inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $A$ be a regular $12$-sided polygon. A new $12$-gon $B$ is constructed by connecting the midpoints of the sides of $A$. The ratio of the area of $B$ to the area of $A$ can be written in simplest form as $\frac{a + \sqrt{b}}{c}$, where $a$, $b$, and $c$ are integers. Find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Inside an unequal-sided triangle \( ABC \), there is a point \( O \) such that \(\angle OBC = \angle OCB = 20^\circ\). Additionally, \( \angle BAO + \angle OCA = 70^\circ \). Find angle \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70^\circ}
Evaluate the integral:

$$ \frac{\exp\left(-\frac{N}{2}\right)}{2\pi N} \iint \exp\left(-\left( \frac{x^2-2xy-2Ny+2y^2}{2N}\right)\right) \,dx\,dy $$

where $N > 0$. Provide the result of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let  $\sigma(n)$  be the number of positive divisors of  $n$ , and let  $\operatorname{rad} n$  be the product of the distinct prime divisors of  $n$ . By convention,  $\operatorname{rad} 1 = 1$ .  Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \]*Proposed by Michael Kural*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{164}
Given that the even function $f(x)$ is increasing on the interval $[0, +\infty)$, determine the relationship between $f(-1)$ and $f(a^2-2a+3)$. Express your answer as an inequality using standard notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(-1) < f(a^2-2a+3)}
In city \( \mathrm{N} \), there are only blondes and brunettes. Blondes always lie, and brunettes always tell the truth. Every day, the residents dye their hair to the opposite color. One Monday in October, all people born in autumn were asked, "Were you born this month?" - 200 residents answered "yes", and no one answered "no". On Friday of the same week, these same people were asked again, "Were you born this month?" - and now only 50 answered affirmatively. How many "yes" answers were given to the same question on the last Monday of September, given that there are exactly four Mondays in October and no one in city \( \mathrm{N} \) was born in November?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The legs of a right triangle are 3 and 4. Find the area of the triangle with vertices at the points of tangency of the incircle with the sides of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6/5}
Petya and Vasya came up with ten quadratic trinomials. Then Vasya sequentially named consecutive natural numbers (starting from some number), and Petya substituted each named number into one of the trinomials of his choice and wrote the obtained values on the board from left to right. It turned out that the numbers written on the board formed an arithmetic progression (in that specific order).

What is the maximum number of numbers Vasya could have named?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the radius of convergence of the series \(\sum_{k=1}^{\infty} \frac{3^n}{z^n+z^{-n}}\) using the ratio test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
A frog sitting at the point \((1, 2)\) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length \(1\), and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices \((0,0), (0,4), (4,4),\) and \((4,0)\). The probability that the sequence of jumps ends on a vertical side of the square is given in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Determine the degree of the minimal polynomial of $\cos\left(\frac{\pi}{13}\right)$ over $\mathbb{Q}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. Find how many pairs of integers $(m,n)$ there are such that $1\leq m\leq 2012$ and  
$$5^n<2^m<2^{m+2}<5^{n+1}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{279}
7. Given the hyperbola $C: x^{2}-\frac{y}{3}=1$, with left and right foci $F_{1}$ and $F_{2}$, respectively. A line is drawn through point $F_{2}$ intersecting the right branch of the hyperbola $C$ at points $P$ and $Q$, such that $\angle F_{1} P Q=90^{\circ}$. Then the inradius of $\triangle F_{1} P Q$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{7}-1}
Determine the largest interval where the initial value problem \( t\dfrac{dy}{dt} = t + \left|y\right| \) with the initial condition \( y(-1) = 1 \) has a unique solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-e, 0)}
What is the probability of finding a trinomial multiple of a given polynomial \( p(x) \) in \( GF(2)[x] \) that satisfies the conditions \( p(1) = 1 \) and \( p(x) \) has a constant term of 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
On an infinite chessboard, some bishops and knights are placed such that:

- Each bishop has a knight on the same diagonal (there may be other pieces between them).
- Each knight has a bishop exactly \( \sqrt{5} \) units away.
- Removing any piece causes at least one of these conditions to fail.

