inclusionAI/Ring-lite-rl-data huggingface dataset:
How many positive integers at most $420$ leave different remainders when divided by each of $5$, $6$, and $7$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
Evaluate the limit \( \lim_{n\to \infty} n^2 \log \left(\dfrac{r_{n}}{r_{n+1}} \right) \) where \( r_{n} \) is given by \( r_{n}=\dfrac{\sqrt{n}}{n!} \left(\dfrac{n}{e} \right)^n. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{12}}
A mineral water bottle consists of a neck and a body. The body can be considered as a cylindrical shape. Now, place a mineral water bottle upright, and the water level is 15cm high; then, invert the bottle, and it is found that there is a 5cm high space left empty. If the bottle originally contained 500ml of mineral water, then the volume of the bottle is     ml. (rounded to 1ml)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{667}
Assume that \(z\) and \(w\) are differentiable functions of \(x\) and \(y\), satisfying the equations \(xw^3+yz^2+z^3=-1\) and \(zw^3-xz^3+y^2w=1.\) Find \(\frac{\partial z}{\partial x}\) at \((x,y,z,w)=(1,-1,-1,1).\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{5}{4}}
3. On the line AB, point O is marked and from it rays OC, OD, OE, OF are drawn in the specified order into one half-plane of line AB (ray OC lies between rays OA and OD). Find the sum of all angles with vertex O, whose sides are rays OA, OC, OD, OE, OF, OB, if $\angle \mathrm{COF}=97^{\circ}, \angle \mathrm{DOE}=35^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1226}
The piece "nedoslon" moves one square diagonally. What is the maximum number of non-attacking nedoslon pieces that can be placed on a $7 \times 7$ chessboard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
A warehouse is located inside a transportation zone, which is a triangle with vertices at three transport nodes. The distances between these nodes are 2 km, $\sqrt{7}$ km, and 3 km. A truck must make three trips to each transport node sequentially, starting from the warehouse and returning back each time. What is the minimum total distance the truck will travel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{19}}
设正整数 n 的正因数为 1=d_1<d_2<\cdots<d_k=n，其中 k\geq6. 求所有的 n 使得 n=d_5^2+d_6^2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
Determine the values of $z$ on the unit circle (i.e., $|z|=1$) for which the series $$\sum_{n=1}^\infty \frac{z^n}{n}$$ converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{z \neq 1}
Determine the values of $p \in [1, +\infty[$ for which the function $f(x) = |x|$ belongs to the Sobolev space $W^{1,p}(]-1, 1[)$. Recall that $W^{m,p}(I)$ consists of functions $u$ such that $u$ and all its derivatives up to order $m$ are in $L^p(I)$, where $D^{\alpha} u$ denotes the derivative in the distribution sense.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, +\infty)}
Example 5 Let $a$ and $d$ be non-negative real numbers, $b$ and $c$ be positive numbers, and $b+c \geqslant a+d$. Find the minimum value of $\frac{b}{c+d}+$ $\frac{c}{a+b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\sqrt{2}-\\frac{1}{2}"]}
Let  $H$  be a regular hexagon of side length  $x$ . Call a hexagon in the same plane a "distortion" of  $H$  if
and only if it can be obtained from  $H$  by translating each vertex of  $H$  by a distance strictly less than  $1$ . Determine the smallest value of  $x$  for which every distortion of  $H$  is necessarily convex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a convex $n$-sided polygon, if there are $k$ sides each equal to the length of the longest diagonal of the polygon, then the maximum value of $k$ is .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that $|\vec{a}|=1$, $|\vec{b}|=\sqrt{2}$, $\vec{c}=\vec{a}+\vec{b}$, and $\vec{c} \perp \vec{a}$, the angle between vectors $\vec{a}$ and $\vec{b}$ is $\_\_\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\pi}{4}}
Given the equations of a hyperbola and an ellipse, $\frac{x^{2}}{1-m} + \frac{y^{2}}{m+2}=1$ and $\frac{x^{2}}{2m} + \frac{y^{2}}{2-m}=1$, respectively, find the range of values for $m$ that satisfy the compound proposition $p \land q$, where $p$ states that the first equation represents a hyperbola and $q$ states that the second equation represents an ellipse with foci on the $x$-axis. Express your answer as an interval or inequality for $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 < m < 2}
Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $r$ be a real number. Two of the roots of $f(x)$ are $r + 2$ and $r + 8$. Two of the roots of $g(x)$ are $r + 5$ and $r + 11$, and
\[f(x) - g(x) = 2r\] for all real numbers $x$. Find $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20.25}
