inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $S(n)$ be the sum of digits for any positive integer $n$ (in decimal notation). Let $N=\sum_{k=10^{2003}}^{10^{2004}-1} S(k)$. Determine $S(N)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Given a Poisson distribution with a double mode at $X=1$ and $X=2$, calculate the probability $P(X=0)$. Express your answer as a decimal or a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1353}
Given $a > 0$, let proposition $p$: the function $y = a^{x}$ is monotonically increasing on $\mathbb{R}$; proposition $q$: the inequality $ax^{2} - ax + 1 > 0$ holds for $\forall x \in \mathbb{R}$. If $p$ and $q$ are false, and $p$ or $q$ is true, find the range of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,1] \cup [4,+\infty)}
In a triangular prism $ABC-A_{1}B_{1}C_{1}$, a line is drawn through the midpoint of any two edges. Among these lines, the number of lines that are parallel to the plane $ABB_{1}A_{1}$ is $\_\_\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the value of \( r \) such that for a simple random walk defined by independent random variables \( Z_1, Z_2, \ldots, Z_n \), where each \( Z_j \) is either 1 or -1 with equal probability, the probability that the sum \( S_n = \sum_{j=1}^n Z_j \) satisfies \(|S_n| \leq r\) is 0.5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.6745 \sqrt{n}}
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $a_2a_4a_6a_8=120$ and $\frac{1}{a_4a_6a_8} + \frac{1}{a_2a_6a_8} + \frac{1}{a_2a_4a_8} + \frac{1}{a_2a_4a_6} = \frac{7}{60}$, then the value of $S_9$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{63}{2}}
Determine the number of homomorphisms from the dihedral group $D_6$ to the dihedral group $D_5$. A homomorphism is a function $\varphi: D_6 \rightarrow D_5$ such that for all $x, y \in D_6$, $\varphi(xy) = \varphi(x)\varphi(y)$. Provide the total number of such homomorphisms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
3. There are 195 different cards with numbers $1, 5, 7, 5^{2}, 7^{2}, \ldots, 5^{97}, 7^{97}$ (each card has exactly one number, and each number appears exactly once). In how many ways can 3 cards be chosen so that the product of the numbers on the chosen cards is a square of an integer divisible by 35?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{223488}
For a list of n elements where n >= 3, determine the maximum number of inversions required to order the list using the inversion technique, where inverting the initial segments of the list is allowed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 3}
Determine the minimum value of \( n \) for which the function \( y(x) = x\sin x \) is a solution to an \( n^{th} \) order linear non-homogeneous differential equation with constant coefficients.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the real value(s) of $a (a \ne -1)$ for which the limit
$$ \lim_{ n \to \infty} \frac{ 1^a+2^a\cdots+n^a}{(n+1)^{a-1}[ (na+1)+(na+2) \cdots+(na+n)]} = \frac{1}{60}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Cagney can frost a cupcake every 15 seconds, while Lacey can frost every 40 seconds. They take a 10-second break after every 10 cupcakes. Calculate the number of cupcakes that they can frost together in 10 minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Evaluate the limit as \( x \to 0 \):
\[
\lim_{x \to 0} \frac{(\sin 2x - 2x\cos x)(\tan 6x + \tan\left(\frac{\pi}{3} - 2x\right) - \tan\left(\frac{\pi}{3} + 4x\right))}{x\sin x \tan x\sin 2x}.
\]
Use the facts that \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) and \( \lim_{x \to 0} \frac{\tan x}{x} = 1 \) to evaluate the limit without using L'Hôpital's rule or Taylor series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find a function \( f:[0,\infty) \rightarrow [0,\infty) \) that is convex and increasing, such that \( (1/z)f(z) \) is non-convex in \((0,\infty)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(z) = z^{3/2}}
Calculate the limit of the function involving the Gamma function for a given integer value of \( n \):

\[ f(n) = \lim_{c \rightarrow 0} \frac{\Gamma(-n + c) + \Gamma(-n - c)}{2} \]