Determine all possible values of \( n \), the total number of pieces on the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the integral \( \int_0^\infty \left(x^3-\frac{1}{x^3}\right)e^{-x^2/2}\,\mathrm{d}x. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the derivative of the convolution operation \( z(t) = (h \ast x)(t) \) with respect to \( x(t) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{h(0)}
A non-zero vector \( \vec{a} \) is parallel to the line of intersection of the plane determined by vectors \( \hat{i} \) and \( \hat{i}+\hat{j} \) and the plane determined by vectors \( \hat{i}-\hat{j} \) and \( \hat{i}+\hat{k} \). Find the angle between \( \vec{a} \) and the vector \( \hat{i}-2\hat{j}+2\hat{k} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
In a convex decagon \(A_{1} A_{2} \ldots A_{10}\), all the sides and diagonals connecting vertices at intervals of one (e.g., \(A_{1} A_{3}, A_{2} A_{4}\), etc.) are drawn, except for the side \(A_{1} A_{10}\) and diagonals \(A_{1} A_{9}, A_{2} A_{10}\). A path from \(A_{1}\) to \(A_{10}\) is defined as a non-self-intersecting broken line with endpoints \(A_{1}\) and \(A_{10}\), where each segment coincides with one of the drawn sides or diagonals. Determine the number of such paths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Determine the spectrum of the operator $T: C[0,1] \to C[0,1]$ defined by $Tf(x) = \int_0^x f(t) \, dt$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0\}}
Find all polynomials \( f(x) \) with integer coefficients such that for any two distinct primes \( p \) and \( q \), the values \( f(p) \) and \( f(q) \) are relatively prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm x^k}
Evaluate the limit: $$\lim_{x\to 0}\frac{\sin 5x - 5\sin 3x + 10\sin x}{\sin(\sin x) + \tan x - 2x}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{480}{7}}
Determine the cardinality of the set of injective functions from the set of natural numbers \( \mathbb{N} \) to the power set of natural numbers \( \mathcal{P}(\mathbb{N}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Calculate the limit: \[ \lim_{n \to \infty} \sum_{k=n}^{5n} \binom{k-1}{n-1} \left(\frac{1}{5}\right)^{n} \left(\frac{4}{5}\right)^{k-n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
A caretaker is giving candy to his two babies. Every minute, he gives a candy to one of his two babies at random. The five possible moods for the babies to be in, from saddest to happiest, are "upset," "sad," "okay," "happy," and "delighted." A baby gets happier by one mood when they get a candy and gets sadder by one mood when the other baby gets one. Both babies start at the "okay" state, and a baby will start crying if they don't get a candy when they're already "upset".

The probability that 10 minutes pass without either baby crying can be expressed as $\frac{p}{q}$. Compute $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{337}
A 2D random walker starts at the origin \((0,0)\) and takes one step randomly per second. There is a square absorbing boundary defined by the points \((2,2)\), \((2,-2)\), \((-2,2)\), and \((-2,-2)\). Determine the average time it takes for the random walker to reach the absorbing boundary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{2}}
Calculate the isothermal compressibility \( k_T \) at the critical temperature \( T = T_c \) and pressure \( p \) using the Van der Waals equation:

\[ p = \frac{R T}{v-b} - \frac{a}{v^2} \]

Given:
- \( p_c = \frac{a}{27b^2} \)
- \( v_c = 3b \)
- \( T_c = \frac{8a}{27bR} \)

The partial derivative of pressure with respect to volume at constant temperature is:

\[ \left(\frac{\partial p}{\partial v}\right)_T = -\frac{R T}{(v-b)^2} + 2\frac{a}{v^3} \]

At \( v = v_c \):

\[ -\frac{R T}{(v-b)^2} + 2\frac{a}{v^3} = -\frac{R(T-T_c)}{4b^2} \]

Hence, the isothermal compressibility is:

\[ k_T(T, v = v_c) = -\frac{1}{v}\left(\frac{\partial v}{\partial p}\right)_T \approx C(T-T_c)^{-\gamma} \]

Determine \( k_T(T = T_c, p) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the condition on positive integers \(m\) and \(n\) such that the quotient rings \(\mathbb{Z}[x,y]/(x^2-y^n)\) and \(\mathbb{Z}[x,y]/(x^2-y^m)\) are isomorphic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m = n}
Given the function \( u = x^2y^2 + yz - z^3 \) and the constraint \( x^2 + y^2 + z^2 = 11 \), find the value of \( \left( \frac{\partial u}{\partial y} \right)_{x} \) at the point \((u, x, y, z) = (5, 1, -3, 1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-23}
Find the value of \( l \) for which the integral \( I(l) = \int_{0}^{\infty} \frac{x^l}{2x^6 + 4x^5 + 3x^4 + 5x^3 + 3x^2 + 4x + 2} \, dx \) is minimized, where \( 0 < l < 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the greatest positive integer  $k$  such that the following inequality holds for all  $a,b,c\in\mathbb{R}^+$  satisfying  $abc=1$  \[ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
XLIII OM - II - Problem 5

