inclusionAI/Ring-lite-rl-data huggingface dataset:
Knowing that:
$$I_n=\int_0^{\frac{\pi}{2}}\cos^n(t) \, dt$$
$$I_{2n}=\frac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq 1$$
$$I_{n}\sim \sqrt{\dfrac{\pi}{2n}}$$
Calculate:
$$ \lim_{n\to+\infty} \frac{1\times 3\times \ldots \times (2n+1)}{2\times 4\times \ldots\times 2n}\times\dfrac{1}{\sqrt{n}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2}{\sqrt{\pi}}\)}
设集合 $M=\{1,2, \cdots, 10\}, T$ 为 $M$ 的某些二元子集构成的集合，满足对 $T$ 中任意两个不同的元素 $\{a, b\} 、\{x, y\}$ ，均有
$11 \nmid(a x+b y)(a y+b x)$ 。
求 $T$ 的元素个数的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
36. Using a protractor with only $0^{\circ}, 4^{\circ}, 10^{\circ}, 16^{\circ}, 43^{\circ}, 89^{\circ}$ and $180^{\circ}$ markings, the maximum number of angles (not greater than $180^{\circ}$) that can be directly measured is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Determine the correct operating point \((x_{10}, x_{20}, u_{0})\) for the dynamic system described by the ODE \(10\ddot{y}(t) + \dot{y}(t) = u^{2}(t)\) when linearizing around \(u=1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1, 1)}
Calculate the definite integral:

$$
\int_{0}^{\frac{\pi}{2}}\left(x^{2}-5x+6\right) \sin 3x \, dx
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{67 - 3\pi}{27}}
Determine the constant $c$ such that the equation \( \sum_{k=2}^{N}c^{\frac{1}{k\log k}}=N \) holds true for any positive integer \( N \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the right triangle \(ABC\), the lengths of the legs are \(AC = 3\) and \(BC = 4\). Point \(A\) is moved to point \(A_1\) parallel to \(BC\). Then point \(B\) is moved to point \(B_1\) parallel to the line \(A_1C\). Finally, point \(C\) is moved to point \(C_1\) parallel to the line \(A_1B_1\) such that the resulting triangle \(A_1B_1C_1\) is a right triangle at \(B_1\) and the length of the leg \(A_1B_1\) is 1 unit. What is the length of the leg \(B_1C_1\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
What is the smallest $k$ such that $k$ cells can be marked on a $10 \times 11$ board so that any placement of a three-cell L-shaped piece will touch at least one marked cell?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
In triangle $\vartriangle ABC$, $D$ is on $AC$ such that $BD$ is an altitude, and $E$ is on $AB$ such that $CE$ is an altitude. Let F be the intersection of $BD$ and $CE$. If $EF = 2FC$, $BF = 8DF$, and $DC = 3$, then find the area of $\vartriangle CDF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{3 \sqrt{3}}{2}$}
Find the limit of the following expression as $n$ approaches infinity:

$$n\left(\sum_{k=1}^n \sqrt{1+\frac{k}{n^2}} - n - \frac{1}{4} - \frac{5}{24n}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the limit $\lim_{n \to \infty} \left(\frac{(kn)!}{n!(kn-n)!}\right)^{1/n}$ using the Stirling approximation and Taylor series expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{k^k}{(k-1)^{k-1}}}
Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability $\frac{2}{3}$. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability $\frac{3}{4}$. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125}
Let \( g(z) = z^4 + iz^3 + 1 \). Determine the number of zeros that \( g \) has in the region \( \{z \in \mathbb{C}: \text{Re}(z), \text{Im}(z) > 0\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11:00}
Suppose $g(y)$ is a differentiable function on $\mathbb{R}$ satisfying the equation \( \{g(y)\}^4 - y\{g(y)\}^3 + y^2\{g(y)\}^2 - 3y^3 + 8y^4 - 8y^5 = 0 \) for each \( y \in \mathbb{R} \). Determine the value of \( g'(1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{14}{9}}
Consider the sequence $\{y_n\}$ defined by the equation $y_n^3 - (n^2 - 5n)y_n + 5n^2 = 0$ for all $n \in \mathbb{N}$. Determine the limit of $y_n$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is $9$, spots a license plate with a $4$-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Find the age that is not one of Mr. Jones's children.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Segment $A B$ is the diameter of a circle. A second circle with center at point $B$ has a radius of 2 and intersects the first circle at points $C$ and $D$. Chord $C E$ of the second circle is part of a tangent to the first circle and is equal to 3. Find the radius of the first circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{\sqrt{7}}}
7. Xiao Ming and Xiao Hua are counting the picture cards in a box together. Xiao Ming is faster, and in the time it takes him to count 6 cards, Xiao Hua can only count 4. When Xiao Hua counted to 48 cards, he forgot the number and had to start over. When he counted to 112 cards, there was only 1 card left in the box. The box originally had ( ) picture cards.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{353}
Determine the asymptotic complexity \( \Theta \) of the recurrence relation: \[ t(n) = 2^n t(n/2) + n. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(2^{2n})}
Given 3 cities, each pair of cities is connected by a road with an exponentially distributed cost with parameter 1. Calculate the expected value of the minimum cost to connect all cities, denoted as \( E(c_3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{6}}
Four, $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns;
$$
\begin{array}{llllll}
a_{11} & a_{12} & a_{13} & a_{14} & \cdots & a_{1 n} \\
a_{21} & a_{22} & a_{23} & a_{24} & \cdots & a_{2 n} \\
a_{31} & a_{32} & a_{33} & a_{34} & \cdots & a_{3 n} \\
a_{41} & a_{42} & a_{43} & a_{44} & \cdots & a_{4 n} \\
\cdots \cdots
\end{array}
$$

where the numbers in each row form an arithmetic sequence, and the numbers in each column form a geometric sequence, with all common ratios being equal. Given $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$, find
$$
a_{11}+a_{22}+a_{33}+a_{44}+\cdots+a_{n n} .
