mammals, and six-legged insects. The total number of legs counted was 980. Calculate the number of two-legged birds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
Find the minimum period of the periodic function $g(x)$ given that for all $x$, \( g(x+2023) = \frac{1-g(x)}{1+g(x)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4046}
Consider the circle defined by the equation \((x - r - 1)^2 + y^2 = r^2\). Let \((x_0, y_0)\) be a point on this circle such that \(4x_0 - y_0^2 \leq 0\). Find the minimum value of \(|r|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the contour integral \( \int_{|z|=1}\frac{\log z}{z}dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi^2}
In triangle \( \triangle ABC \), \( O \) is the circumcenter of \( \triangle ABC \), and \( D \) is the midpoint of \( AC \). It is given that \( \overrightarrow{OB} \cdot \overrightarrow{OD} = \overrightarrow{OA} \cdot \overrightarrow{OC} \). Determine the minimum value of \( \frac{\sin B}{\tan A} + \frac{\sin B}{\tan C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the double integral of the function \( f(x,y) = (x+y)^9(x-y)^9 \) over the region defined by \( 0 \leq x \leq \frac{1}{2} \) and \( x \leq y \leq 1-x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{400}}
Let \( n \) be a positive integer. Consider the points \((n, 0)\), \((n-1, n)\), \((n+1, n)\), and \((0, n)\). Connect the point \((n, 0)\) to \((n-1, n)\) and \((n+1, n)\), and connect the point \((0, n)\) to \((n-1, n)\) and \((n+1, n)\). What is the area of the quadrilateral enclosed by these four line segments?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
For large $n$, approximate the probability that the expression \( Y = \frac{V_1 + V_2 + \cdots + V_n} {\sqrt{V_1^2 + V_2^2 + \cdots + V_n^2}} \) is greater than 0.5, where \( V_i \) are independent and identically distributed random variables with a uniform distribution over \([-2, 2]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.3085}
The positive integer divisors of 252, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 14?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Calculate the variance of the function \( f(X) = 100 \times \exp(x) + 100 \times \exp(2x) \) using the formula for variance:
\[
\operatorname{Var}(f(X))=[f'(EX)]^2\operatorname{Var}(X)+\frac{[f''(EX)]^2}{4}\operatorname{Var}^2(X)+\tilde{T}_3
\]
where the expected value of \( x \) is \( 0.05 \) and the standard deviation is \( 0.1 \). Assume \( x \) follows a normal distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1071.30}
The curve defined by the differential equation \( \frac{dy}{dx}=\frac{y^3}{e^x +y^2} \) passes through the point \((0,2)\). The line \(x=\ln 5\) intersects the curve at points where \(y=a\) and \(y=b\). Calculate the value of \(\frac{4(a^2+b^2)}{53}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the integral \( \int_{0}^{1} (1-x^{\frac{1}{a}})^{-\frac{1}{2}} \, dx \) as \( a \to 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the value of the expression \(2(|AB - I_5| + |BA - I_4|) + 1\), where \(A\) is a \(5 \times 4\) matrix and \(B\) is a \(4 \times 5\) matrix. Here, \(I_n\) denotes the \(n \times n\) identity matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine how many zeros of the function \( f(z) = z^5 + 3z^4 + 6 \) lie within the circle \( |z| = 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the arithmetic sequence $\{a\_n\}$, $a_{66} < 0$, $a_{67} > 0$, and $a_{67} > |a_{66}|$. $S_{n}$ denotes the sum of the first $n$ terms of the sequence. Find the smallest value of $n$ such that $S_{n} > 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
Determine the number of integer solutions for the equation \(x_1 + x_2 + x_3 + x_4 + x_5 = 41\), where \(0 \leq x_i \leq 20\) for all \(i\), and \(x_i\) is even when \(i\) is even, while \(x_i\) is odd when \(i\) is odd. Use a generating function to find the coefficient of \(x^{41}\) in the expanded form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5720}
Evaluate the limit: $$\lim_{x \to 1}\frac{\Gamma(1+x)\Gamma(2+x)\Gamma(3+x) - 12}{x-1}$$ without using L'Hôpital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52 - 36\gamma}