A geometric shape with $5$ faces could be ______$(only fill in one type)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{square pyramid}}
Given that the equation \(|z|^2+\bar{A}z^2+A(\bar{z})^2+B\bar{z}+\bar{B}z+c=0\) represents a pair of intersecting lines with an angle of intersection \(\theta\), find the value of \(|A|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2 \cos \theta}}
Evaluate the line integral of \( \exp(2\overline{z}) \) over the unit circle \( C \) in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi i}
If $$\sin\theta= \frac {3}{5}$$ and $$\frac {5\pi}{2}<\theta<3\pi$$, then $$\sin \frac {\theta}{2}$$ equals \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-\\frac {3 \\sqrt {10}}{10}"]}
Determine the time complexity of the conjugate gradient method for solving the system of linear equations \(Ax = b\), where \(A\) is a sparse, symmetric, positive definite matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^2)}
Find the radius of convergence of the series: $(-1)^{n+1} \frac{\bigl(\bigl|\tfrac{1-x}{1+x}\bigr|-1\bigl)^n} n$ representing the equation $\ln|(1-x)/(1+x)|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
Given 50 feet of fencing, where 5 feet is used for a gate that does not contribute to the enclosure area, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the remaining fencing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126.5625}
Determine all integers $k$ such that there exists infinitely many positive integers $n$ [b]not[/b] satisfying 
\[n+k |\binom{2n}{n}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \neq 1}
On a straight road, there are an odd number of warehouses. The distance between adjacent warehouses is 1 kilometer, and each warehouse contains 8 tons of goods. A truck with a load capacity of 8 tons starts from the warehouse on the far right and needs to collect all the goods into the warehouse in the middle. It is known that after the truck has traveled 300 kilometers (the truck chose the optimal route), it successfully completed the task. There are     warehouses on this straight road.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
5. In each cell of a $10 \times 10$ table, a natural number was written. Then, each cell was shaded if the number written in it was less than one of its neighbors but greater than another neighbor. (Two numbers are called neighbors if they are in cells sharing a common side.) As a result, only two cells remained unshaded. What is the minimum possible sum of the numbers in these two cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
For each positive integer $n$, define $s(n)$ to equal the sum of the digits of $n$. The number of integers $n$ with $100 \leq n \leq 999$ and $7 \leq s(n) \leq 11$ is $S$. What is the integer formed by the rightmost two digits of $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
10.4. On the cells of an 8 by 8 board, chips are placed such that for each chip, the row or column of the board in which it lies contains only one chip. What is the maximum possible number of chips on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
A sphere is inscribed in a right cone with base radius $15$ cm and height $30$ cm. The radius of the sphere can be expressed as $b\sqrt{d} - g$ cm, where $g = b + 6$. What is the value of $b + d$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.5}
A can do a piece of work in 12 days. When he had worked for 2 days, B joins him. The complete work was finished in some days. B alone can finish the work in 27.99999999999998 days. In how many days was the complete work finished when A and B worked together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
在 $\triangle A B C$ 中, $\cos A, \sin A, \tan B$ 依次构成以 $\frac{3}{4}$ 为公比的等比数列。请给出 $\cot C$ 的值，其原始答案为 \frac{m}{n} 格式，请给出 m + n 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
A rectangle with integer length and width has a perimeter of 120 units. What is the number of square units in the least possible area, assuming at least one of the dimensions is a prime number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{116}
已知下列结论成立:
任意给定复平面 $\mathbb{C}$ 上的多项式 $f(z)$ 和 $g(z)$ ，以及实数 $r>0$ 。若不等式
$$
|g(z)|<|f(z)|
$$
对所有满足 $|z|=r$ 的复数 $z$ 都成立，则 $f(z)$ 与 $g(z)+f(z)$ 在区域 $\{z \in \mathbb{C}||z|<r\}$ 中有相同的零点个数（此处计算零点的重数）。
根据上述结论求多项式 $z^{9}+3 z^{5}-8 z^{3}+2 z+1$ 在区域 $\{z \in \mathbb{C}|1<|z|<2\}$ 中的零点个数（此处计算零点的重数).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
There are 2021 light bulbs in a row, labeled 1 through 2021, each with an on/off switch. They all start in the off position. When 1011 people walk by, they interact with the bulbs as follows:

1. The first person flips the switch on every bulb.
2. The second person flips the switch on every 3rd bulb (bulbs 3, 6, etc.).
3. The third person flips the switch on every 5th bulb.
4. In general, the $k$th person flips the switch on every $(2k - 1)$th light bulb, starting with bulb $2k - 1$.

After all 1011 people have gone by, how many light bulbs are on?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
在直三棱柱 $A_{1} B_{1} C_{1}-A B C$ 中, $\angle B A C=\frac{\pi}{2}, A B=A C=A A_{1}=1$. 已知 $G$与 $E$ 分别为 $A_{1} B_{1}$ 和 $C C_{1}$ 的中点, $D$ 与 $F$ 分别为线段 $A C$ 和 $A B$ 上的动点 (不包括端点). 若 $G D \perp E F$ ，求线段 $D F$ 的长度的取值范围。答案的格式为 \left[\frac{\sqrt{m}}{n}, a\right)，请给出m+n+a的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
设 S 是边长为 1 的正六边形内及边上的点构成的集合. 求最小的实数 r，使得存在一种将 S 中的点三染色的方法，满足任意两个同色点之间的距离小于 r.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
7.5. Two spheres are inscribed in a dihedral angle, touching each other. The radius of one sphere is 1.5 times that of the other, and the line connecting the centers of the spheres forms an angle of $30^{\circ}$ with the edge of the dihedral angle. Find the measure of the dihedral angle. Write the cosine of this angle in your answer, rounding it to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.68}
How many ways are there to remove an $11 \times 11$ square from a $2011 \times 2011$ square so that the remaining part can be tiled with dominoes ($1 \times 2$ rectangles)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2002001}
Express the complex number \( \dfrac{(\text{cis}\ 75^\circ-\text{cis}\ 155^\circ)(1-\cos 8^\circ+i \sin 8 ^\circ)}{2-2\cos 8^\circ} \) in the form \( r\ \text{cis}\ \theta \), where \( 0 \le \theta < 360^\circ \). Determine the value of \( \theta \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{111}
Use a set of sample data $8$, $x$, $10$, $11$, $9$ to estimate the population standard deviation. If the average of this set of sample data is $10$, then the population standard deviation $s=$ __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt {2}}
An 8th circular disk with radius \( r \) is rolled around 7 fixed disks, each with radius \( r \), arranged such that their centers form a regular hexagon with side length \( 2r \). The 8th disk touches one of the 6 outer disks and rolls around the 7 fixed disks once without slipping, returning to its starting position. How many times does the 8th disk rotate about its own center during this process?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $ABCD$ be a square with side length $1$. How many points $P$ inside the square (not on its sides) have the property that the square can be cut into $10$ triangles of equal area such that all of them have $P$ as a vertex?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Determine the values of $T$ for which the differential equation $x''(t) = -x(t)$ has non-unique solutions given the boundary conditions $x(0) = a_1$ and $x(T) = a_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{T = n\pi}
In a $n^2 \times n^2$ grid divided into $n^2$ subgrids of size $n \times n$, how many ways can you select $n^2$ cells from this grid such that there is exactly one cell coming from each subgrid, one from each row, and one from each column?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n!)^{2n}}
Evaluate the double integral \( \iint_A xy^3e^{x^2-y^4} \,dx\,dy \) where \( A \) is the region bounded by \( x=4, \; y=0, \; x^2-y^4=9, \) and \( x^2-y^4=1 \). Use the substitution \( x^2-y^4=v, \; x^2=u. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^9 - 2e}
Given the function y = (m^2 + 2m)x + m^2 + m - 1 + (2m - 3) is a linear function of x, find the value of the constant m. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( a, b, c, d \) be distinct real numbers such that \( a + b + c + d = 3 \) and \( a^2 + b^2 + c^2 + d^2 = 45 \). Find the value of the expression:
\[
\frac{a^{5}}{(a-b)(a-c)(a-d)}+\frac{b^{5}}{(b-a)(b-c)(b-d)} +\frac{c^{5}}{(c-a)(c-b)(c-d)}+\frac{d^{5}}{(d-a)(d-b)(d-c)}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