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2-\frac{1}{2^{n-1}}-\frac{n}{2^{n}}}
Given the functions $f(x)=x^{2}-2x+2$ and $g(x)=-x^{2}+ax+b- \frac {1}{2}$, one of their intersection points is $P$. The tangent lines $l_{1}$ and $l_{2}$ to the functions $f(x)$ and $g(x)$ at point $P$ are perpendicular. Find the maximum value of $ab$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{4}}
The chord $AB$ of the ellipse $O: \frac{x^2}{12} + \frac{y^2}{6} = 1$ satisfies $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$. Let the projection of the origin $O$ onto $AB$ be $M$. The number of points $M$ at a distance of $1$ from the line $x+y=1$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A motorist travels for some hours, the first half at 60 kmph and the rest at 48 kmph. The motorist traveled a distance of 324 km. How many hours did the motorist travel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.075}
Five standard six-sided dice are rolled.  We are told there is no three-of-a-kind, but there is a pair of dice that show the same number.  These two dice are set aside, and the other three dice are re-rolled.  What is the probability that after re-rolling these three dice, at least three of the five dice show the same value?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{9}}
Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and the altitude on side $BC$ is $\frac{a}{2}$. Determine the maximum value of $\frac{c}{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \sqrt{2}}
Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of $3:2:1$, respectively. Due to some confusion they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{5}{18}$}
Given two singular matrices $X$ and $Y$ such that $XY = Y$ and $YX = X$, express $X^2 + Y^2$ in terms of $X$ and $Y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X + Y}
Shapovalov A.V.

A banker learned that among identical-looking coins, one is counterfeit (lighter). He asked an expert to identify this coin using a balance scale without weights, and required that each coin participate in weighings no more than twice. What is the maximum number of coins the banker can have so that the expert can definitely identify the counterfeit one in $n$ weighings?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n^2+1}
Determine the asymptotic behavior of the sequence \( n - \left( {\sum\limits_{i = n+1}^\infty  {\frac{1}{{i^2 }}} } \right)^{ - 1} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
$\left.\begin{array}{c}\text { Sphere inscribed in a pyramid } \\ \text { Regular pyramid }\end{array}\right]$

In a regular quadrilateral pyramid with a height not less than $h$, a hemisphere of radius 1 is placed such that it touches all the lateral faces of the pyramid, and the center of the hemisphere lies on the base of the pyramid. Find the smallest possible value of the total surface area of such a pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
How many factors of $$(20^{12})^2$$ less than $$20^{12}$$ are not factors of $$20^{12}$$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
On the lateral side \( CD \) of the trapezoid \( ABCD (AD \parallel BC) \), point \( M \) is marked. From vertex \( A \), a perpendicular \( AH \) is dropped onto segment \( BM \). It turns out that \( AD = HD \). Find the length of segment \( AD \) if it is known that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Evaluate the integral $$\int_{0}^{\infty}e^{-\sqrt 3 x^{\frac{2}{3}}}\sin(x^{\frac{2}{3}})\mathrm{d}x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{\pi}}{16}}
How many real numbers $x$ satisfy the equation $\frac{1}{5}\log_2 x = \sin (5\pi x)$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{159}
In order to compute the area of a particular circle, Juan first measures the length of its diameter. The actual diameter is 20 cm, but Juan's measurement has an error of up to $20\%$. What is the largest possible percent error, in percent, in Juan's computed area of the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
设 (a_1,b_1),(a_2,b_2),\cdots,(a_{100},b_{100}) 是不同的非负整数对. 用 N 表示满足 1\leq i<j\leq100 且 |a_ib_j-a_jb_i|=1 的数对 (i,j) 的个数. 求 N 的最大可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{197}
