inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the largest possible value of \(\alpha\) such that a grid graph \(M_{n,n}\) may not have an \(\alpha\)-spanning tree.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \( D \subset \mathbb{R}^3 \) be the set \( D = \{(x, y, z) \mid x^2 + y^2 = 1~\text{and}~z = 0\} \). Let \( T :  \mathbb{R}^3 \to \mathbb{R}^3 \) be the linear transformation defined by \( T(x, y, z) = (2x − y + z, x − y + 10z, 3x − 7y + 2z) \). Find the area of \( T(D) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \sqrt{138}}
Evaluate the sum \( \sum_{i=1}^{2021} \frac{z_i^3}{z_i+1} \), where \( z_1, z_2, \ldots, z_{2021} \) are the roots of the polynomial \( z^{2021}+z-1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1347}
Suppose $x$ and $y$ are positive integers such that $x^y = y^x$. If the prime factorization of $x$ includes a prime $p$ with exponent $m$, determine the exponent of $p$ in the prime factorization of $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{m y}{x}}
Given that $y_1$ and $y_2$ are a fundamental set of solutions of the differential equation $(\mathrm{e}^x y')' + y' + y = 0$ for $x > 0$, and the Wronskian $W(y_1, y_2)(0) = 1$, find the value of $W(y_1, y_2)(-1)$. Use Abel's formula to determine your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^e}
A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section, the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Given two lines $l_{1}:(3+m)x+4y=5-3m$ and $l_{2}:2x+(5+m)y=8$. If line $l_{1}$ is parallel to line $l_{2}$, then the real number $m=$              .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
Find the spectrum of an operator $S: C([0,2]) \to C([0,2])$ defined by $(Sf)(t) = \int_0^{t+1} f(x) dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{0}}
Petya can draw only 4 things: a sun, a ball, a tomato, and a banana. Today he drew several things, including exactly 15 yellow items, 18 round items, and 13 edible items. What is the maximum number of balls he could have drawn?

Petya believes that all tomatoes are round and red, all balls are round and can be of any color, and all bananas are yellow and not round.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given that a three-digit positive integer (a_1 a_2 a_3) is said to be a convex number if it satisfies (a_1 < a_2 > a_3), determine the number of three-digit convex numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Given the set \( A = \{0, 1, 2, 3, 4, 5, 6, 7\} \), how many mappings \( f \) from \( A \) to \( A \) satisfy the following conditions?
1. For all \( i, j \in A \) with \( i \neq j \), \( f(i) \neq f(j) \).
2. For all \( i, j \in A \) with \( i + j = 7 \), \( f(i) + f(j) = 7 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{384}
Given the hyperbola \( C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) with \( a > 0 \) and \( b > 0 \), the eccentricity is \( \frac{\sqrt{17}}{3} \). Let \( F \) be the right focus, and points \( A \) and \( B \) lie on the right branch of the hyperbola. Let \( D \) be the point symmetric to \( A \) with respect to the origin \( O \), with \( D F \perp A B \). If \( \overrightarrow{A F} = \lambda \overrightarrow{F B} \), find \( \lambda \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Determine all values of the parameter \( a \) for which the system of equations \( \left\{\begin{array}{l} |x-6-y|+|x-6+y|=12 \\ (|x|-6)^{2}+(|y|-8)^{2}=a \end{array}\right. \) has exactly two solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the smallest integer \( n \) such that the function \( g(x) = \ln \ln \ldots \ln (|x| + 3) \), where the natural logarithm is applied \( n \) times, is surjective onto \((-\infty, \infty)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Consider a three-person game involving the following three types of fair six-sided dice.
- Dice of type $A$ have faces labelled 2, 2, 4, 4, 9, 9.
- Dice of type $B$ have faces labelled $1, 1, 6, 6, 8, 8$.
- Dice of type $C$ have faces labelled $3, 3, 5, 5, 7, 7$.
All three players simultaneously choose a die (more than one person can choose the same type of die, and the players don't know one another's choices) and roll it. Then the score of a player $P$ is the number of players whose roll is less than $P$'s roll (and hence is either 0, 1, or 2). Assuming all three players play optimally, what is the expected score of a particular player?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{9}}
Evaluate the limit: $$L=\lim_{ x\to \infty} x^2 \times \log \left(x \cot^{-1}x\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{3}}
Twelve balls are numbered by the numbers $1,2,3,\cdots,12$. Each ball is colored either red or green, so that the following two conditions are satisfied:

(i) If two balls marked by different numbers $a$ and $b$ are colored red and $a+b<13$, then the ball marked by the number $a+b$ is colored red, too.

(ii) If two balls marked by different numbers $a$ and $b$ are colored green and $a+b<13$, then the ball marked by the number $a+b$ is also colored green.

How many ways are there of coloring the balls?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6
}
Wendy takes Honors Biology at school, a small class with only fourteen students (including Wendy) who sit around a circular table. Wendy's friends Lucy, Starling, and Erin are also in that class. Last Monday, none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarn would intersect if possible. If all possible positions of the students around the table are equally likely, let $\frac{m}{n}$ be the probability that the yarns intersect, 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{4}
5. On a chessboard of size $2015 \times 2015$, dominoes are placed. It is known that in every row and every column there is a cell covered by a domino. What is the minimum number of dominoes required for this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1344}
29 boys and 15 girls attended a ball. Some boys danced with some girls (no more than once with each partner). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Mary typed a six-digit number, but the two  $1$ s she typed didn't show.  What appeared was  $2002$ .  How many different six-digit numbers could she have typed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Jack walked a total distance of 8 miles in 1 hour and 15 minutes. The route included an elevation gain of 600 feet, and he walked at varying speeds during this duration. Initially, he walked at a speed of 4 miles per hour for the first 30 minutes, then increased his speed to 6 miles per hour for the next 20 minutes. Finally, he slowed down his pace and walked the remaining distance at a rate of 3 miles per hour. What was his average speed for the entire walk, in miles per hour?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.4}
求和$\sum _ { n = 1 } ^ { 2 0 0 0 } ( - 1 ) ^ { n } \frac { n ^ { 2 } + n + 1 } { n ! }$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ $\frac { 2 0 0 1 } { 2 0 0 0 ! } - 1 .$}
A sample consists of 1, 3, 2, 2, a, b, c. It is known that the mode of this sample is 3, and the mean is 2. Calculate the variance of this sample. Express your answer as a fraction or decimal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{7}}
Let $g(x) = x^3 - x$, and $f(x)$ be a polynomial of degree at most 100. If $f(x)$ and $g(x)$ have no common factor, and \( \frac{d^{100}}{dx^{100}}\left[\frac{f(x)}{g(x)}\right] = \frac{p(x)}{q(x)} \), determine the degrees of the polynomials $p(x)$ and $q(x)$. Provide your answer as a pair of integers representing the degrees of $p(x)$ and $q(x)$. \( \boxed{(\text{degree of } p(x), \text{degree of } q(x))} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(202, 303)}
Let \( ABC \) be a triangle, and let \( D, E \), and \( F \) be the midpoints of sides \( BC, CA \), and \( AB \), respectively. Let the angle bisectors of \( \angle FDE \) and \( \angle FBD \) meet at \( P \). Given that \( \angle BAC = 37^\circ \) and \( \angle CBA = 85^\circ \), determine the degree measure of \( \angle BPD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61^\circ}
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that \( x f(x) - y f(y) = (x - y) f(x + y) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = ax + b}
