inclusionAI/Ring-lite-rl-data huggingface dataset:
Consider all the natural numbers from $000000$ to $999999$. A number is called "lucky" if the sum of its first three digits equals the sum of its last three digits. A number is called "medium" if the sum of all its digits equals $27$. Find the difference between the number of lucky numbers and medium numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the values of $a$ for which the improper integral $$\int_0^{\infty}\frac{\ln(1+x^2)}{x^a}dx$$ converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 3)}
Suppose $X_1, \ldots, X_{100}$ are independent and identically distributed as $N(\mu,1)$, but only the values of $X_i$ that are less than 0 are recorded. If 40 observations are less than 0, find the maximum likelihood estimator (MLE) of $\mu$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Phi^{-1}(0.6)}
What is the minimum number of direct connections required to ensure that any group of 8 or fewer workstations out of 12 can access different servers simultaneously, given that there are 7 servers available?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Determine the automorphism group of the group \(\mathbb{Z}_2 \oplus \mathbb{Z}_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_3}
Find a single element in the ring \( \mathbb{Q}[x] \) that can generate both \( 2x^2 \) and \( 2x^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2}
Let $S = \{m + \frac{1}{n} \mid m, n \in \mathbb{N}\}$. Determine the derived set (set of cluster points) of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{N}}
Given positive real numbers $a, b, c$ such that $a^2 + b^2 + c^2 = 3$, determine the minimum value of the expression \[ P = \frac{1}{(a-b)^2} + \frac{1}{(b-c)^2} + \frac{1}{(c-a)^2}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{4}}
Determine the number of possible derived trees for the expression $1-2-(3-4)*5*6$ using the following grammar:

$$\begin{align}V_n&=\{expr,term,factor,number\}\\ V_t&= \{(,),-,*,0...9\}\\ P&=\left \{ \begin{aligned} expr&\to expr-expr\;\mid\;term\\ term&\to term*factor\;\mid\;factor\\ factor&\to number \;\mid\; (expr) \\ number&\to 0|1|2|3|4|5|6|7|8|9 \end{aligned} \right \}\\ S&=expr \end{align} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Compute the limit: \( \lim_{x \to +\infty} f(x) \), where \( f(x) = \frac{1}{\Gamma(x)} \int_x^{+\infty} t^{x-1} e^{-t} \, dt \) for all \( x \in \Bbb R_+^* \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Consider the matrix \( A = I_9 - 2u^Tu \) with \( u = \frac{1}{3}[1, 1, 1, 1, 1, 1, 1, 1, 1] \). If \( p \) and \( q \) are two distinct eigenvalues of \( A \), find \( |p-q| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Simplify the expression: $L_1^2L_1^*L_1 \cap L_1^*L_1^2\Sigma^*$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{L_1^3 L_1^*}
Determine the base $d$ of a numbering system that allows writing the largest number of numbers using a given number of digits, where digits in different positions are considered different. Consider only sets of $m$ digits, where $m$ is divisible by both bases being compared.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find a maximal ideal in \( \mathbb{Z}[x] \) that properly contains the ideal \((x-1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, x - 1)}
Evaluate the limit: $$L=\lim_{x\to 0}\frac{\sin(\ln(x+1))-\ln(\sin(x)+1)}{\sin^4\left(\frac{x}{2}\right)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Evaluate the limit: $$\lim_{n\rightarrow\infty} \frac{(n+1)^{2n^2+2n+1}}{(n+2)^{n^2+2n+1}n^{n^2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e}}
Calculate the line integral \( \int_r (xy^2z^2-1, x^2yz^2, x^2y^2z)\cdot ds \) where \( r = r_1 \cup r_2 \). The parametric representation of \( r_1 \) is \( \sigma_1 (t) = (\sin t, \cos t, t-\frac{\pi}{2}), t \in [0,\frac{\pi}{2}] \) and the parametric representation of \( r_2 \) is \( \sigma_2 (t) = (\cos^3 \phi, \sin^3 \phi, 0), \phi \in [0,\pi] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the line integral \( \int_C \vec{F}(x,y) \cdot \,d\vec{r} \) where \( \vec{F}(x,y) = \begin{bmatrix} 5y \\ 10x \end{bmatrix} \) and \( C \) is the quarter-circle arc centered at \((0,2)\) going from the origin to \((2,2)\). Use polar coordinates to solve the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20 + 5\pi}
Evaluate the integral \( \int_0^{2\pi} \cos(\cos(z))\cosh(\sin(z)) \, dz \) using a corollary of Cauchy's Integral Formula or Gauss's Mean Value Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Determine the Galois group of the polynomial \((x^2-1)(x^2-2)\ldots(x^2-p+1)\) over the finite field \(\mathbb{Z}_p\), where \(p\) is an odd prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/2\mathbb{Z}}
Evaluate $a+2b+3c$ given that \( \lim_{h \to 0}\frac { \int _{ 0 }^{ h }{ f(x)dx-h(af(0)+bf(h/3)+cf(h)) } }{ { h }^{ 4 } } \) is a finite non-zero number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{4}}
$A, B, C, D$ all lie on a circle with $\overline{AB} = \overline{BC} = \overline{CD}$. If the distance between any two of these points is a positive integer, what is the smallest possible perimeter of quadrilateral $ABCD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