Find \( f(n) \) for \( n = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\gamma}
There are 20 piles of stones, each with 2006 stones. An operation involves taking one stone from each of any 19 piles and placing them into the remaining pile. After fewer than 20 operations, one pile has 1990 stones, and another pile has between 2080 and 2100 stones. How many stones are in this pile?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2090}
A metal bar at a temperature of $20^{\circ} \mathrm{C}$ is placed in water at a temperature of $100^{\circ} \mathrm{C}$. After thermal equilibrium is established, the temperature becomes $80^{\circ} \mathrm{C}$. Then, without removing the first bar, another identical metal bar also at $20^{\circ} \mathrm{C}$ is placed in the water. What will be the temperature of the water after thermal equilibrium is established?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68^{\circ} \mathrm{C}}
Using the numbers $2, 3, \cdots, 2019$ to form 1009 fractions, with each number appearing only once in either the numerator or the denominator, select the largest fraction among these 1009 fractions. Find the minimum value of this largest fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1010}{2019}}
Let $X$ and $Y$ be independent random variables, each uniformly distributed on $[0,1]$. Define $Z = \max\{X, Y\}$. Calculate the conditional expectation $\mathbb{E}[X|Z]$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}Z}
Evaluate the sum of the shifted sinc function defined as follows: $$g(y) = \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the area of the portion of the circle defined by \(x^2 - 10x + y^2 = 9\) that lies above the \(x\)-axis and to the left of the line \(y = x-5\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["4.25\\pi"]}
The average age of a certain number of members of a committee is the same as it was 4 years ago, because an old member has been replaced by a young member. The new member is 40 years younger. How many members are in the committee?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Evaluate the triple integral \( \iiint_\Omega{\sqrt{x^2+y^2+z^2}dxdydz} \) where \( \Omega=\{x^2+y^2+z^2\le y\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{10}}
Determine the condition under which the $2\times 2$ stochastic matrix \(A\) given by \(A=\left( \begin{array}{cc} 1-a & a \\ b & 1-b\\ \end{array}\right)\) can be expressed as the square of a transition probability matrix \(P\) for a Discrete-Time Markov Chain, i.e., find the condition such that \(P^{2}=A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a + b \leq 1}
In a 4x4 magic square where all rows, columns, and the two main diagonals must sum to the same given value, what is the minimum number of squares that need to be filled to ensure a unique solution for the entire square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Paco had 39 sweet cookies and some salty cookies. He ate 23 salty cookies and 32 sweet cookies. He ate 9 more sweet cookies than salty cookies. How many salty cookies did Paco have initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
1. Given 50 quadratic trinomials. The graphs of these trinomials have their vertices on the $x$-axis. No three graphs intersect at the same point. What is the minimum number of intersection points they can form?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
Let $X_i$, $i = 1,\ldots,n$ be i.i.d. Poisson random variables with parameter $\lambda = 2$. Determine the minimum value of $n$ such that $P(1.9 \leq \overline{X} \leq 2.1) \geq 0.90$, where $\overline{X}$ is the sample mean.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{542}
Evaluate the limit: $$\lim_{x \rightarrow 0^{+}} \frac{1}{9} \left(\frac{1}{\arctan x} - \frac{1}{x}\right).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of zero divisors in the ring \( \mathbb{Z}_5[x]/(x^3-2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Find the splitting field of the polynomial \( f(x) = x^3 + x^2 + 1 \) over the field \( \mathbb{Z}_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_8}
It is known that the occurrence of a certain symbol in the text is between \(10.5\%\) and \(11\%\) of the length of the text (where the length of the text is understood as the total number of characters in the text). Find the minimum possible length of the text.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
设无穷等比数列 $\left\{a_n\right\}$ 的公比 $q$ 满足 $0<|q|<$ 1 。若 $\left\{a_n\right\}$ 的各项和等于 $\left\{a_n\right\}$ 各项的平方和，将公比 $q$ 表示为 $\frac{m}{n}$ 的形式（其中 $m, n$ 为整数且 $\operatorname{gcd}(m, n)=1$ ), 设 $a_2$ 的取值范围是两个区间，将这两个区间的端点分别表示为 $\frac{a_1}{b_1}$ （左区间左端点）、 $\frac{a_2}{b_2}$ （左区间右端点）、 $\frac{a_3}{b_3}$ （右区间左端点）、 $\frac{a_4}{b_4}$ （右区间右端点），且所有分数都是既约分数（分子分母只有公因数1）。请提供 $a_1+b_1+a_2+b_2+a_3+b_3+a_4+b_4$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. The area of $\triangle ABC$ is $S$, and $(a^{2}+b^{2})\tan C=8S$, also $\sin A\cos B=2\cos A\sin B$. Find the value of $\cos A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{30}}{15}}
If a sequence $\{a_n\}$ satisfies: $a_1 < a_2 > a_3 < a_4 > \ldots > a_{2n-1} < a_{2n} > a_{2n+1} \ldots$, then the sequence $\{a_n\}$ is called a "sine sequence". Now, arrange the numbers 1, 2, 3, 4, 5 into a "sine sequence". Let the total number of such arrangements be $a$. The coefficient of the $x^2$ term in the expansion of the binomial $(\sqrt{x} - \frac{a}{\sqrt{x}})^6$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-96}
Find the limit: \[ \lim_{x\to\infty} \frac{\displaystyle \sum\limits_{i = 0}^\infty \frac{x^{in}}{(in)!}}{\displaystyle\sum\limits_{j = 0}^\infty \frac{x^{jm}}{(jm)!}} \] for natural numbers \(n\) and \(m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{m}{n}}
Sandy grew 8 carrots and some turnips. Mary grew 6 carrots. They grew 14 carrots in all. How many turnips did Sandy grow?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the number of groups of non-negative integers \(b_1, b_2, \ldots, b_m\) such that the bitwise XOR of all the elements in the group equals a given integer \(k\), where \(0 \leq b_i \leq 2^{20}-1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{20(m-1)}}
I have two random variables $X$ and $Y$ with joint PDF $$ f_{x,y}(x,y) = e^{-y}1_{0<x<y} ; 0<x<y<\infty $$ and 0 otherwise. Given $Z=X+Y$, I need to find the PDF of $Z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-\frac{z}{2}}-e^{-z}}
A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$.  Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$.  Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges.  What is the minimal number of cycles possible in $G$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1001}
There is a solution of table salt in a flask. From the flask, $\frac{1}{5}$ of the solution is poured into a test tube and evaporated until the salt concentration in the test tube doubles. After that, the evaporated solution is poured back into the flask. As a result, the salt concentration in the flask increases by $3\%$. Determine the initial percentage concentration of salt.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
The Venusian prophet Zabruberson sent his pupils a $10000$-letter word, with each letter being either $A$ or $E$. This is known as the Zabrubic word. The pupils consider that for $1 \leq k \leq 10000$, each word comprised of $k$ consecutive letters of the Zabrubic word is a prophetic word of length $k$. It is known that there are at most $7$ prophetic words of length $3$. Find the maximum number of prophetic words of length $10$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
Determine the smallest positive integer \( m \) such that the symmetric group \( S_n \) can be embedded into the general linear group \( GL_m(F) \), where \( F \) is a field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - 1}
Find the Hausdorff dimension of the set \( F \), where \( F \) consists of numbers \( x \in [0,1] \) with base 3 expansions \( 0.a_1a_2\ldots \) such that there exists an integer \( k \) for which \( a_i \neq 1 \) for all \( i \geq k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\log 2}{\log 3}}
Let \(ABCD\) be a trapezoid such that \(AB \parallel CD, \angle BAC=25^{\circ}, \angle ABC=125^{\circ}\), and \(AB+AD=CD\). Compute \(\angle ADC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70^{\circ}}
A $3 \times 6$ grid is filled with the numbers in the list $\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9\}$ according to the following rules:  