Example - (2004 American Invitational Mathematics Examination) A circle of radius 1 is randomly placed inside a $15 \times 36$ rectangle $ABCD$. Find the probability that this circle does not intersect with the diagonal $AC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{375}{442}}
The domain of the function $y=\sin x$ is $[a,b]$, and its range is $\left[-1, \frac{1}{2}\right]$. Calculate the maximum value of $b-a$. Express your answer as a numerical value in terms of pi if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\pi}{3}}
Evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \left[ e^{\ln\left(\cos x \cdot \frac{d(\cos x)}{dx}\right)} \right]dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Given two matrices A and B, where S denotes the column space of A and T denotes the column space of B, find the matrix M whose column space is the sum of S and T. Assume both A and B have columns in \( \mathbb{R}^m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[A \quad B]}
[asy]
size(130);
pair A = (2, 2.4), C = (0, 0), B = (4.3, 0),
E = 0.7*A, F = 0.57*A + 0.43*B, D = (2.4, 0);
draw(A--B--C--cycle);
draw(E--D--F);
label("$A$", A, N);
label("$B$", B, E);
label("$C$", C, W);
label("$D$", D, S);
label("$E$", E, NW);
label("$F$", F, NE);
//Credit to MSTang for the diagram[/asy]
In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D$, $E$, and $F$ lie on sides $BC$, $AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$50^\circ$}
At the first site, higher-class equipment was used, and at the second site, first-class equipment was used. There was less higher-class equipment than first-class equipment. Initially, 40% of the equipment from the first site was transferred to the second site. Then, 20% of the equipment at the second site was transferred back to the first site, with half of the transferred equipment being first-class. After these transfers, the amount of higher-class equipment at the first site was 26 units more than at the second site, and the total amount of equipment at the second site increased by more than 5% compared to the original amount. Determine the total amount of first-class equipment.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Find the minimum number of prime factors that a negative square-free even integer must have to ensure that the class number of \( \mathbb{Q}(\sqrt{n}) \) is even.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Question 104, Given complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}$ satisfy the following conditions: $\left|\mathrm{z}_{1}\right|=2,\left|\mathrm{z}_{2}\right|=3,3 \mathrm{z}_{1}-2 \mathrm{z}_{2}=2-\mathrm{i}$, then $\mathrm{z}_{1} \mathrm{z}_{2}=$ $\qquad$ -
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{18}{5}+\frac{24}{5}i}
Calculate the length of the module $A = \mathbb{R}[X] / (X^3 - X^2 + X -1)$ as an $\mathbb{R}[X]$-module.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a 4-state Markov Chain with state space \(\{1,2,3,0\}\), represented by the transition matrix:
\[
\begin{pmatrix}
0.2 & 0.5 & 0.2 & 0.1 \\
0.2 & 0.6 & 0.1 & 0.1 \\
0.2 & 0.3 & 0.3 & 0.2 \\
0 & 0 & 0 & 0.1 \\
\end{pmatrix}
\]
calculate the probability that state 2 is never reached, meaning the process is absorbed into state 0 without transitioning to state 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{52}}
Find the remainder when \(7^{2023}\) is divided by 1001, given the following congruences:

\[7^{2023} \equiv 3 \pmod{7}\]
\[7^{2023} \equiv 4 \pmod{11}\]
\[7^{2023} \equiv 8 \pmod{13}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{983}
Compute the genus of the graph with vertex set \( V(G) = \{u_1, \cdots, u_7, v_1, \cdots, v_9, w_1, \cdots, w_5\} \) and edge set \( E(G) = \bigcup_{i=1}^{11} E_i \), where:

\[ E_1 = \{u_1u_j \mid 2 \leq j \leq 7\} \cup \{u_1v_j \mid j=1,3,5,6,8,9\} \cup \{u_1w_3, u_1w_4\} \; ; \]
\[ E_2 = \{u_2u_j \mid j=3,4,6,7\} \cup \{u_2v_j \mid j=4,5,6,7\} \cup \{u_2w_5\} \; ; \]
\[ E_3 = \{u_3u_j \mid j=4,5,7\} \cup \{u_3v_j \mid j=1,7,8,9\} \cup \{u_3w_2\} \; ; \]
\[ E_4 = \{u_4u_j \mid j=5,6\} \cup \{u_4v_j \mid j=1,2,3,4\} \cup \{u_4w_1\} \; ; \]
\[ E_5 = \{u_5u_j \mid j=6,7\} \cup \{u_5v_j \mid j=4,5,6,7\} \cup \{u_5w_5\} \; ; \]
\[ E_6 = \{u_6u_7\} \cup \{u_6v_j \mid j=1,7,8,9\} \cup \{u_6w_2\} \; ; \]
\[ E_7 = \{u_7v_j \mid j=1,2,3,4\} \cup \{u_7w_1\} \; ; \]
\[ E_8 = \{v_1v_5, v_1v_6\} \; ; \]
\[ E_9 = \{v_2v_7\} \; ; \]
\[ E_{10} = \{v_3v_7\} \; ; \]
\[ E_{11} = \{v_4v_8, v_4v_9\} \; . \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral \( \int_0^{2\pi} \sin(a\sin \theta - b \theta)\,d\theta \) where \(a \in \mathbb{R}\) and \(b \geq 0\) is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $C$ be an arbitrary point on the circle $k$ with diameter $AB$, different from points $A$ and $B$. Draw the perpendicular from point $C$ to the diameter $AB$, with the foot of the perpendicular being point $D$ on segment $AB$, and let the second intersection of this perpendicular with circle $k$ be point $E$. The circle with center $C$ and radius $CD$ intersects the circle $k$ at points $P$ and $Q$. Let $M$ be the intersection point of segments $CE$ and $PQ$. Determine the value of $\frac{P M}{P E}+\frac{Q M}{Q E}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find an approximation for the constant coefficient in the asymptotic expansion of $k\log(x)$ as $k$ tends to infinity, where $x$ is the solution to the equation $p_K(x)=\frac{1}{2}$. Provide your answer as a single value or expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{3}}
Find the limit \( \lim_{x\to e}\frac{x^{e^x}-e^{x^e}}{x-e} \) without using L'Hospital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{e^e + e - 1}}
A continuous-time Markov chain $X_t$ with state space $\{1,2,3\}$ has an infinitesimal generator matrix:
\[ A= \begin{bmatrix}-6 & 2 & 4\\2 &-5 &3 \\2 & 3& -5\end{bmatrix} \]
If the chain starts in state 2, what is the expected total time spent in state 2 before the chain first enters state 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{13}}
On a blackboard, two three-digit numbers are multiplied together. If the multiplication sign is replaced with a 0, the resulting seven-digit number is an integer multiple of the product of the two numbers. Determine the exact factor by which the seven-digit number is a multiple of the product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
Determine the number of \(2 \times 2\) matrices \(A\) with elements from the set \(\{-1, 0, 1\}\) such that \(A^2 = I\), where \(I\) is the identity matrix of order 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given the equations $x^{5} y^{2} + y^{5} z^{2} + z^{5} x^{2} = q^{3}$, $xy = yz = zx = r$, and $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{q}$, find $\sqrt{x} + \sqrt{y} + \sqrt{z}$ in terms of $q$ and $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt[4]{r}}
Determine the minimum number of crossings required for a non-alternating knot.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the integral \( \int_{|z|=1}\frac{z^2-z+1}{z-1}dz \) using Cauchy's Integral Formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi i}
An equilateral cone contains an equilateral cylinder. Determine the ratio of the volume of the cone to the volume of the cylinder.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{26 + 15\sqrt{3}}{18}}
Given a positive integer  $n>1$ . In the cells of an  $n\times n$  board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least two cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 - n}
Evaluate the following sum
$$\frac{1}{\log_2{\frac{1}{7}}}+\frac{1}{\log_3{\frac{1}{7}}}+\frac{1}{\log_4{\frac{1}{7}}}+\frac{1}{\log_5{\frac{1}{7}}}+\frac{1}{\log_6{\frac{1}{7}}}-\frac{1}{\log_7{\frac{1}{7}}}-\frac{1}{\log_8{\frac{1}{7}}}-\frac{1}{\log_9{\frac{1}{7}}}-\frac{1}{\log_{10}{\frac{1}{7}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1}
Given that triangle XYZ is a right triangle with two altitudes of lengths 6 and 18, determine the largest possible integer length for the third altitude.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In a study of product lifespan, consider the hypothesis test:

\[H_0: \mu = 2\]
\[H_1: \mu < 2\]

where \(\mu\) is the expected lifetime of the product, assumed to follow an exponential distribution with parameter \(\lambda\). The probability density function is:

\[f_X(x) = \begin{cases} \lambda e^{-\lambda x} & \text{for } x \geq 0 \\ 0 & \text{otherwise} \end{cases}\]

