inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the integral \( \int_R (y+2x^2)(y-x^2) \, dA \) where \( R \) is the region bounded by the curves \( xy=1 \), \( xy=2 \), \( y=x^2 \), and \( y=x^2-1 \) with \( x \) and \( y \) positive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Find the simplest possible relation between the holomorphic functions $f$ and $g$ in a certain region of the complex plane, given that $$|f(z)|^2 + |f(z)| = |g(z)|^2 + |g(z)|.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(z) = e^{i\theta} g(z)}
Find the limit of the expression \( \sum_{k = n}^{5n} \binom{k-1}{n-1} \frac{1}{5^n} \left(\frac{4}{5}\right)^{k-n} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Suppose $X_1, X_2, \ldots, X_n$ are independent and identically distributed random variables, where $X_i > 0$ and $E[X_i] = \mu$, $\text{Var}(X_i) = \sigma^2$. If $m < n$, find $E\left(\frac{X_1 + X_2 + \cdots + X_m}{X_1 + X_2 + \cdots + X_n}\right)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{m}{n}}
Calculate the expected number of picks required to end a game where you start by randomly selecting a number $K$ from the set $[0, N-1]$. The game ends if $K = 0$. If $K \neq 0$, you continue selecting numbers from the set $[0, K-1]$ until you eventually pick 0. What is this expected number of picks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H_N}
Determine the values of \( k \) for which the matrix \( A = \begin{bmatrix} 1 & 1 & k \\ 1 & 1 & k \\ 1 & 1 & k \end{bmatrix} \) has an eigenbasis in \( \mathbb{R}^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \neq -2}
Let $\gamma$ be a closed and continuously differentiable path in the upper half-plane $\{z \in \mathbb{C} : z = x + iy, x, y \in \mathbb{R}, y > 0\}$, excluding the point $i$. Determine the value of the contour integral:
\[ \frac{1}{2\pi i} \oint_\gamma \frac{2i}{z^2 + 1} \, dz \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A pyramid is inscribed in a cone such that the base of the pyramid is a quadrilateral with pairs of adjacent sides equal, and the angle between one pair of adjacent sides is \( \alpha \). Determine the ratio of the volume of the pyramid to the volume of the cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2 \sin \alpha}{\pi}}
Determine the conditions on \(m\) and \(n\) such that the integral \(\int_{0}^{\infty} \frac{x^{n}}{(1+x)^{m}}dx\) converges, where \(m,n \geq 0\) and \(m,n \in \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m > n + 1}
Evaluate the limit: \[ \lim_{n\rightarrow \infty}\frac{n!n^{x}}{x(x+1)...(x+n)} \] given that \( \text{Re}(x) > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Gamma(x)}
Evaluate the limit: \[ \lim_{n\to\infty}\dfrac{1}{n}\left\{\dfrac{(2n)!}{n!}\right\}^{\dfrac{1}{n}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{e}}
What is the minimum number of values needed to uniquely determine a line in three-dimensional space \(\mathbb{R}^3\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Compute the expected value \( E[e^{vS}] \), where \( v \) and \( S \) are two independent random variables, each exponentially distributed with parameters \( \lambda_1 \) and \( \lambda_2 \), respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given $n$ different wine bottles numbered $i=1, 2, \ldots, n$, where the $i$-th bottle is $i$ years old and is still good with a probability of $1/i$, find the expected value of the age of a randomly selected bottle that is still good.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n}{H_n}}
Determine the conditions under which the matrix $X = A^\top Y A$ is invertible, given that $A \in F^{m \times n}$ is full rank and $Y$ is positive definite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \geq n}
Evaluate the limit \(\lim_{r \to \infty} \frac {r^C \int_0^{\frac{\pi}{2}} x^r \sin(x)\, dx}{\int_0^{\frac{\pi}{2}} x^r \cos(x)\, dx}\) where \(C\) is a real constant. Determine the value of \(\pi L - C\), given that \(L > 0\) is the limit's value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the intersection point of the projective lines corresponding to \(y = mx\) and \(y = mx + c\) (where \(c \neq 0\)) in the projective plane, given the embedding \((x,y) \to [1,x,y]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1, m]}
Evaluate the limit of the sequence given by:

$$\lim_{n\to \infty} \frac{1}{n^4}\left(\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+\cdots+\frac{n}{a_n}\right)$$

where $a_n$ is defined recursively by $a_1=1$ and $a_{n+1}=\dfrac{a_n}{1+na_n}$. Find the value of this limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
Determine the maximum number of distinct labels that can be assigned to the vertices of a graph such that each vertex is adjacent to at least one vertex with a different label.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Given a gamma distribution with shape \(\alpha=2\) and rate \(\lambda=10\), find an expression for \(\mathbb{E}[X^{-2}]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the minimum achievable accuracy for an algorithm $\psi$ that approximates an elementary function $f$ within a floating-point system characterized by $e$ bits for the exponent and $m$ bits for the mantissa, given that the domain $D(f) = [1,2)$ and the range $f(D(f)) = [1,2)$. Provide your answer in terms of $e$ and $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{-(m+1)}}