In a queuing system, there are two facilities, A and B, each with a single server. The service times are exponentially distributed with mean service times of 1 minute at facility A and 4 minutes at facility B. Sixty customers per hour arrive at the facilities, and each customer independently chooses a facility to minimize their expected time in the system. What is the long-run probability that a random customer chooses facility A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{8}}
A square with sides initially measuring $4m$ increases at a constant rate of $3m$ per second. One corner of the square moves along a line $L$ at a speed of $2m/s$ for $5$ seconds. During this time, the square makes one complete revolution about $L$, revolving at a constant rate and remaining always perpendicular to $L$. Calculate the volume of the solid generated by this motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1510}
Find the exact area under the curve defined by the equation \([x+y] + [x-y] = 5\) with the constraints \(x \geq y\), \(x \geq 0\), and \(y \geq 0\), where \([\cdot]\) denotes the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
The set of real numbers $ x$ for which
\[ \frac{1}{x-2009}+\frac{1}{x-2010}+\frac{1}{x-2011}\ge 1\]
is the union of intervals of the form $ a<x\le b$. What is the sum of the lengths of these intervals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3

}
Factor the polynomial $2t^4 + t^2 + 2$ into prime polynomials over the finite field $\mathbb{F}_3$. Provide the factorization.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(t^2 + 1)^2}
Assume the Generalized Continuum Hypothesis (GCH) holds and \(\kappa\) is a regular cardinal. Determine the number of non-isomorphic partial orders of size less than \(\kappa\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\kappa}
Given positive integers \( x, y, z \) and real numbers \( a, b, c, d \), which satisfy \( x \leq y \leq z \), \( x^a = y^b = z^c = 70^d \), and \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d} \), determine whether \( x + y \) is greater than, less than, or equal to \( z \). Fill in the blank with ">", "<" or "=".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{=}
Find the length of the graph of the function \( y = f(x) \) over the interval \([0, 3]\), where \( f(0) = 0 \) and \( f'(x) = \frac{\ln(x+2)}{x^2+1} \) for \( x > -2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.32}
Evaluate the limit \( \lim_{n \to \infty} \int_0^1 \frac{\sqrt{n}}{1 + n \ln(1 + x^2)} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Find the form of matrices that commute with all matrices of the form $E_{ij}$, where $E_{ij}$ is the matrix with a $1$ in the $(i,j)$ entry and $0$s elsewhere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda I}
Given an AR(1) model with 144 observations, where \( y_t = \phi y_{t-1} + \epsilon_t \) and \( \epsilon_t \) is white noise with mean zero and variance \( \sigma^2 \), find an estimate for \( \phi \). The initial and final observations are \( y_1 = -1.7 \) and \( y_{144} = -2.1 \). Additionally, \( \sum_{t=2}^{144} y_t y_{t-1} = -128.6 \) and \( \sum_{t=1}^{144} y_t^2 = 246.4 \). Use the formula for the first sample autocorrelation: \[ \hat{\phi} = \frac{\sum_{t=2}^{144} (y_t - \overline{y})(y_{t-1} - \overline{y})}{\sum_{t=1}^{144} (y_{t-1} - \overline{y})^2} \] to find \( \hat{\phi} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-0.53}
Determine the number of equivalence classes of propositional terms, using a set of 5 propositional variables, that evaluate to TRUE under exactly 13 distinct valuations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dbinom{32}{13}}
For how many angles $x$, in radians, satisfying $0\le x<2\pi$ do we have $\sin(14x)=\cos(68x)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
134
}
Provide an example of a sequence of numbers \( v_n \) such that its mean tends to a limit, but \( v_n \) itself does not converge to a limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^n}
Let $a, b, c$ be distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$ have a common real root, and the equations $x^2 + x + a = 0$ and $x^2 + cx + b = 0$ also have a common real root. Compute the sum $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
You toss a weighted coin 100 times. The probability of getting heads, $p$, is either 0.1 or 0.01, but you don't know which. Determine the decision boundary for the number of heads obtained that would lead to the maximum likelihood estimate (MLE) of $p$ being 0.1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the cardinality of the product \( \prod_{0<\alpha<\omega_1}\alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_1}}
Suppose \( P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0 \) where \( a_0, a_1, \ldots, a_{n-1} \) are real numbers for \( n \geq 1 \) (monic \( n \)th-degree polynomial with real coefficients). If the inequality
\[ 3(P(x)+P(y)) \geq P(x+y) \] holds for all real numbers \( x,y \), determine the minimum possible value of \( P(2024) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1536216}
