inclusionAI/Ring-lite-rl-data huggingface dataset:
Alexander is eating sliced olives for dinner. Olives are sliced into thirds so that each olive is composed of two indistinguishable “end“ pieces and one “middle“ piece. For each olive slice that Alexander puts on his plate, there is a $\frac23$ chance that it is an end piece and a $\frac13$ chance that it is a middle piece. After placing a randomly selected olive slice on his plate, he checks to see if he can rearrange the slices on his plate so that he has at least one whole olive, which consists of any middle piece and any two end pieces. He can successfully rearrange the olive slices on his plate into at least one whole olive after randomly selecting $n$ olive slices with probability of at least $0.8$. What is the minimum n for which this is true?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Initially, several different natural numbers are written on a board. The average magic of a set of numbers is defined as the ratio of their sum to their product. After erasing the smallest number from the board, the average magic of the remaining numbers increased threefold. Determine the original set of numbers on the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{4, 5, 7\}}
Given skew lines \( a \) and \( b \), with \( AB \) as the common perpendicular segment where \( A \) is on line \( a \) and \( B \) is on line \( b \), and \( AB = 2 \). The angle between the skew lines is \( 30^{\circ} \). If \( AP = 4 \) on line \( a \), find the distance from point \( P \) to line \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Find a general formula for the denominator of the coefficients in the Taylor expansion of \( \sqrt{1-x} \), given by \( \sum_{n=0}^{\infty} x^n (-1)^n \binom{1/2}{n} \). The denominators are of the form \( 2^m \). Determine \( m \) in terms of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - s_2(n)}
Find the supremum of the function \( \frac{|x|^{2/3}-|y|^{2/3}}{|x-y|^{2/3}} \) within the unit ball centered at the origin, where \( x \neq y \) and \( x, y \in \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find an integer $x \in \mathbb{Z}[i]$ such that $(1+2i)x \equiv 1 \pmod{3+3i}$. Provide your answer in the form of a Gaussian integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - i}
Determine the condition on the coefficients \(a\) and \(b\) such that the cubic curve \(bx^3 + y^3 = 3axy\) has a node at the origin with tangent directions along the coordinate axes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \neq 0}
If the representation of a positive number as a product of powers of distinct prime numbers contains no even powers other than $0$s, we will call the number singular. At most how many consequtive singular numbers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7
}
In a truth table with $n^2$ entries, an algorithm takes two rows, $a$ and $b$, and performs $s$ logical operations, resulting in Boolean values. If each iteration returns $c$ TRUE values, where $c > b - a + 1$, determine the number of additional operations required before selecting a new pair of rows.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c - (b - a + 1)}
Let $G$ be a non-abelian group of order $21$. Determine the number of automorphisms of $G$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
How many polynomials of the form \( z^3 + az^2 + bz + c \), where \( a \), \( b \), and \( c \) are integers, have all roots \( z \) satisfying either \( \left\lvert z \right\rvert = 10 \) or \( \left\lvert z \right\rvert = 15 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{412}
Every square of a $2020 \times 2020$ chess table is painted in red or white. For every two columns and two rows, at least two of the intersection squares satisfies that they are in the same column or row and they are painted in the same color. Find the least value of number of columns and rows that are completely painted in one color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2
}