Evaluate the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) using Stokes' Theorem, where \( S \) is the surface defined by the graph of \( f(x,y) = 4x - 8y + 30 \) over the rectangle \( R = \{ (x,y) \mid -2 < x < 3, 0 < y < 2 \} \). The vector field is given by \( \mathbf{F}(x,y,z) = -x^2 \mathbf{i} + xz \mathbf{j} + yx \mathbf{k} \). The curve \( C \) is the boundary of \( S \), oriented counterclockwise when viewed from above.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Determine the number of solutions to the equation \(x^{2} - 2x \cdot \sin(x \cdot y) + 1 = 0\) that lie within the circle defined by \(x^{2} + y^{2} \leq 100\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the interval of convergence for the series \( \sum_{n=1}^{\infty} \left(\frac{n - i}{i - 2n}\right)^{n} (z-2)^{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 4)}
Calculate the line integral \( \int_K \underline{u} \bullet \underline{t} \, ds \) for the vector field \( \underline{u}(x,y,z) = (x^3+2y+z,\,y^3+2x+z,\,1/2(x^2+y^2)+x+y) \), where \( K \) is the curve of intersection of the surfaces \( V: x+y-z=1 \) and \( S: z = 1-x^2-y^2, \ z \geq 0 \), along the path from \((1,0,0)\) to \((0,1,0)\). Use Stokes' theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find a non-constant function \( x(t) \) that satisfies the differential equation \( (x')^2 + x^2 = 9 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \sin t}
Determine the set of positive real numbers \(x\) for which the series \(\sum_{n=1}^\infty \frac{x^{a(n)}}{n^3}\) converges, where \(a(n)\) is the number of zeroes in the base 3 representation of \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 25)}
Given the ordinary differential equation $\dot{\vec{x}} = A\vec{x}$, where $A$ is an $n \times n$ matrix, and the first coordinate of the solution is $x_1(t) = 5t^2\cos(3t) + 2\sin(3t) - e^{2t}\sin(3t) + 4t$, determine the minimum dimension $n$ of the matrix $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $P(x)$ be a polynomial of degree 3 such that $P(-1) = 10$, $P(1) = -6$, $P(x)$ has a local maximum at $x = -1$, and $P'(x)$ has a local minimum at $x = 1$. Determine the distance between the local maximum and local minimum of the curve $y = P(x)$. Express your answer as a positive number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{65}}
Find the sum of the roots of the equation \( 3 \cos \frac{4 \pi x}{5} + \cos \frac{12 \pi x}{5} = 2 \cos \frac{4 \pi x}{5} \left(3 + \operatorname{tg}^{2} \frac{\pi x}{5} - 2 \operatorname{tg} \frac{\pi x}{5}\right) \) within the interval \([-11, 19]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{225}{2}}
Find the greatest common divisor of all binomial coefficients \( \binom{n}{k} \) where \( k \) is odd, for a given integer \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\nu_2(n)}}
Given that \( a, b, c \) are positive real numbers satisfying \( a + b + c = 1 \), evaluate the expression:
\[ f(a, b, c) = \frac{a^4}{a^3 + b^2 + c^2} + \frac{b^4}{b^3 + a^2 + c^2} + \frac{c^4}{c^3 + b^2 + a^2} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{7}}
Find the gradient \( \nabla_A \) of the function \( f(A) = ||Ax - y||^2 \), where \( A \in \mathbb{R}^{n\times n} \) and \( x, y \in \mathbb{R}^n \), and the norm is the Euclidean norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(Ax - y)x^\top}
Given column vectors $v_1, v_2, v_3 \in \mathbb{R}^2$, if for all distinct $i, j \in \{1, 2, 3\}$, the minimum eigenvalue of the matrix $v_iv_i^T + v_jv_j^T$ satisfies $\lambda_{\min}(v_iv_i^T+v_jv_j^T) \leq 1$, determine an upper bound for $\lambda_{\min}(v_1v_1^T+v_2v_2^T+v_3v_3^T)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the lower bound on the number of distinct angles formed by all possible 3-point subsets of a set of $m$ non-collinear points in the plane $\mathbb{R}^2$. Provide your answer as a function of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m - 2}
Three planets revolve about a star in coplanar circular orbits with the star at the center. All planets revolve in the same direction, each at a constant speed, and the periods of their orbits are 60, 84, and 140 years. The positions of the star and all three planets are currently collinear. They will next be collinear after $n$ years. Find $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
n = 105
}
Find the smallest positive integer \( x \) such that \( 10^x \equiv 1 \pmod{1997} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{998}
In a 29x29 table, the numbers 1 through 29 are each written 29 times. The sum of the numbers above the main diagonal is three times the sum of the numbers below this diagonal. Determine the number written in the central cell of the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Find the largest natural number \( m \) such that for all non-negative real numbers \( a_1 \ge a_2 \ge \ldots \ge a_{2014} \ge 0 \), the following inequality holds:
\[ \frac{a_1 + a_2 + \ldots + a_m}{m} \ge \sqrt{\frac{a_1^2 + a_2^2 + \ldots + a_{2014}^2}{2014}}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Let $T \sim \mathcal{U}(0,1)$ and define the random variable $Y$ as follows: \[ Y = \begin{cases} -\Phi^{-1}(T+\frac{1}{2}) & \text{if} & 0 \leq T \leq \frac{1}{2} \\ -\Phi^{-1}(T-\frac{1}{2}) & \text{if} & \frac{1}{2} < T \leq 1, \end{cases} \] where $\Phi^{-1}$ is the inverse of the normal distribution function. Calculate the probability $P(Y \leq y)$ for a given $y \in \mathbb{R}$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Phi(y)}
On line \( m \), three equilateral triangles are placed: \(\triangle ABC\), \(\triangle HFG\), and \(\triangle DCE\). It is given that \(BC = \frac{1}{3} CE\), and \(F\) and \(G\) are the midpoints of \(BC\) and \(CE\) respectively. Additionally, \(FM \parallel AC\) and \(GN \parallel DC\). Let the areas of the three parallelograms in the figure be \(S_{1}, S_{2}, S_{3}\), respectively. If \(S_{1} + S_{3} = 10\), find the value of \(S_{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the number of elements of order 9 in the group \(C_{135} \times C_6 \times C_9\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
Find the cardinality of the set of all algebraic extensions of an infinite field $k$ that have the same cardinality as $k$. Provide your answer as a cardinal number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{|k|}}
ABC is an acute-angled triangle with area 1. A rectangle R has its vertices on the sides of the triangle, with two vertices on BC, one on AC, and one on AB. Another rectangle S has its vertices on the sides of the triangle formed by the points A, R3, and R4, with two vertices on R3R4 and one on each of the other two sides. What is the maximum total area of rectangles R and S over all possible choices of triangle and rectangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Evaluate the Stieltjes integral \( \int_{0}^3 x^2 \, d\alpha \), where \( \alpha(x) = x+1 \) for \( 0 \le x \le 1 \) and \( \alpha(x) = x^2+2 \) for \( 1 < x \le 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{124}{3}}
Evaluate the integral \( \int_{C} f(z) \, dz \) over the upper half-circle \( C \) with \( 0 \leq \arg z \leq \pi \), starting from \( z=1 \), using the parametrization \( z = e^{ix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i\pi}
In a room with 3 bulbs, each bulb has a lifetime that is independently and identically distributed according to an exponential distribution with parameter \( \lambda = 1 \). Bulbs are replaced only when all bulbs burn out. What fraction of the time is the room only one-third lit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{6}{11}}
Let \( a = (123)(456) \) in \( S_{10} \). Determine the highest possible order of a permutation \( b \) in \( S_{10} \) such that \( b^k = a \) for some integer \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In trapezoid \(ABCD\), the bases \(AB\) and \(CD\) measure 41 and 24, respectively. The diagonals of the trapezoid are perpendicular to each other. Calculate the dot product of the vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{984}
In a 2-player game, each player rolls k six-sided dice on their turn. The first player to accumulate a total of k or more sixes over all their turns wins the game. Compute the expected number of turns needed to complete the game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Compute the residue of the function \( \frac{e^{1/z}}{1-z} \) at the point \( z_0 = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e - 1}
Evaluate the improper integral \( \int_0^{\infty} \frac{ \sin{x}}{x^2+1}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2e}}
Determine the asymptotic behavior of the sum \( \sum_{k = 1}^{\frac{n}{2} - \sqrt{n}} n \binom{n - 1}{k - 1} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n 2^n)}
A gun is located at the origin of an infinite number line and starts shooting bullets along the positive x-axis at a rate of one bullet per second. Each bullet's velocity is randomly chosen from a uniform distribution between 0 and 1 m/s. If two bullets collide, they explode and disappear. What is the probability that at least one bullet will travel infinitely without colliding with another bullet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given $S_0 = \{A, B\}$, where $A$ and $B$ are subsets of a set $M$ with a non-empty intersection, define $S_{i+1}$ recursively as the set of all subsets of $M$ that can be obtained by taking the union, intersection, or complement of any two elements in $S_i$. Determine the maximum possible cardinality of $\bigcup S_i$ as $i$ tends to infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find the range of values of \(\lambda\) such that the quadratic equation \((1-\mathrm{i}) x^{2}+ (\lambda+\mathrm{i}) x+(1+\mathrm{i} \lambda)=0\) has two imaginary roots, where \(\lambda\) is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 2) \cup (2, \infty)}