Determine the supremum of the volumes of spheres contained in tetrahedra with all altitudes no longer than $1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi/48}
What is the maximum number of rooks that can be placed on a $300 \times 300$ chessboard such that each rook attacks at most one other rook? (A rook attacks all the squares it can reach according to chess rules without passing through other pieces.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
For how many values of $x$ in $[0,\pi]$ is $\sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What values of \( d \), where \( 0 < d < 1 \), result in a generalized Cantor set with Lebesgue measure zero, when at each stage intervals of size \( 1 - 2d \) are retained at both ends and intervals of size \( 1 - 2d^n \) are removed, with \( 2^{n-1} \) such intervals at the \( n^{th} \) stage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < d < 1/2}
How many numbers of the form $\overline{a b c d a b c d}$ are divisible by 18769?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Krishna is preparing a pizza with 8 slices and has 10 different toppings available. Each slice can have one topping, and the same topping cannot be used on adjacent slices. How many different ways can Krishna arrange the toppings on the pizza slices?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43046730}
已知多项式
$$
f(z)=z^{3}+a z^{2}+b z+c(a, b. c \in \mathbf{Z})
$$
的所有根的模都是 20 或 15. 则这样的多
项式有 $\qquad$个。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{572}
Evaluate the surface integral \( \int_{S} \vec{F} \cdot d\vec{S} \) where \( \vec{F} = y\hat{i} + 2x\hat{j} - z\hat{k} \) and \( S \) is the surface of the plane \( 2x + y = 4 \) in the first octant, cut off by the plane \( z = 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
In a 100x100 grid, each unit square contains a natural number. A rectangle with sides parallel to the grid is called "good" if the sum of the numbers inside the rectangle is divisible by 17. We can paint all unit squares in a "good" rectangle, and no unit square can be painted more than once. Determine the maximum number of unit squares, denoted as \(d\), that can be guaranteed to be painted.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9744}
In a fair coin game, you win $1 if a head shows up and lose $1 if a tail shows up. What is the expected number of rounds you will play before you lose the first dollar from your own pocket?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the number of positive integers $n \le 600$ whose value can be uniquely determined when the values of $\left\lfloor \frac{n}{4} \right\rfloor$, $\left\lfloor \frac{n}{5} \right\rfloor$, and $\left\lfloor \frac{n}{6} \right\rfloor$ are given, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to the real number $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Given that $f(z)$ satisfies the condition $\Im (f'(z)) = 6x(2y-1)$, with $f(0) = 3-2i$ and $f(1) = 6-5i$, find the value of $f(1+i)$. Provide your answer in the form $a + bi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 + 3i}
Compute the maximum integer value of $k$ such that $2^k$ divides $3^{2n+3} + 40n - 27$ for any positive integer $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the tightest possible constant \( \epsilon_1 \) such that the inequality \( \left| 1 - \frac{1}{1+E[x]} - E \left[ 1 - \frac{1}{1+x} \right] \right| \leq \epsilon_1 \cdot \text{Var}(x) \) holds for a real-valued random variable \( X \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
At Jefferson High School, there are 500 students enrolled. One hundred twenty students are in the orchestra, 190 are in band, and 220 are in chorus. If only 400 students are in orchestra, band, and/or chorus, how many students are in exactly two of these groups?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130}
? % of 480 = 193.2
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40.25\%}
Suppose we have $10$ balls and $10$ colors. For each ball, we independently color it with one of the $10$ colors, then group the balls together by color at the end. If $S$ is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of $S$ written as a decimal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
How many ways can you line up the players of two soccer teams in a row so that no two players from the same team stand next to each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \times (11!)^2}
Andy is mixing blue, green, white, and red paint in a 1 : 2 : 5 : 3 ratio. If he uses 6 liters of green paint and needs to convert the total amount of paint used into gallons, how many gallons of paint does he use total? (Note: 1 liter = 0.264172 gallons)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.72}
Given the set \( S = \{1, 2, \cdots, 2005\} \), and a subset \( A \subseteq S \) such that the sum of any two numbers in \( A \) is not divisible by 117, determine the maximum value of \( |A| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}
Given the sequence ${a_n}$ that satisfies the equation $a_{n+1}+(-1)^{n}a_{n}=3n-1,(n∈N^{*})$, determine the sum of the first 40 terms of the sequence ${a_n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1240}