The convex quadrilateral  $ABCD$  has area  $1$ , and  $AB$  is produced to  $E$ ,  $BC$  to  $F$ ,  $CD$  to  $G$  and  $DA$  to  $H$ , such that  $AB=BE$ ,  $BC=CF$ ,  $CD=DG$  and  $DA=AH$ . Find the area of the quadrilateral  $EFGH$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the value of the constant \( \lambda > 0 \) such that \( 2\lfloor \lambda n \rfloor = 1 - n + \lfloor \lambda \lfloor \lambda n \rfloor \rfloor \) holds for all positive integers \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \sqrt{2}}
Calculate the expected value of the squared integral of the function \( f(t) = (W_2 - W_1)1_{[2,3)}(t) + (W_3 - W_1)1_{[3,5)}(t) \) with respect to Brownian motion \((W_t)_{t\ge0}\), where \(1_{[a,b)}(t)\) is the indicator function for the interval \([a, b)\). Use the properties of Brownian motion to find the most efficient method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
How many rearrangements of the letters of "HMMTHMMT" do not contain the substring "HMMT"? For instance, one such arrangement is $HMMHMTMT$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{361}
3. Calculate $\frac{1}{1 \times 3 \times 5}+\frac{1}{3 \times 5 \times 7}+\frac{1}{5 \times 7 \times 9}+\cdots+\frac{1}{2001 \times 2003 \times 2005}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1004003}{12048045}}
Calculate the volume of the solid in the first octant bounded by the surfaces \(z = x + y\), \((x^2 + y^2)^2 = 2xy\), and \(z = 0\) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{8}}
Masha wrote the numbers $4, 5, 6, \ldots, 16$ on the board and then erased one or more of them. It turned out that the remaining numbers on the board cannot be divided into several groups such that the sums of the numbers in the groups are equal. What is the greatest possible value that the sum of the remaining numbers on the board can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Find the least possible value of \( \sum_{n=1}^{696} |nx - 1| \) for \( x \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
In a $25 \times n$ grid, each square is colored with a color chosen among $8$ different colors. Let $n$ be as minimal as possible such that, independently from the coloration used, it is always possible to select $4$ columns and $4$ rows such that the $16$ squares of the intersections are all of the same color. Find the remainder when $n$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{601}
Determine the values of \(a \in \mathbb{R}\) for which the function \(f_a(x,y) = (x^2 + y^2)^a\) is Lebesgue-integrable over the set \(\mathbb{R}^2 \setminus E\), where \(E = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \le 1, y \le \sqrt{|x|} \}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\frac{3}{2}, -1)}
Evaluate the integral \( \int_{0}^{2\pi} e^{-\sin\theta} \cos(\theta-\cos\theta) \, d\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
If Person B trades all their chairs for the same number of tables as Person A, Person B needs to pay an additional 320 yuan. If Person B does not pay the extra money, they would receive 5 fewer tables. It is known that the price of 3 tables is 48 yuan less than the price of 5 chairs. How many chairs does Person B originally have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
For a positive integer  $n$ , let  $f(n)$  be the sum of the positive integers that divide at least one of the nonzero base  $10$  digits of  $n$ . For example,  $f(96)=1+2+3+6+9=21$ . Find the largest positive integer  $n$  such that for all positive integers  $k$ , there is some positive integer  $a$  such that  $f^k(a)=n$ , where  $f^k(a)$  denotes  $f$  applied  $k$  times to  $a$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Find the coordinates of all points P from which two tangent lines to the ellipse \( \frac{x^2}{17} + \frac{y^2}{8} = 1 \) are perpendicular to each other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + y^2 = 25}