Evaluate the expression \( \sum_{cyc} \frac{a^4}{(a-b)(a-c)} \) where \( a = -\sqrt{3} + \sqrt{5} + \sqrt{7} \), \( b = \sqrt{3} - \sqrt{5} + \sqrt{7} \), and \( c = \sqrt{3} + \sqrt{5} - \sqrt{7} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Determine the maximum number of vertices a planar graph and its complement can have while ensuring both graphs remain planar.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n} \, dS \) for the vector field \( \mathbf{F}(x,y,z) = xi + yj + zk \), where \( S \) is the plane surface bounded by the triangle with vertices at \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\). The unit normal \( \mathbf{n} \) to \( S \) has a nonnegative z-component. Use the vector representation \( \mathbf{r}(u,v) = (u+v)i + (u-v)j + (1-2u)k \) to perform the evaluation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the points where the curve $x^y = y^x$ intersects itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(e, e)}
An $8 \times 10$ rectangular board is to be completely covered without overlapping using a combination of $S$-shaped tiles and $T$-shaped tiles, where each tile consists of 4 unit squares. Both types of tiles can be rotated or flipped. Determine the minimum number of $T$-shaped tiles needed to cover the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the transition matrix for a Markov chain:
\[
P = \begin{bmatrix}
0 & 0.3 & 0 & 0.7 \\
0 & 0.8 & 0 & 0.2 \\
0.4 & 0 & 0.6 & 0 \\
0.4 & 0 & 0.6 & 0
\end{bmatrix}
\]
with states 0, 1, 2, 3 representing (win, lose), (lose, lose), (win, win), and (lose, win), respectively, determine the probability that Jones won the game on Sunday given that he won on Monday.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{5}}
Determine the genus of a coffee mug with two tubes going through the middle, where one tube goes through the other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Consider a regular pentagon circumscribed in a circle. Connect each vertex of this pentagon to every other non-adjacent vertex with a straight line segment to form a pentagram, which contains a smaller pentagon. What is the ratio of the area of the original (large) pentagon to the smaller one in terms of the golden ratio?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\phi^4}
Evaluate the limit \( L = \lim_{x \to 0} \frac{2 + \tan(e^x - \cos x) - e^x - \cosh x}{x(\sqrt{1+2x} - \sqrt[3]{1+3x})} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
How many distinct non-isomorphic graphs can be constructed using a set with 7 elements?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1044}
Calculate the limit \( \lim_{n\rightarrow\infty}\frac{\int_{0}^{1}f^n(x)\ln(x+2)dx}{\int_{0}^{1}f^n(x)dx} \) where \( f(x)=1-x^2+x^3 \) and \( f^n(x) \) is the function \( f(x) \) raised to the power of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
In $\triangle ADE$, $\measuredangle ADE=140^\circ$, points $B$ and $C$ lie on sides $AD$ and $AE$, respectively, and points $A,~B,~C,~D,~E$ are distinct.$^\star $  If lengths $AB,~BC,~CD,$ and $DE$ are all equal, then the measure of $\measuredangle EAD$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$10^{\circ}$}
How many distinct bracelets can be formed using 8 red beads, 6 blue beads, and 4 yellow beads, given that all beads must be used and two bracelets are considered identical if one can be obtained from the other through rotation or reflection? Beads of the same color are indistinguishable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255920}
Let $A$ be an $11\times 11$ diagonal matrix with the characteristic polynomial $(x-3)^4(x-4)(x-7)^4(x-9)^2$. Find the dimension of the space of $11\times 11$ matrices $V$ such that $AB=BA$ for all $B \in V$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Determine the probability density function for the y-coordinates of a random point uniformly sampled on the triangle with vertices A = (1,0,0), B = (0,1,0), and C = (0,0,1).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(1 - y)}
Given three different prime numbers \( p, q, r \) such that \( pqr = 1899\ldots962 \) (where the number contains \( n \) nines), define \( f(n) \) as the sum of the digits of the natural number \( n \). Calculate the value of \( f(p) + f(q) + f(r) - f(pqr) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Compute the measure of the set \( A = \{ (x, x): x \in [0, 1] \} \) under the product measure \( \mu_1 \times \mu_2 \), where \( \mu_1 \) is the Lebesgue measure and \( \mu_2 \) is the counting measure on \([0, 1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $x, y, z$ be positive real numbers such that $x^2 + xy + y^2 = \frac{25}{4}$, $y^2 + yz + z^2 = 36$, and $z^2 + zx + x^2 = \frac{169}{4}$. Find the value of $xy + yz + zx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$10 \sqrt{3}$}