Evaluate the integral in closed form: \[ \mathfrak{R} \int_{0}^{\pi/2} \frac{x^2}{x^2 + \log^2(-2\cos x)} \, \mathrm{d}x \] where \(\mathfrak{R}\) denotes the real part, and \(\log(z)\) denotes the principal value of the logarithm for \(z \neq 0\) given by \[\log(z) = \ln |z| + i \arg z, \quad -\pi < \arg z \leq \pi.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Let \( k \) be a positive integer. A sequence \( {a}_{0},{a}_{1},\ldots ,{a}_{n},n > 0 \) of positive integers satisfies the following conditions:
 (i) \( {a}_{0} = {a}_{n} = 1 \) ;
 (ii) \( 2 \leq  {a}_{i} \leq  k \) for each \( i = 1,2, \) ... \( ,n - 1 \) ;
 (iii)For each \( j = 2,3,\ldots ,k \) ,the number \( j \) appears \( \phi \left( j\right) \) times in the sequence \( {a}_{0},{a}_{1},\ldots ,{a}_{n} \) ,where \( \phi \left( j\right) \) is the number of
 positive integers that do not exceed \( j \) and are coprime to \( j \) ;
 (iv) For any \( i = 1,2,\ldots ,n - 1,\gcd \left( {{a}_{i},{a}_{i - 1}}\right)  = 1 = \gcd \left( {{a}_{i},{a}_{i + 1}}\right) \) ,and \( {a}_{i} \) divides \( {a}_{i - 1} + {a}_{i + 1} \) .
 Suppose there is another sequence \( {b}_{0},{b}_{1},\ldots ,{b}_{n} \) of integers such that \( \frac{{b}_{i + 1}}{{a}_{i + 1}} > \frac{{b}_{i}}{{a}_{i}} \) for all \( i = 0,1,\ldots ,n - 1 \) . Find the minimum
 value of \( {b}_{n} - {b}_{0} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Differentiate the following integral with respect to the function $f(t)$, where the upper bound of the integral is $t$:
\[
\frac{\partial}{\partial f(t)} \int_0^t f^3(\tau) \cdot g(\tau) \, d\tau
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 f(t)^2 g(t)}
Assume that $a$ and $b$ are nonnegative real numbers such that \(\sin{(ax+b)}=\sin{29x}\) for all integers \(x\). Find the smallest possible value of \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\pi - 29}
Given the differential equation \( \frac{dy}{dx} = 1-y^2 \) and the condition that the function \( y \) is symmetric about the point \( (\ln(9),0) \), find the value of \( y(\ln(4)) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{65}{97}}
Calculate the expected number of Bernoulli trials required to achieve a streak of 10 consecutive successes, given that the probability of success for each trial is 0.2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12207030}
Given the line $y=ax$ intersects the circle $C:x^2+y^2-2ax-2y+2=0$ at points $A$ and $B$, and $\Delta ABC$ is an equilateral triangle, then the area of circle $C$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\pi}
Given a function $f_0(t)$ that is pointwise discontinuous at a finite number of points and continuous everywhere else, and a sequence of functions $f_1, f_2, \cdots$ generated by the recursive process:
$$f_n(t) = \frac{1}{2\epsilon}\int_{t-\epsilon}^{t+\epsilon}f_{n-1}(\tau)d\tau, \quad \text{for } n = 1, 2, \cdots \text{ and some } \epsilon > 0$$
Determine the maximum number of times the function $f_k(t)$ can be continuously differentiated.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k}
An insurance policy covers two employees of a company, reimbursing no more than one loss per employee per year, up to a company-wide maximum of $8000. The probability of an employee incurring a loss is $40\%$, with losses uniformly distributed between $(1000, 5000)$, independent of other employees' losses. Given that one employee has incurred a loss exceeding $2000$, what is the probability that the total losses will exceed the policy's reimbursement limit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{51}}
Evaluate the limit: \[ \lim_{n \to \infty} \left( \sin^4{x} + \frac{1}{4} \sin^4{2x} + \cdots + \frac{1}{{4}^n} \sin^4{2nx} \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin^2 x}
Evaluate \( g'(\frac{1}{2}) \) where \( g(a) = \lim_{h \to 0^+} \int_h^{1-h} t^{-a}(1-t)^{a-1} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the smallest value of \( T \) such that \( |y(x)| \leq 0.1 \) for all \( x > T \), where \( y(x) = e^{-\frac{x}{5}} \left[2\cos\left(\frac{\sqrt{34}}{5} x\right) + \frac{7}{\sqrt{34}}\sin\left(\frac{\sqrt{34}}{5}x\right)\right] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15.75}
Evaluate the limit: $$ \lim_{m\to\infty} \frac{\int_0^{\pi/2}(\sin x)^{2m}dx}{\int_0^{\pi/2}(\sin x)^{2m+1}dx} $$ using integral properties or special functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A physical system consists of four identical particles. The total energy of the system is \(4E_0 > 0\). Each particle can have an energy level equal to \(kE_0\) for \(k \in \{0, 1, 2, 3, 4\}\). A particle with energy \(kE_0\) can occupy any one of the \(k^2 + 1\) distinct energy states at that energy level. How many distinct energy configurations can the system have, considering the particles obey Bose-Einstein statistics?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Evaluate the integral \(\int \int_R |xy|\,dx\,dy\) over the region \(R\) bounded by \(x^2+4y^2=1\) and \(x^2+y^2=1\) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