Four glasses are arranged in a row: the first and third contain orange juice, while the second and fourth are empty. Each time, you can take a full glass and pour all its contents into one of the two empty glasses. What is the expected number of pourings required for the first and third glasses to be empty, while the second and fourth glasses are full?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Raymond wants to travel in a car with $3$ other (distinguishable) people. The car has $5$ seats: a driver’s seat, a passenger seat, and a row of $3$ seats behind them. If Raymond’s cello must be in a seat next to him, and he can’t drive, but every other person can, how many ways can everyone sit in the car?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Determine the order of the group $G = \{A \in M_2(\mathbb{Z}_p): \mathrm{det}A = \pm 1 \}$, where $p > 2$ is a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2p(p^2 - 1)}
Evaluate the sum \( a_k = \sum_{j=\lceil k/2\rceil}^k {j \choose k-j} \) in closed form using a generating function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_{k+1}}
Determine the range of values for $b$ such that for all solutions $y(x)$ of the differential equation $y'' + y' + by = 0$, the limit \( \lim_{x \to \infty} \frac{y(x)}{e^x} = 0 \) is satisfied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, \infty)}
Find the least positive integer \( k \) such that \( 10^k \equiv 1 \pmod{2003} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1001}
Evaluate the integral \( \int_\Omega xye^{x^2-y^2}dxdy \) over the domain \( \Omega = \{(x,y)\mid 1\leq x^2-y^2\leq 9, 0\leq x \leq 4, y\geq 0\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2e^9 - 4e}
Mary chose an even 4-digit number \(n\). She wrote down all the divisors of \(n\) in increasing order from left to right: \(1, 2, \ldots, \frac{n}{2}, n\). At some moment Mary wrote 323 as a divisor of \(n\). What is the smallest possible value of the next divisor written to the right of 323?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{340
}
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 \(S=\{w, w^2\}\), and \(w\) is a complex cube root of unity with \(w \neq 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( G = (V, E) \) be a simple connected graph with exactly one cycle. The graph has four vertices of degree 2, five vertices of degree 3, three vertices of degree 4, and one vertex of degree 5. If the maximum degree \( \Delta = 5 \), determine the number of pendant vertices in \( G \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Determine the conditions on the complex numbers $a$, $b$, and $c$ such that the matrix \(\begin{bmatrix}2&0&0\\a&2&0\\b&c&-1\end{bmatrix}\) has a minimal polynomial of \((x-2)(x+1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a = 0}
Evaluate the double integral \( \iint_L (xy - x^2) \, dx - (x+y)^2 \, dy \) where \( L \) is the triangle with vertices \( O(0,0) \), \( A(3,1) \), and \( B(2,2) \). Solve this without using Green's Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-14}
Eric rolls a ten-sided die (with sides labeled $1$ through $10$) repeatedly until it lands on $3, 5$, or $7$. Conditional on all of Eric’s rolls being odd, the expected number of rolls can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
9}
Determine the values of $x \in \mathbb{R}$ for which the series \( \sum_{n=1}^\infty \frac{n^{nx}}{n!} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1)}
Determine the domain of the function \( f(x) = \int_{1}^{x} \frac{e^t}{t} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \infty)}
Find the cumulative distribution function (CDF) of the absolute value of the difference between two independent random variables, each uniformly distributed on the interval [0,1].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2z - z^2}
Given that triangle $PQR$ is a right triangle, each side being the diameter of a semicircle, the area of the semicircle on $\overline{PQ}$ is $18\pi$, and the arc of the semicircle on $\overline{PR}$ has length $10\pi$, calculate the radius of the semicircle on $\overline{QR}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{136}}
A Martian writes down the equation \(x^2 - 19x + 76 = 0\). When asked to write down the difference between the larger and smaller root, the Martian writes 9. How many fingers do Martians have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Evaluate the triple integral \( \iiint_D |z| \, dx \, dy \, dz \) over the domain \( D = \{(x,y,z) \in \mathbb{R}^3 : x^2+y^2-16 \leq z \leq 4 - \sqrt{x^2+y^2} \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{704\pi}
Evaluate the integral \( \int_1^4 \frac{x^2}{\sqrt{1-x^3}}\, dx. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2i\sqrt{7}}
Given a random sample \(X_1\) and \(X_2\) of size \(n=2\) from an exponential distribution with parameter \(\theta\), find the efficiency of the estimator \(2Y_1\) relative to \(\bar{X}\), where \(Y_1\) is the first order statistic. Both \(2Y_1\) and \(\bar{X}\) are unbiased estimators of \(\theta\). Calculate the variance of \(2Y_1\) and use it to determine the relative efficiency \(\frac{V(\bar{X})}{V(2Y_1)}\), given that \(V(\bar{X})=\frac{\theta^2}{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
In square ABCD, M and N are the midpoints of BC and CD respectively. Find sin(theta), where theta is the angle between AM and AN. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{5}}
For how many pairs of integers $(m, n)$ with $0 < m \le n \le 50$ do there exist precisely four triples of integers $(x, y, z)$ satisfying the following system?