Let \( f(x) = \lim_{n\rightarrow \infty}\frac{x^2+2(x+1)^{2n}}{(x+1)^{2n+1}+x^2+1} \), where \( n \in \mathbb{N} \). Define \( g(x) = \tan \left(\frac{1}{2}\arcsin\left(\frac{2f(x)}{1+(f(x))^2}\right)\right) \). Determine the number of points where \( g(x) \) is not differentiable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $f: \mathbb{C}^5 \rightarrow \mathbb{C}^7$ be a linear function such that $f(2i e_1 + e_3) = f(e_2)$ and $\mathbb{C}^7 = X \oplus \text{Im}(f)$. Determine the dimension of $X$. \( \text{Im}(f) \) denotes the image of $f$. \( e_1, e_2, e_3 \) are standard basis vectors in \( \mathbb{C}^5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the maximum number of positive roots that the equation \( \sum_{i=1}^N \frac{a_i}{x+b_i} = 0 \) can have, where each \( b_i \) is a positive constant and \( a_i \) can be either positive or negative.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N-1}
Consider a set $D \subset \mathbb{R}$ and a point $c \in D$. Determine the number of sequences $f: \mathbb{N} \rightarrow D$ such that $\lim\limits_{n \rightarrow \infty} f(n) = c$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Let $a$ be the integer determined by \(\frac{1}{1}+\frac{1}{2}+\ldots+\frac{1}{23}=\frac{a}{23!}\). Find the remainder when $a$ is divided by 13.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the improper integral: $$\int_{0}^{\infty}\frac{\cos(x)-\cos(2x)}{x}dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $ 3$ times their perimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6
}
Find the value of $a$ for which the integral \[ \int \limits^{\infty }_{0}\frac{dx}{a^{2}+(x-\frac{1}{x})^{2}} =\frac{\pi}{5050} \] holds, given that $a^{2}\geq0$. \(a\) is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2525}
Find the radius of convergence of the power series: $$\sum_{n=1}^\infty \frac{x^n}{\cos\left(\frac{n\pi}2\right)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the differential equation \((xy^3 + x^2y^7)\frac{dy}{dx} = 1\) with the initial condition \(y \left(\frac{1}{4}\right)=1\), find the value of \(\frac{dy}{dx}\) when \(y = -1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{16}{5}}
Given a 3x3 invertible matrix $A$ with $\operatorname{tr}(A) = \operatorname{tr}(A^2) = 0$ and $\det(A) = 1$, determine the characteristic polynomial of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda^3 - 1}
Evaluate the contour integral \( \oint_{|z|=4} \frac{\sin^2(z)}{(z-\frac{\pi}{6})^2(z+\frac{\pi}{6})} \, dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{3}i}
Given a discrete-time Markov chain defined by the equation $X(n+1) = X(n) - 1 + A(n)$, where $A(n)$ is a random variable taking values $0, 1, 2$ with probabilities $p_0$, $p_1$, and $p_2$, respectively, and $X(n) = n$ with $X(n) \geq 1$, find the relationship between $p_0$, $p_1$, and $p_2$ that ensures the stability of the Markov chain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p_1 + 2p_2 < 1}
Given a Doppler shift channel where the input signal is $x(t)$ with Fourier transform $X(f)$, and the output is $Y(f) = X(f-f_1)$, find the channel response $H(f) = \frac{Y(f)}{X(f)} = \frac{X(f-f_1)}{X(f)}$. Determine the inverse Fourier transform of $H(f)$, denoted as $h(t)$. Provide your answer as $h(t)$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{h(t) = e^{j2\pi f_1 t}}
Given the group $G=\mathbb{Z}_6 \times \mathbb{Z}_3$, the element $a=(4,1)$, the subgroup $H=\langle a\rangle$, and the quotient group $G_1=G/H$, determine $(k+\ell) \mod{5}$, where $k$ is the order of the class of $(3,1)$ in $G_1$ and $\ell$ is the order of the class of $(2,2)$ in $G_1$. What is the value of $(k+\ell) \mod{5}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A customer wants to purchase exactly four items from a store, with a total cost of 1000 rubles, where the cheapest item costs at least 99 rubles. The store offers two special deals:
1) Buy at least three items and get one item for free, with the free item's cost not exceeding the minimum price of the paid items.