In the 3rd grade, the boys wear blue swim caps, and the girls wear red swim caps. The male sports commissioner says, "I see 1 more blue swim cap than 4 times the number of red swim caps." The female sports commissioner says, "I see 24 more blue swim caps than red swim caps." Based on the sports commissioners' statements, calculate the total number of students in the 3rd grade.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Evaluate the integral \( \frac{1}{2 \pi i} \int_{|z|=2} \frac{4z^{50}}{15z^{51}-3z^{30}+z^2-9} \, dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{15}}
Pentagon ABCDE is such that all five diagonals AC, BD, CE, DA, and EB lie entirely within the pentagon. If the area of each of the triangles ABC, BCD, CDE, DEA, and EAB is equal to 1, calculate the area of the pentagon ABCDE. Express your answer as a decimal value rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.62}
Given that \( 5^{\log 30} \times \left(\frac{1}{3}\right)^{\log 0.5} = d \), find the value of \( d \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $(X_n)_{n\geq1}$ be a sequence of independent random variables such that for every positive integer $k$, the following probabilities hold: $P(X_k=k^2)=\frac{1}{k^2}$, $P(X_k=2)=\frac{1}{2}$, and $P(X_k=0)=\frac{1}{2}-\frac{1}{k^2}$. Define $S_n=X_1+X_2+\dots+X_n$. Determine if there exists a real number $c$ such that $\frac{S_n}{n}\rightarrow c$ almost surely, and if so, find the value of $c$. \(\text{If } c \text{ exists, express it as } \boxed{c}.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a point  $p$  and a line segment  $l$ , let  $d(p,l)$  be the distance between them.  Let  $A$ ,  $B$ , and  $C$  be points in the plane such that  $AB=6$ ,  $BC=8$ ,  $AC=10$ .  What is the area of the region in the  $(x,y)$ -plane formed by the ordered pairs  $(x,y)$  such that there exists a point  $P$  inside triangle  $ABC$  with  $d(P,AB)+x=d(P,BC)+y=d(P,AC)?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{288}{5}}
Every day, you put either 1 or 2 dollars into a piggy bank, each with a probability of \( \frac{1}{2} \). What is the expected number of 2-dollar bills in the piggy bank when the total amount reaches at least 100 dollars for the first time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{301}{9}}
Investment funds A, B, and C claim that they can earn profits of 200%, 300%, and 500% respectively in one year. Tommy has $90,000 and plans to invest in these funds. However, he knows that only one of these funds can achieve its claim while the other two will close down. He has thought of an investment plan which can guarantee a profit of at least \$n in one year. Find the greatest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30000}
Let  $p$  be a prime.  It is given that there exists a unique nonconstant function  $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$  such that  $\chi(1) = 1$  and  $\chi(mn) = \chi(m)\chi(n)$  for all  $m, n \not\equiv 0 \pmod p$  (here the product  $mn$  is taken mod  $p$ ). For how many positive primes  $p$  less than  $100$  is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod p?\] Here as usual  $a^{-1}$  denotes multiplicative inverse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Compute the Kullback-Leibler divergence for $P \sim \mathrm{Bin}(n,p)$ and $Q \sim \mathrm{Bin}(n,p+h)$, where $h$ is a small positive number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{nh^2}{2p(1-p)}}
Calculate the area inside the circle given by the polar equation \( r = 3a\cos(\theta) \) and outside the cardioid given by \( r = a(1+\cos(\theta)) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi a^2}
Find the residue of the function \( \sin\left(\frac{1}{z}\right) \) at its poles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Construct an explicit isomorphism between the algebraic group $\mathbb{G}_m/\{\pm 1\}$ and $\mathbb{G}_m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x \mapsto x^2}
Let $T=TNFTPP$. Points $A$ and $B$ lie on a circle centered at $O$ such that $\angle AOB$ is right. Points $C$ and $D$ lie on radii $OA$ and $OB$ respectively such that $AC = T-3$, $CD = 5$, and $BD = 6$. Determine the area of quadrilateral $ACDB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Determine the number of $2 \times 2$ involutory matrices over the ring $\mathbb{Z}_{26}$, where an involutory matrix satisfies $M^2 = I$, with $I$ being the identity matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{736}