Evaluate the integral \( \int_{C} (z^2 - 3z + \text{Im} \, z)\,dz \) where \( C \) is the square with vertices \( z=0 \), \( z=1 \), \( z=1+i \), and \( z=i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given a function \( f(x, y) \) with continuous partial derivatives everywhere, and \( \nabla f(0, -18) = -2i + 3j \), define a new function \( g(x, y) = f(xy + x^2, xy - y^2) \). Calculate \( \nabla g(3, -3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-15\mathbf{i} + 21\mathbf{j}}
Evaluate the limit using the Central Limit Theorem: 
\[ \lim_{n\to\infty}p^{n}\sum_{k \geqslant{n(p^{-1}-1)}}^{\infty}\binom{n+k-1}{n-1}(1-p)^{k}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
In triangle ABC, three lines divide it into four congruent triangles and three pentagons. One of the triangles has all three sides along the new lines, while the other three triangles each have two sides along the new lines. If all four triangles are congruent, express the area of each triangle in terms of the area of triangle ABC.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{7}}
You are tossing an unbiased coin. The last 28 consecutive flips have all resulted in heads. Let \( x \) be the expected number of additional tosses you must make before you get 60 consecutive heads. Find the sum of all distinct prime factors of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65821}
Let $X_1, X_2, \ldots,$ be independent random variables with distribution $P(X_k=2^k)=P(X_k=-2^k)=\frac{1}{2}$ for all $k$. Define $T_n = X_1 + X_2 + \cdots + X_n$. Determine if $\frac{T_n}{2^{n}}$ converges in distribution, and if so, find the limiting distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{U(-2, 2)}
Given a square matrix $A$ of order $n$ where each element is either -1, 0, or 1, determine the maximum possible value of the determinant of $A$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^{n/2}}
Let \( \{ a_n \} \) be a sequence of non-negative real numbers such that the series \( \sum_{n=1}^{\infty} a_n \) is convergent. Determine the range of values of the real number \( p \) for which the series \( \sum_{n=1}^{\infty} \frac{\sqrt{a_n}}{n^p} \) diverges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, \frac{1}{2}]}
A rock contains $10^{20}$ atoms of a substance with an exponentially distributed lifetime and a half-life of 100 years. Determine the number of centuries that must pass until it is most probable that only 100 atoms remain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59.8}
Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a$, $b$, $c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the number of ways in which the number 200 can be expressed as a sum of positive integers, denoted as \( p(200) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3972999029388}
Determine the number of terms required in the alternating series representation of the integral \( \int_{0}^{\frac{1}{2}} \frac{1}{1+x^6} \, dx \) to ensure the error is less than \( 10^{-5} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the improper integral: \[ \int_{-\infty}^{+\infty} \frac{x^2}{(x^2-8x+20)^2} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5\pi}{4}}
An open can of oil is accidentally dropped into a lake, causing the oil to spread over the surface as a circular disc of uniform thickness. The radius of the disc increases at a steady rate of 10 cm/sec. At the moment when the radius is 1 m, the thickness of the oil slick is decreasing at a rate of 4 mm/sec. Given that the volume of oil remains constant, how fast is the thickness of the oil slick decreasing when the radius is 2 m?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
For a positive integer $n$, let $A_n$ and $B_n$ be the families of $n$-element subsets of $S_n=\{1,2,\ldots ,2n\}$ with respectively even and odd sums of elements. Compute $|A_n|-|B_n|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the integer value of \( \alpha = \angle BAC \) for which it is possible to find a right-angled triangle ABC with at least two integer-valued sides.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Find the sum of the series \( \sum_{n=1}^{\infty} \left( n\cdot \ln\frac{2n+1}{2n-1} - 1\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1 - \ln 2}{2}}
Evaluate the double integral \( I = \int_{-1}^{1}\int_{0}^{x^2} \sqrt{x^2-y}\,dy\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
There are $14$ players participating at a chess tournament, each playing one game with every other player. After the end of the tournament, the players were ranked in descending order based on their points. The sum of the points of the first three players is equal with the sum of the points of the last nine players. What is the highest possible number of draws in the tournament.(For a victory the player gets $1$ point, for a loss $0$ points, in a draw both players get $0,5$ points.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
40
}
Given that the line l with a slope of α is perpendicular to the line x+2y-λ=0, find the value of tan(π + α).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
If 5% more is gained by selling an article for Rs. 600 than by selling it for a certain amount, the cost of the article is Rs. 400. What is the lower selling price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{580}