2) Buy exactly one item costing at least $N$ rubles and receive a 20% discount on the next purchase.

Determine the maximum value of $N$ for which the second offer is more advantageous than the first.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
Find a subspace $W$ of the vector space $V$ with maximal dimension such that for all vectors $w$ in $W$, the quadratic form $$ q(x,y,z) = 2x^2 - \frac{1}{2}y^2 + 4z^2 $$ satisfies $q(w) \geq 0$. Determine the dimension of this subspace.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For a random walk in $\mathbb{Z}^2$ with increments having finite support and an expectation vector on the horizontal axis, determine whether the probability of first intersecting a vertical line at a given height scales with $t$ or $\sqrt{t}$ as $t$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{t}}
Three coins are initially showing "Heads" on a table. In each round, Danny turns one coin upside down: the left coin with a probability of \( \frac{1}{2} \), the middle coin with a probability of \( \frac{1}{3} \), and the right coin with a probability of \( \frac{1}{3} \). Let \( X_n \) represent the number of coins showing "Heads" after \( n \) rounds. Determine \( \lim_{n \to \infty} \Pr(X_n = 0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
Ben chooses random numbers from the set \( \{0, 1, \ldots, 9\} \) independently and with uniform probability, arranging them in a sequence in the order they are picked. What is the expected number of distinct numbers that appear between the first two occurrences of the number 0?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{2}}
Given that $X = h(Z, H)$, where $h$ is a measurable function, and $Z$ is a random variable independent of $H$, and knowing that $\mathbb{E}[X | H = s] = f(s)$, find the value of $\mathbb{E}[h(Z, s)]$. Express your answer in terms of $f(s)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(s)}
What is the minimal number of vertices required for a triangulation of the Klein bottle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In triangle \(ABC\), \(\angle A = 45^\circ\) and \(M\) is the midpoint of \(\overline{BC}\). The line \(\overline{AM}\) intersects the circumcircle of \(\triangle ABC\) at a second point \(D\), such that \(AM = 2MD\). Determine \(\cos \angle AOD\), where \(O\) is the circumcenter of \(\triangle ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{8}}
Determine the maximum number of inflection points that a cubic planar curve, given by the equation $F(x,y)=0$ where $\deg F=3$, can have. Consider the real roots of the determinant $F_{xx}F_y^2-2F_{xy}F_xF_y+F_{yy}F^2_x$ in your analysis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the radius of convergence for the Taylor series of the function \( f(z) = \frac{z}{\sin(z^2 + z)} \) at \( z = 0 \), given that \( f(0) = 1 \) to make \( f \) analytic at \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the expected value of $X_k 1_{\{|X_k| > n\}}$ for $x \geq e$, where $X_1, X_2, \ldots$ are independent and identically distributed random variables with $P(X_i > x) = \frac{e}{x \log(x)}$. Provide your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the number of abelian groups \( A \) of order \( p^4 \), where \( p \) is a prime number, such that there exists a subgroup \( N \) with \( A/N \cong \mathbb{Z}/p^3\mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Suppose that \( h: \mathbb{R} \to \mathbb{R} \) is a function with a fixed point at \( x^* \). If \( h'(x^*) = 0 \) and \( |h''(x)| < 2 \), determine the smallest value of \( n \) such that the sequence \( x_0, x_1, x_2, \ldots \) defined by \( x_n = h(x_{n-1}) \) satisfies \( |x_n - x^*| < 10^{-50} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A person has seven friends and invites a different subset of three friends to dinner every night for one week (seven days). How many ways can this be done so that all friends are included at least once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31169376000}