\[\begin{cases} x^2 + y+ z = m \\ x + y^2 + z = n\end{cases}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
188
}
Evaluate $ \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}+\sqrt[3]{x})}\ dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$1-6\ln(1+\sqrt[6]{e})+6\ln 2$}
Determine the number of steps required to reduce \( n = 2016^{155} \) to 1 using the algorithm where, at each step, the largest divisor of the current number (other than the number itself) is subtracted from it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
Determine the order of the entire function \(f(z) = \cos\sqrt{z}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given an exponential distribution with the PDF of the minimum of a sample as \( f_{\min}(x) = n\cdot\lambda\cdot(e^{-\lambda x})^n \), calculate the probability \( P\left(\frac{\lambda}{n} \le \frac{\ln(3)}{\min(x)}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - 3^{-n^2}}
Given the function \( z(x,y) \) defined implicitly by the equation \( F(x,y,z) = 5x + 2y + 5z + 5\cos(5z) + 2 = 0 \), find the mixed partial derivative \( \frac{\partial^2 z}{\partial x \partial y} \) at the point \( \left(\frac{\pi}{5}, \frac{3}{2}, \frac{\pi}{5}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-10}
Given three independent events $E_1, E_2, E_3$, the probability that only $E_1$ occurs is $\alpha$, only $E_2$ occurs is $\beta$, and only $E_3$ occurs is $\gamma$. The probability $p$ that none of the events occurs satisfies the equations $(\alpha-2\beta)p=\alpha\beta$ and $(\beta-3\gamma)p=2\beta\gamma$. If the probability of occurrence of $E_1$ is $X$ and the probability of occurrence of $E_3$ is $Y$, find the ratio $\frac{X}{Y}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Minimize the expression \( \{tr^Trt^T\}_{ii} \) subject to the constraint \( r^Tt = 1 \), where \( r, t \in \mathbb{R}^{n \times 1} \). Assume that the diagonal elements of \( t \) are ordered in nondecreasing order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
You flip a fair coin more than 11 times. Calculate the probability that there is at least one occurrence of a sequence of 6 consecutive heads immediately followed by 6 consecutive tails.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4096}}
Find the limit of the expression as $x$ approaches $0$ from the right: $$\lim_{x \to 0^+} \frac{\sum_{n=1}^\infty \frac{\sin(nx)}{n^2}}{\sin(x)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the minimum value of the integral \( \int_{0}^{\infty} |x^3 - a - bx - cx^2|^2 e^{-x} \, dx \) with respect to the real numbers \(a, b, c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Determine the cardinality of the set $X$, where $X$ is the set of all relations $R$ over $\mathbb{N}$ such that the transitive closure $R^{*}$ equals $\mathbb{N} \times \mathbb{N}$. What is the cardinality of $X$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Let $a$ be a positive real number. In the coordinate plane, denote by $S$ the area of the region bounded by the curve $y = \sin x$ for $0 \leq x \leq \pi$ and the $x$-axis. Denote by $T$ the area of the region bounded by the curves $y = \sin x$ and $y = a \cos x$ for $0 \leq x \leq \frac{\pi}{2}$ and the $x$-axis. Find the value of $a$ such that the ratio $S:T = 3:1$. \( \boxed{a} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Given distinct prime numbers $p$ and $q$, find the number of positive integer solutions to the equation:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{pq} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the greatest common divisor (GCD) of expressions of the form \(p^6 - 7p^2 + 6\), where \(p\) is a prime number greater than or equal to 11.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{672}
Given the equations $x + y + z = 0$, $x^2 + y^2 + z^2 = 1$, and $x^4 + y^4 + z^4 = 2$, find the value of $x^5 + y^5 + z^5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the values of \( p \) for which the double series \( \sum_{i, j=1}^\infty a_{i+j}^p \) converges, given that \( (a_i)_{i=1}^\infty \) is a sequence of positive numbers such that \( \sum_{i=1}^\infty a_i < \infty \). Specifically, investigate the case when \( p = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p \geq 2}
In a grid representing numbers in base 8, select numbers such that there is exactly one number with each possible last digit (0-7) and one with each possible second digit (0-7). Note that numbers ending with 0 represent a value 8 greater than their residue class. Calculate the sum of these selected numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}