inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the minimum number of coin flips $n$ required to ensure that the estimated probability $\hat{p} = X/n$ of a biased coin is within $5\%$ of the true probability $p$, given that $0.4 \leq p \leq 0.6$. Ensure that the probability of this confidence interval, $\mathbb{P}[0.95p \leq \hat{p} \leq 1.05p]$, is at least 0.95.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2305}
Let \( S \) be a set of 2017 points in the plane. Define \( R \) as the radius of the smallest circle that contains all points in \( S \) either on its interior or boundary. Let \( D \) be the longest distance between any two points in \( S \). Determine the pair \((a, b)\) of real numbers such that \( a \le \frac{D}{R} \le b \) for all possible sets \( S \), where \( a \) is as large as possible and \( b \) is as small as possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\sqrt{3}, 2)}
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{x\sin(kx)}{x^2+a^2} \, \mathrm{d}x \) using real analysis techniques, where \( k > 0 \) and \( a \) is a positive constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi e^{-a k}}
Evaluate the limit: $$\lim_{n\to\infty}\left(\frac{(n+1)^{n+1}}{n^n}-\frac{n^n}{(n-1)^{n-1}}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
In a $2 \times 3 \times 3$ array of letters, how many five-letter strings can be formed if you are allowed to repeat letters but cannot stall and repeat? An example of an acceptable string is "DETER", while an unacceptable string is "ADDER". Disregard the requirement that the string be an actual English word.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1503378}
Find the area of the region in the xy-plane defined by the following conditions:
\[
\left\{
\begin{array}{l}
2x + y \leq 2, \\
x \geq 0, \\
\left( x + \sqrt{x^2 + 1} \right) \left( y + \sqrt{y^2 + 1} \right) \geq 1,
\end{array}
\right.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the value of \(a \in \mathbb{R}\) for which all nontrivial solutions of the differential equation \(x'' + ax' + 4x = 0\) are oscillatory about zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-4, 4)}
Let \( p, q \) be two distributions over the same finite set \( X \) with a total variation distance \( TV(p,q) = \epsilon \). Given a function \( f: X \to \mathbb{R} \) such that \( 0 \leq \sum_{x \in X} f(x) p(x) \leq 1 \) and \( 0 \leq \sum_{x \in X} f(x) q(x) \leq 1 \), determine the best upper bound of \( \left| \sum_{x \in X} f(x)(p(x) - q(x)) \right| \) as a function of \( \epsilon \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit: $$\lim_{x\to0}{\frac{\sqrt{\cosh{(5x^2)}}\cdot e^{2x^3}-1}{x^2\tan(3x)}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Calculate the limit: $$\lim_{n \to \infty} \sqrt[n]{|\sin n|}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the smallest integer \( m \) such that the inequality \( 0.\overline{2}_b < \log_{b}{2} \) holds for all bases \( b > m \), where \( 0.\overline{2}_b \) represents a repeating decimal in base \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $g: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function with 12 distinct zeros. Determine the values of $n \ge 1$ for which the $n$-th derivative $g^{(n)}$ has at least 2 zeros.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 \leq n \leq 10}
Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{1}{\sqrt[4]{n^4+n+2}} + \frac{1}{\sqrt[4]{n^4+n+3}} + \cdots + \frac{1}{\sqrt[4]{n^4+5n-1}} \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the line integral \( \int_{L}{(2x-y)dx+(x-y)dy} \) without using Green's theorem, where the curve \( L \) is defined as the union of the semicircle \( \{(x,y): x^{2}+y^{2}=2y, x\geq0\} \) and the quarter-circle \( \{(x,y):x^{2}+y^{2}=4, x\leq0,y\geq0\} \), oriented counter-clockwise starting from the point \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\pi + 4}
Use Green's Theorem to calculate the work done by the force field \( \mathbf{F}(x,y)=x(x+y)\mathbf{i}+xy^2\mathbf{j} \) as a particle moves along the path from the origin along the \( x \)-axis to \((1,0)\), then along the line segment to \((0,1)\), and back to the origin along the \( y \)-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{12}}
Given that $w$ is a complex cube root of unity with $w \neq 1$, determine the number of non-singular matrices of the form:
\[A=\begin{bmatrix} 1 & a & b\\ w & 1 & c\\ w^2 & w & 1 \end{bmatrix}\]
where $a, b, c$ are elements of the set $S=\{w, w^2\}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Solve for x in the equation: 
\[ 2^{x^{2^{x^{\cdots}}}} = 4 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[4]{2}}
Given that two congruent 30°-60°-90° triangles with hypotenuses of 12 are overlapped such that their hypotenuses exactly coincide, calculate the area of the overlapping region.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9 \sqrt{3}}
When \((x^2 + x + 1)^{54}\) is expanded, how many coefficients of the resulting polynomial are divisible by 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Find $x$ if\[\cfrac{1}{\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}+\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}}=\frac{x}{36}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70
}