Find the particular solution $p_n$ for the nonhomogeneous recurrence relation $a_n = 2a_{n-1} + 3a_{n-2} + 8n - 4$, given the initial conditions $a_0 = 1$ and $a_1 = 3$. The non-homogeneous term is $f(n) = 8n - 4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2n - 3}
Find the formula for the number of reflection subgroups of the dihedral group $D_{2n}$, where the group has order $2n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sigma(n)}
Use the residue theorem to evaluate the integral \( \int_{0}^{2\pi}  \frac {27} {(5+4\sin\theta)^2} d\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\pi}
Determine the maximum number of axes of symmetry that the union of \( k \) line segments on a plane can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2k}
Three cyclists started simultaneously: the first and second from point A, and the third towards them from point B. After 1.5 hours, the first cyclist was equidistant from the other two, and 2 hours after departure, the third cyclist was equidistant from the first and second. Determine how many hours after departure the second cyclist was equidistant from the first and third.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $\mathcal{F}$ be the family of all analytic functions $f: D_1(0) \to D_1(0)$ such that $f(\frac{i}{3}) = 0$. Determine the supremum $M = \sup\{\operatorname{Im} f(0) : f \in \mathcal{F}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Determine a region in the xy-plane where the differential equation \( \frac{dy}{dx} = y^{4/5} \) has a unique solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y \neq 0}
Find the degree of the splitting field of the polynomial $x^6 + 1$ over the field $\mathbb{Z}_3$. Express your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the tangent map of the function \( f: O(n) \rightarrow O(n) \) defined by \( f(M) = M^3 \) at the identity matrix \( I \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3X}
A tank is full of oil weighing 20 lb/ft³. The tank is an inverted right rectangular pyramid with a base width of 2 feet, a depth of 2 feet, and a height of 5 feet. Calculate the work required to pump the oil to a height of 3 feet above the top of the tank.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1700}{3}}
Evaluate the contour integral \( \oint_{C}\frac{dz}{z^2-1} \) around the unit circle \( C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the minimum number of vertices in a connected graph where each vertex has a degree of at least 3, and there is no path that visits each vertex exactly once.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the limit: $$\lim_{x\rightarrow +\infty}{8x(x+1)-\sqrt{4x^2+2x}\cdot\sqrt[3]{64x^3+144x^2+90x+17}}$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Your favorite baseball team is playing against your uncle's favorite team in the World Series. At the beginning of each game, you and your uncle bet on the game's outcome. If you bet \( b \) dollars and your team wins the game, your uncle gives you an IOU for \( b \) dollars. But if they lose the game, you give him an IOU for \( b \) dollars. When the series is over, all outstanding IOUs are settled in cash. You would like to walk away with \( \$100 \) in cash if your team wins the series, and lose \( \$100 \) if your team loses the series. How much should you bet on the opening game to achieve this outcome? (The first team to win a total of four games wins the series.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31.25}
A set \( M \) of points on a line can be covered by three segments of length 1. Determine the smallest number of segments of length 1 required to cover all midpoints of segments with endpoints in the set \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the integral: \[ \int_{-\infty}^\infty \exp(-\sqrt{1+x^2})\,dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 K_1(1)}
Evaluate the integral: $$\int_{-1}^1 \frac{d}{dx}\ \frac{1}{1 + 2^{1/x}}\ dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8, 4$. Define $n = (10a + b)^{10c+d}$. Calculate the probability that $1984!$ is divisible by $n$. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{6}}