Given a set of unique numbers \(\{1, 2, 3, \ldots, n\}\), determine the number of binary search trees (BSTs) that can be formed with a height not exceeding a given height \(h\). For instance, if \(n = 4\) and \(h = 2\), calculate the number of BSTs with height at most 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Farmer James sends his favorite chickens, Hen Hao and PEAcock, to compete at the Fermi Estimation All Star Tournament (FEAST). The first problem at the FEAST requires the chickens to estimate the number of boarding students at Eggs-Eater Academy given the number of dorms $D$ and the average number of students per dorm $A$. Hen Hao rounds both $D$ and $A$ down to the nearest multiple of $10$ and multiplies them, getting an estimate of $1200$ students. PEAcock rounds both $D$ and $A$ up to the nearest multiple of $10$ and multiplies them, getting an estimate of $N$ students. What is the maximum possible value of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2600}
Given five distinct positive integers $x_1, x_2, x_3, x_4, x_5$ such that $x_1 + x_2 + x_3 + x_4 + x_5 = 100$, find the maximum value of the expression:

\[
\frac{(x_2 x_5 + 1)(x_3 x_5 + 1)(x_4 x_5 + 1)}{(x_2 - x_1)(x_3 - x_1)(x_4 - x_1)} \\
+ \frac{(x_1 x_5 + 1)(x_3 x_5 + 1)(x_4 x_5 + 1)}{(x_1 - x_2)(x_3 - x_2)(x_4 - x_2)} \\
+ \frac{(x_1 x_5 + 1)(x_2 x_5 + 1)(x_4 x_5 + 1)}{(x_1 - x_3)(x_2 - x_3)(x_4 - x_3)} \\
+ \frac{(x_1 x_5 + 1)(x_2 x_5 + 1)(x_3 x_5 + 1)}{(x_1 - x_4)(x_2 - x_4)(x_3 - x_4)}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{729000}
Let $G$ be a graph with $n$ vertices where each edge is chosen with probability $\frac{1}{2}$. A graph $G$ is separated if we can divide the vertices into two equal-sized sets with no edges between them. It is almost-separated if we can divide the vertices into two equal-sized sets with fewer than $10n$ edges between them. Determine the probability that $G$ is either separated or almost-separated.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Express the operation \( C_{ij} = B_{ik} B_{jl} A_{kl} \) in terms of standard matrix multiplication using Einstein's notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C = B A B^\top}
A tower is constructed with a square base and square horizontal cross-sections. Viewed from any direction perpendicular to a side, the tower has base $y = 0$ and profile lines $y = (x - 1)^2$ and $y = (x+1)^2$. Find the volume of the solid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Let \(m\) be a positive integer, and let \(T\) denote the set of all subsets of \(\{1, 2, \ldots, m\}\). A subset \(S\) of \(T\) is called \(\delta\)-good if for all distinct \(s_1, s_2 \in S\), the symmetric difference \(|\Delta(s_1, s_2)| \geq \delta m\). Find the largest possible integer \(s\) such that there exists an integer \(m\) and a \(\frac{1024}{2047}\)-good set of size \(s\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2048}
Determine the order of the pole of the function \(\frac{\mathrm{Log}(z)}{(z-1)^4}\) at \(z=1\), where \(\mathrm{Log}(z)\) is the principal branch of the logarithm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A black bishop and a white king are placed randomly on a $2000 \times 2000$ chessboard (in distinct squares). Let $p$ be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). If $p$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find the value of $m$. \( \text{(Proposed by Ahaan Rungta)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1333}
Find a solution to the differential equation \( ty'' - (t+1)y' + y = 0 \) that is not a constant multiple of the solution \( y(t) = e^t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t + 1}
Minimize the determinant of the matrix \( A = (VV^T + \lambda I)^{-1} \), where \( V \) is an \( m \times n \) matrix and \( \lambda \) is a very small positive scalar.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}