1. Both the first three columns and the last three columns contain the integers 1 through 9.  
2. No numbers appear more than once in a given row.  

Let $N$ be the number of ways to fill the grid and let $k$ be the largest positive integer such that $2^k$ divides $N$. What is $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given the functional equation \( f(f(x))[1+f(x)] = -f(x) \), find the value of \( f(3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{3}{4}}
A set of 8 problems was prepared for an examination. Each student was given 3 of them. No two students received more than one common problem. What is the largest possible number of students?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right, where each integer is squared. For example, $b_3 = 149$ (since $1^2 = 1$, $2^2 = 4$, $3^2 = 9$), and $b_5 = 1491625$. For $1 \le k \le 100$, determine how many $b_k$ are divisible by 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given that $p$, $q$, $r$, and $s$ are distinct integers, and the polynomial equation $(x-p)(x-q)(x-r)(x-s)=16$ has an integral root $t$, find the value of $p+q+r+s$ in terms of $t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4t}
Determine the number of edges in the line graph L(G) of a graph G, given that G is regular, planar, and its line graph is planar but not isomorphic to G.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Mike is saving up to buy a house. He puts away 15% of his $120,000 a year salary. He needs to save up 25% of the cost of a $550,000 house for a downpayment. How long will it take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.64}
Find the minimum value of the expression:
\[ \frac{1}{|x_1-x_2|\cdots|x_1 - x_{n}|} + \frac{1}{|x_2-x_1||x_2-x_3|\cdots|x_2 - x_n|}+\cdots+\frac{1}{|x_n-x_1|\cdots|x_n-x_{n - 1}|} \]
where \(x_1, x_2, \ldots, x_n \in [-1, 1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-2}}
Find the largest integer $k$ ($k \ge 2$), for which there exists an integer $n$ ($n \ge k$) such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions:

1. each chosen number is not divisible by $6$, by $7$, nor by $8$;
2. the positive difference of any two distinct chosen numbers is not divisible by at least one of the
numbers $6$, $7$, and $8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Evaluate the integral \( \int_4^6 \sin^4 (\pi x) \cdot \cos^2 ( 2 \pi x) \,\mathrm{d}x \) using trigonometric identities.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{16}}
Evaluate the limit: $$ \lim_{t\to 1^-} (1-t) \sum_{n=0}^\infty \frac{t^n}{1+t^n} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2 \equiv a \pmod{n}$, $x^3 \equiv a \pmod{n}$, or $x^5 \equiv a \pmod{n}$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of residues modulo $n$.