A soccer player takes 19 consecutive penalty shots. The probability of failing the $n$-th shot is $\frac{1}{2n+1}$. What is the probability that the player succeeds in an even number of these shots?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{19}{39}}
Given two independent random variables $X$ and $Y$ that are both normally distributed with mean 0 and variance 1, $\mathcal N(0,1)$, find the expected value of $X^4 + Y^3$ conditioned on the sum $X + Y$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the volume of the solid obtained by revolving the area enclosed by the curve \(27ay^{2} = 4(x-2a)^{3}\), the \(x\)-axis, and the parabola \(y^{2} = 4ax\) about the \(x\)-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80\pi a^3}
Using the four arithmetic operators and parentheses, find a way to combine the numbers 10, 10, 4, and 2 such that the result is 24. What is the arithmetic expression?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2 + 4 \div 10) \times 10}
Calculate the limit: \[ \lim_{n \to \infty} \frac{2\cdot6\cdot10\cdot\cdot\cdot(4n-2)}{1\cdot5\cdot9\cdot\cdot\cdot(4n-3)} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given the coin denominations 1 cent, 5 cents, 10 cents, and 50 cents, determine the smallest number of coins Lisa would need so she could pay any amount of money less than a dollar.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given the function \( f(x) = \cos x + \log_2 x \) for \( x > 0 \), if the positive real number \( a \) satisfies \( f(a) = f(2a) \), then find the value of \( f(2a) - f(4a) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the sum of the three smallest integer values of the parameter \(a\) for which the inequality \(\frac{x^{2}+(a+1) x+a}{x^{2}+5 x+4} \geq 0\) results in the union of three non-overlapping intervals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let \( s_n \) be the number of solutions to the equation \( a_1 + a_2 + a_3 + a_4 + b_1 + b_2 = n \), where \( a_1, a_2, a_3, \) and \( a_4 \) are elements of the set \( \{2, 3, 5, 7\} \) and \( b_1 \) and \( b_2 \) are elements of the set \( \{1, 2, 3, 4\} \). Determine the number of values of \( n \) for which \( s_n \) is odd.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The real function $g$ has the property that, whenever $x,$ $y,$ $m$ are positive integers such that $x + y = 3^m,$ the equation
\[g(x) + g(y) = 2m^2\]holds. What is $g(2187)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
Determine the number of elements in the centralizer of the permutation \((1234)\) in the alternating group \(A_4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the circle equation x^{2}+y^{2}-4x-4y+7=0, find the number of tangent lines that have equal intercepts on the x and y axes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Fran's school just instituted a school uniform policy. Each student needs to buy five complete uniforms, each consisting of pants, shirt, tie and socks. The pants cost $20, the shirt costs a certain ratio as the pants, the tie costs 1/5 as much as the shirt and the socks cost $3/pair. Each student needs to spend $355. What is the ratio of the cost of the shirt to the cost of the pants?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
In a binomial random graph $G_{n,p}$ with $n$ vertices, where each edge is present with probability $p$, consider the event that $k$ distinct sets of 3 vertices each form a triangle. Express the probability of this event, $\mathbb{P}\left(\bigcap_{j=1}^k \mathcal{E}_j\right)$, in terms of $p$. Here, $\mathcal{E}_j$ denotes the event that a set of 3 vertices forms a triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^{3k}}
Find the surface area of the solid generated by revolving the polar curve \( r = \sqrt{\cos 2\theta} \) around the y-axis from \( \theta = 0 \) to \( \theta = \frac{\pi}{4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \sqrt{2}}
Given that $X$ and $Y$ are independent and identically distributed random variables, each following an exponential distribution with parameter $\lambda = 1$, compute the expected value $E(\sqrt{X+Y})$. Provide your answer to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.33}
An urn contains a large number of balls with 50% black, 30% white, and 20% red. Balls are extracted randomly, one at a time, recorded, and returned to the urn. What is the average number of balls one should extract until all three colors have appeared?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{559}{84}}
Evaluate the limit: $$\lim_{x\to 0} \frac{\sqrt[5]{1+\frac{x}{5}}-\sqrt[6]{1+\frac{x}{6}}}{1-\sqrt{1-\frac{x}{3}}}$$ without using L'Hospital's rule, power expansions, or derivatives.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{150}}
Compute the value of \(1234^{1234} \pmod{5379}\) given that \(5379 = 3 \times 11 \times 163\). Do not use Euler's Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4603}
Let \(\Gamma\) be the set of functions \(f_n: \mathbb{C} \to \mathbb{C}\) defined by \(f_n(z) = \frac{\pi^2}{24}z^2 - \frac{1}{n^2}z + \frac{589}{20} \cdot \frac{1}{n^3}\) for \(n \in \mathbb{N}\). Let \(\Lambda\) be the set of all roots with positive imaginary part of the elements of \(\Gamma\). Find the sum of all elements of \(\Lambda\), rounding the real and imaginary parts to the nearest integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + 22i}
For any two different real numbers \( x \) and \( y \), define \( D(x, y) \) as the unique integer \( d \) that satisfies
\[ 
2^{d} \leq |x-y| < 2^{d+1}.