Given that \(E[X] = \mu = \frac{1}{\lambda}\), a sample of 10 products yields a sample mean \(\bar{X}\). Define \(c > 0\) such that if \(W \geq c\), the null hypothesis is accepted; otherwise, it is rejected. Determine the critical value \(c\) for a significance level of 95%.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.085}
Evaluate the limit \( \lim\limits_{x\rightarrow 0}\frac{\ln\left( 3+4x\right) -3{\ln\left( 3+x\right)} }{x^3} \) using the standard result \( \lim\limits_{x\rightarrow 0}\frac{\ln(1+x)}{x}=1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
Evaluate the integral \( I = \int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\pi \ln 2}
Given a sample of 7 elements with an inter-quartile range (IQR) of 4, find the minimum possible value of the variance of the sample.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{3}}
In a Poisson process with a rate of 0.1 observations per minute, calculate the expected time of the 2nd arrival, given that the 2nd arrival occurs within the first 2.5 minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.63}
There are 10 dozen mangoes in a box, with each box of mangoes having a varying price per dozen. The prices per dozen are as follows: $4, $5, $6, $7, $8, $9, $10, $11, $12, and $13. If there are 36 such boxes, what is the total cost of all the mangoes in all the boxes together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$3060}
Determine the spectrum of the weighted right shift operator \(T\) on \(\ell^{\infty}(\mathbb{N})\), defined by:
\[ T(a_1,a_2,a_3,...) = \left(0, \frac{a_1}{1}, \frac{a_2}{2}, \frac{a_3}{3}, ...\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0\}}
Evaluate the limit without using L'Hopital's rule: \[ \lim_{x\rightarrow b}\frac{b^{x^b}-b^{b^x}}{x^b-b^x} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b^{b^b} \ln b}
Given \(x \geq 1\) and \(y \geq 1\), for real number \(k\), it holds that 
\[
\frac{x^{2}}{1+x}+\frac{y^{2}}{1+y}+(x-1)(y-1) \geq k x y
\]
Find the maximum value of \(k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{13 - 5\sqrt{5}}{2}}
Determine the range of values of \( b \) for which the solution \( u(t) \) of the initial value problem \( u'(t) = Bu(t), \; t > 0 \) and \( u(0) = u_0 \), where \( u_0 \) is a given vector in \( \mathbb{R}^2 \) and \( B = \begin{bmatrix} -1 & 2 \\ -1 & b \end{bmatrix} \), satisfies \( |u(t)| \to 0 \) as \( t \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1)}
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). Given that the system of equations \(2x + y = 2019\) and \(y = |x-a| + |x-b| + |x-c|\) has exactly one solution, find the minimum possible value of \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1010}
Let \( {ABC} \) be a right triangle with \( m\left( \widehat{A}\right)  = {90}^{ \circ  } \) . Let \( {APQR} \) be a square
 with area 9 such that \( P \in  \left\lbrack  {AC}\right\rbrack  ,Q \in  \left\lbrack  {BC}\right\rbrack  ,R \in  \left\lbrack  {AB}\right\rbrack \) . Let \( {KLMN} \) be
 square with area 8 such that \( N,K \in  \left\lbrack  {BC}\right\rbrack  ,M \in  \left\lbrack  {AB}\right\rbrack \) ,and \( L \in  \left\lbrack  {AC}\right\rbrack \) .
 What is \( \left| {AB}\right|  + \left| {AC}\right| \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
12}