How many integers $1 \le a \le 2015$ are primitive residues modulo $2015$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{694}
Given a list of integers of size \( n \), determine the minimum number of moves required to convert it into a permutation of the numbers from 1 to \( n \). In one move, you may increase or decrease any element of the list by one. For example, if \( n = 3 \) and the list is \((-1, -1, 2)\), the minimum number of moves required is 6 to convert it to the permutation \((1, 3, 2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the values of \( \alpha > 0 \) for which the integral \( \int_{0}^{\infty}\exp\left ( \frac{1}{2}x-x^\alpha \right )\,\mathrm{d}x \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha \geq 1}
Given \( x \in \left(0, \frac{\pi}{2}\right) \), what is the integer part of the number \( m = 3^{\cos^2 x} + 3^{\sin^5 x} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In $\triangle PQR,$ where $PQ=PR=17$ and $QR=15.$ Points $G,H,$ and $I$ are on sides $\overline{PQ},$ $\overline{QR},$ and $\overline{PR},$ respectively, such that $\overline{GH}$ and $\overline{HI}$ are parallel to $\overline{PR}$ and $\overline{PQ},$ respectively. What is the perimeter of parallelogram $PGHI$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Evaluate the integral \( \int_{0}^{2\pi}\exp(\ic\cos(\theta))\cos(\theta)\,\dd\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i J_1(1)}
In the Cartesian coordinate system, the vertices of the isosceles trapezoid \(ABCD\) are given by \(A(1,1)\), \(B(2,-1)\), \(C(-2,-1)\), and \(D(-1,1)\). A point \(P(0,2)\) is rotated \(180^\circ\) around point \(A\) to obtain point \(P_1\). Subsequently, point \(P_1\) is rotated \(180^\circ\) around point \(B\) to obtain point \(P_2\), and this process continues cyclically around points \(C\) and \(D\). Determine the coordinates of point \(P_{2021}\) after 2021 such rotations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2018, 0)}
Determine the annihilator of the tensor $e_1\wedge e_2 + e_3\wedge e_4$ in the vector space $V$ with basis $\{e_1, e_2, e_3, e_4\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0\}}
If $(3x-1)^7 = a_7x^7 + a_6x^6 + \cdots + a_0$, then $a_7 + a_6 + \cdots + a_0$ equals
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
A magician has a hat that contains $a$ white rabbits and $b$ black rabbits. The magician repeatedly draws pairs of rabbits chosen at random from the hat, without replacement. Call a pair of rabbits *checkered* if it consists of one white rabbit and one black rabbit. Given that the magician eventually draws out all the rabbits without ever drawing out an unpaired rabbit and that the expected value of the number of checkered pairs that the magician draws is $2020$, compute the number of possible pairs $(a,b)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Side $AB$ of triangle $ABC$ has length 8 inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is:


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[ {\dfrac{40}{3}} \]}
Given that the lengths of the sides $a$, $b$, $c$ opposite to the angles $A$, $B$, $C$ in triangle $ABC$ form an arithmetic sequence, find the maximum value of angle $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Side $AB$ of triangle $ABC$ has length 8 inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{40}{3}$}
If p, some digit, and r are distinct positive digits and the product of the two-digit integers pq and pr is 221, then the sum of the digits p, this digit, and r is 11. What is the digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let the operation $x*y$ be defined as $x*y = (x+1)(y+1)$. The operation $x^{*2}$ is defined as $x^{*2} = x*x$. Calculate the value of the polynomial $3*(x^{*2}) - 2*x + 1$ when $x=2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Three vehicles, "GAZ", "MAZ", and "KamAZ" simultaneously departed from Zlatoust to Miass. "KamAZ", upon reaching Miass, immediately turned back and met "MAZ" 18 km from Miass and "GAZ" 25 km from Miass. "MAZ", upon reaching Miass, also immediately turned back and met "GAZ" 8 km from Miass. What is the distance from Zlatoust to Miass?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60 \text{ km}}
Determine the value of gain \( K \) at which the root locus of the closed-loop system \( T(s) = \frac{-K(s+1)^2}{(1-K)s^2 + 2(1-K)s + (2-K)} \) intersects with the imaginary axis. Use the provided Routh table:
\[
\begin{bmatrix}
s^2 & (1-K) & (2-K) \\
s^1 & 2(1-K) & 0 \\
s^0 &(2-K) & 0
\end{bmatrix}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $P(x)=x^3+ax^2+bx+c$ where $a,b,c$ are positive real numbers. If $P(1)\geq 2$ and $P(3)\leq 31$, how many of integers can $P(4)$ take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4
}