\]
Given a set of real numbers \( \mathscr{F} \), for \( x \in \mathscr{F} \), the "scale" of \( x \) in \( \mathscr{F} \) is defined as the value of \( D(x, y) \) for some \( y \in \mathscr{F} \) where \( x \neq y \).

Let \( k \) be a given positive integer. Suppose each \( x \) in \( \mathscr{F} \) has at most \( k \) different scales in \( \mathscr{F} \) (these scales may vary depending on \( x \)). Find the maximum number of elements in the set \( \mathscr{F} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k}
设 $\omega$ 为正实数, 若存在 $a, b(\pi \leqslant a<b \leqslant 2 \pi)$, 使得 $\sin \omega a+\sin \omega b=2$, 则 $\omega$的取值范围是. 请提供满足条件的区间所有整数分子分母的和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Given the differential equation \( y'' + q(t)y = 0 \) where \( q(t) \) is a continuous periodic function with period \( T \), and \( y_1 \) and \( y_2 \) are linearly independent solutions, calculate the value of \( y_1(T) + y_2'(T) \) if there exists a non-trivial solution \( y_* \) that is periodic with period \( T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find a maximal ideal \(I\) in the ring \(\mathbb{Z}[i]\) such that \(\mathbb{Z}[i]/I\) is isomorphic to \(\mathbb{Z}/521\mathbb{Z}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(20 + 11i)}
Evaluate the contour integral \( \int_{\gamma} \frac{\log(z)}{z} \, dz \) where \( \gamma(t) = e^{it} \) for \( 0 \le t \le 2\pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi^2}
Evaluate the integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\ln(1+b\sin x)}{\sin x} \, dx \) where \(|b|<1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \arcsin b}
Evaluate the expression \( \frac{100^2}{100!} + \sum_{k=1}^{100} \left| (k^2 - 3k + 1) S_k \right| \), where \( S_k \) is the sum of the infinite geometric series with first term \( \frac{k-1}{k!} \) and common ratio \( \frac{1}{k} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
For a given positive integer $k$, find the values of $\alpha$ for which the series \[ \sum_{a\in \mathbb{N}^n} \left( \sum_{i=1}^n a_i^k\right)^{-\alpha} \] converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > \dfrac{n}{k}}
Determine the range of values of $p$ for which the series \( \sum_{k = 1}^\infty \frac{\exp(-p k)}{k^{1-p}} \) converges, where \( p \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \infty)}
Compute the limit: \[ \lim_{n \to \infty} \int_0^{n^2} e^{-x^2}n\sin\frac{x}{n} \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the following expression:
\[
1+\frac{1+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}}{2+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}}.
The original answer is in the form \(\frac{k+\sqrt{m}}{n}\). Please find the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the number of triples(x,a,b)where \( x \) is a real number and \( a,b \) belong to the set \( \{ 1,2,3,4,5,6,7,8,9\} \) such that
 \( {x}^{2} - a\{ x\}  + b = 0. \)
 where \( \{ x\} \) denotes the fractional part of the real number \( x \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
112
}
Let \( S \) be a subgroup of the group of positive rational numbers under multiplication, which is dense in the positive reals. Define the S-restricted irrationality measure by confining \( p/q \) to \( S \). If \( S \) is finitely generated, particularly if \( S \) is generated by the first \( n \) primes, determine the S-restricted irrationality measure \( \xi \) such that almost all \( x \) have measure \( \xi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the double integral \( \int_0^2 \int_{\frac{y}{2}}^1 e^{x^2} \, dx \, dy \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e - 1}
Given that $a > 0$, $b > 0$, $c > 1$, and $a + b = 1$, find the minimum value of $( \frac{a^{2}+1}{ab} - 2) \cdot c + \frac{\sqrt{2}}{c - 1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 + 2\sqrt{2}}
Bob is making partitions of 10, but he hates even numbers, so he splits 10 up in a special way. He starts with 10, and at each step, he takes every even number in the partition and replaces it with a random pair of two smaller positive integers that sum to that even integer. For example, 6 could be replaced with 1+5, 2+4, or 3+3, all with equal probability. He terminates this process when all the numbers in his list are odd. The expected number of integers in his list at the end can be expressed in the form \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Compute \(100m+n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7120}
On the diagonals $A C$ and $B D$ of trapezoid $A B C D$ with bases $B C=a$ and $A D=b$, points $K$ and $L$ are located respectively, such that

$C K: K A=B L: L D=7: 4$. Find $K L$.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{11}|7b-4a|}