Determine the values of \( a > 0 \) for which the improper integral \( \int_{0}^{\infty}\frac{\sin x}{x^{a}} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2)}
Determine the integer $d$ such that the homotopy equivalence $\mu: S^m \wedge S^n \rightarrow S^{m+n}$ satisfies $\mu^*(\sigma^{m+n}1) = d \cdot \sigma^m1 \wedge \sigma^n1$, where $\mu^*: H^{m+n}(S^{m+n}) \rightarrow H^{m+n}(S^m \wedge S^n)$ is the induced homomorphism. Here, $\sigma^n 1$ is the image of the unit $1 \in H^0(S^0)$ under the $n$-fold suspension isomorphism $\sigma^n: H^0(S^0) \rightarrow H^n(S^n)$. Find the value of $d$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that the domain of function $f(x)$ is $\mathbf{R}$, $f''(x)$ is the derivative of $f(x)$, and $f''(x) - 2f(x) > 0$, with $f(\frac{1}{2}) = e$, where $e$ is the base of the natural logarithm, determine the solution set for the inequality $f(\frac{1}{2}\ln{x}) < x$. Express your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, e)}
Determine the greatest positive integer \( n \) for which there exist \( n \) nonnegative integers \( x_1, x_2, \ldots, x_n \), not all zero, such that for any choice of \( \varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n \) from the set \( \{-1, 0, 1\} \), not all zero, the sum \( \varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n \) is not divisible by \( n^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the number of matrices \( A \) in \( F_4^{5 \times 3} \) such that \( \text{rank}_{F_4}(A) = 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1051807680}
Consider numbers of the form \( {1a1} \) , where \( a \) is a digit. How many pairs of such numbers are there such that their sum is also a
 palindrome?
 Note: A palindrome is a number which reads the same from left to right and from right to left. Examples:353,91719.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
30
}
Let $x, y, a \in \mathbb{R}^*$, and when $x + 2y = 1$, the minimum value of $\frac{3}{x} + \frac{a}{y}$ is $6\sqrt{3}$. Determine the minimum value of $3x + ay$ when $\frac{1}{x} + \frac{2}{y} = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\sqrt{3}}
Given the function \(f(x) = \lim_{n\to \infty}\dfrac{\left(1- \cos \left(1- \tan \left(\dfrac {\pi}{4}-x\right)\right)\right)(x+1)^n+ \lambda\sin((n - \sqrt{n^2 -8n})x)}{x^2(x+1)^n+x}\), where \(x \neq 0\), determine the value of \(f(0) + 2\lambda\) if \(f(x)\) is continuous at \(x=0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the automorphism $\psi$ of the cyclic group $\Bbb Z_{30}$ such that $\psi(7) = 11$. What is the value of $\psi(x)$ for any $x \in \Bbb Z_{30}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23x}
In triangle $ABC$, points $D$ and $E$ are on side $BC$. Perpendiculars from $D$ and $E$ to $BC$ intersect $BC$ at points $K$ and $L$, respectively. Given that $BK = R$ and $KL = 2R$, and that $DK$ and $EL$ are tangent to a circle centered at $B$, find the length of $BC$. Express your answer in terms of $R$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4R}
Calculate the surface area of the portion of the cylinder defined by \(y^2+z^2=2z\) that is intersected by the cone \(x^2=y^2+z^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Among all fractions of the form $\frac{m}{n}$, where $m$ and $n$ are four-digit numbers with the same sum of digits, the greatest one was selected. Find it. Record your answer as an improper fraction (do not round or simplify!).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9900}{1089}}
In triangle \( \triangle ABC \), the sides opposite to the angles \( \angle A \), \( \angle B \), and \( \angle C \) are denoted as \( a \), \( b \), and \( c \) respectively. If \( b^{2}=a^{2}+c^{2}-ac \), and \( c-a \) is equal to the height \( h \) from vertex \( A \) to side \( AC \), then find \( \sin \frac{C-A}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let $f(x,y,z,u)=\frac{24}{(1+x+y+z+u)^5}, x>0,y>0,z>0,u>0$ be a density function. Find $P(X+Y+Z+U \ge 1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15/16}
An article is sold for Rs. 350, and 5% more is gained than by selling it for another price. If the cost of the article is Rs. 200, what is the lower selling price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{342.86}
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A sequence of 2020 natural numbers is written in a row. Each of them, starting from the third number, is divisible by the previous one and by the sum of the two preceding ones.

What is the smallest possible value for the last number in the sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019!}
In a course conducted by Professor Jones, each student is on average absent for one day out of a 40-day course. What is the probability that out of any two randomly selected students, one student will be absent while the other is present? Express your answer as a percent rounded to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.9\%}
Find the sum of all positive divisors of $40081$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
40500
}
3. In how many ways can the set $\{1,2, \ldots, 12\}$ be divided into six disjoint two-element subsets such that each of them contains coprime numbers (i.e., numbers that do not have a common divisor greater than one)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}