Let $ABC$ be a scalene triangle with inradius $1$ and exradii $r_A$, $r_B$, and $r_C$ such that $20(r_B^{2r}C^2+r_C^2r_A^2+r_A^2r_B^2)=19(r_Ar_Br_C)^2.$
If 
$tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019,$ 
then the area of $\triangle{ABC}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Proposed by Tristan Shin
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
201925
}
4.1. Let \( f(x) = x^2 - x + 1 \), \( g(x) = -x^2 + x + 1 \). Find a non-constant polynomial \( h \) such that \( h(f(x)) = h(g(x)) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x-1)^2}
令S=\ds{m=1}{\infty}\ds{n=1}{\infty}\df{m^2n}{2^m(n2^m+m2^n)},则 [100S]=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
On a backpacking trip with 10 people, in how many ways can I choose 2 cooks and 1 medical helper if any of the 10 people may fulfill these roles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
Calculate the covariance of the random variables X and Y given the joint density function:
\[ f_{X,Y}(x,y) = \frac{1}{2} \int_{0}^{\infty} x^3 e^{-x(y+2)} \text{ d}x \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Problem 4. Simplify the expression

$$
M=\frac{2}{\sqrt{4-3 \sqrt[4]{5}+2 \sqrt[4]{25}-\sqrt[4]{125}}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[4]{5}+1}
Find the norm of the operator $N_x: l_2 \to l_2$ defined by $N_x(y) = (x_n y_n)$, where $x = (x_n) \in l_\infty$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|x\|_\infty}
In a city with 10 parallel streets and 10 streets intersecting them at right angles, what is the minimum number of turns a closed bus route passing through all intersections can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given a machine with two components, it will fail if either component fails. Let $X$ and $Y$ be the time-to-failure random variables for the first and second components, respectively, measured in hours. The joint moment generating function of $X$ and $Y$ is given by: $$ M_{X,Y}(s,t)=\frac{1}{(1-3s-2t+6ts)}. $$ Determine the probability that the machine will fail within the first hour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - e^{-5/6}}
Three faces of a right rectangular prism have areas of 54, 56, and 60 square units. Calculate the volume of the prism in cubic units and round it to the nearest whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{426}
Each of the symbols $\diamond$ and $\circ$ represents an operation in the set $\{+,-,\times,\div\}$, and $\frac{15 \diamond 3}{8 \circ 2} = 3$. What is the value of $\frac{9 \diamond 4}{14 \circ 7}$? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{13}{7}}
Find a $n\in\mathbb{N}$ such that for all primes $p$, $n$ is divisible by $p$ if and only if $n$ is divisible by $p-1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1806}
Given that θ ∈ [0, 2π), if the circle (x - cosθ)² + (y - sinθ)² = r² (r > 0) intersects with the line 2x - y - 10 = 0, find the minimum value of r.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt {5} - 1}
Evaluate the integral
$$ \int_0^1 \frac{\ln\left(\left(x+\sqrt{1-x^2}\right)^2 / \left(x-\sqrt{1-x^2}\right)^2\right)}{1-x^2} x \, dx $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi^2}{2}$}
The sum of the digits of [(10 ^ x) ^ a] - 64 = 279, where a is a positive integer. The value of xy is 32. What is the value of a?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A book contains 30 stories. Each story has a different number of pages under 31. The first story starts on page 1 and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Calculate the value of the expression: \[ \arctan \frac{21 \pi}{\pi^2-54} + \arctan \frac{\pi}{18} + \arctan \frac{\pi}{3} \] without using a calculator or Maclaurin series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In the Cartesian coordinate plane, the circle \( C_1: x^2 + y^2 - a = 0 \) is symmetric about a line \( l \). The resulting circle is \( C_2: x^2 + y^2 + 2x - 2ay + 3 = 0 \). Determine the equation of the line \( l \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2x - 4y + 5 = 0}
For a set $P$ of five points in the plane, with no three points being collinear, determine the maximum number of acute triangles that can be formed using the points in $P$ as vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
An urn contains 5 balls numbered 0, 1, 2, 3, and 4. A ball is drawn from the urn, its number is recorded, and then it is returned to the urn. This process is repeated until the balls numbered 1, 2, and 3 have each been drawn at least once. What is the probability that this process is repeated exactly 5 times?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{66}{625}}