In a $2009 \times 2009$ grid, each small square is filled with a real number whose absolute value does not exceed 1. In addition, for every $2 \times 2$ square, the sum of the 4 real numbers is 0. Find the maximum sum of all the numbers in the grid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2009}
rainbow is the name of a bird. this bird has  $n$  colors and it's colors in two consecutive days are not equal. there doesn't exist  $4$  days in this bird's life like  $i,j,k,l$  such that  $i<j<k<l$  and the bird has the same color in days  $i$  and  $k$  and the same color in days  $j$  and  $l$  different from the colors it has in days  $i$  and  $k$ . what is the maximum number of days rainbow can live in terms of  $n$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 1}
What is the exact distinguishing number of the Fano plane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of subgroups in the group \(A=\mathbb{Z}_{360}\oplus\mathbb{Z}_{150}\oplus\mathbb{Z}_{75}\oplus\mathbb{Z}_{3}\) that are isomorphic to \(\mathbb{Z}_{5}\oplus\mathbb{Z}_{5}\). Each subgroup is generated by two elements \(a\) and \(b\) with \(\text{ord}(a) = \text{ord}(b) = 5\) and \(a \notin \langle b \rangle\). How many such subgroups exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Find the radius of convergence of the complex power series $$\sum_{n=2}^\infty \frac{1}{n!}(z-2n\pi i)^{n-2}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the hyperbola $E: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and the circle $(x-4)^2 + y^2 = 16$, find the eccentricity of the hyperbola if a chord cut by the circle has a length of 4. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the greatest natural number $n$ such that for every set $S$ of 2015 different integers, there exist two subsets of $S$ (each subset can have at least one element and the subsets do not need to be disjoint) such that the sum of the elements in each subset is divisible by $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
Define a sequence $\{a_{n}\}_{n=1}^{\infty}$ with $a_{1} = 1$ and $a_{n+1} = a_{n} + \lfloor \sqrt{a_{n}} \rfloor$ for all $n \geq 1$. Determine the smallest $N$ such that $a_{N} > 2017$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
What is the maximum value of $$\underset{\pi \in S_n}{\max} \biggl( \underset{1\leq i\leq n}{\sum} \big|i-\pi(i)\big|\biggr)\,,$$ where $S_n$ is the set of permutations of $(1,2,\ldots,n)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\left\lfloor\frac{n^2}{2}\right\rfloor\)}
Determine the number of distinct ways to paint and dry 10 differentiable boxes, given that a box must be fully painted before it can be dried, and each box can be painted and dried only once. For example, if the first box is painted, it must be dried before moving to the second box.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3628800}
For some integers that are not palindromes, like 73, a person can create a palindrome by repeatedly reversing the number and adding the original number to its reverse. For example, $73 + 37 = 110$. Then $110+011 = 121$, which is a palindrome. Of all positive integers between 10 and 100, what is the sum of the non-palindrome integers that take exactly five steps to become palindromes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Two equilateral triangles are contained in a square whose side length is $2\sqrt{3}$. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\sqrt{3} - 12}
The government decided to privatize civil aviation. For each of the 127 cities in the country, the connecting airline between them is sold to one of the private airlines. Each airline must make all acquired airlines one-way but in such a way as to ensure the possibility of travel from any city to any other city (possibly with several transfers). What is the maximum number of companies that can buy the airlines?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Points \( D \) and \( E \) are marked on the sides \( AC \) and \( BC \) of triangle \( ABC \) respectively, such that \( AD = EC \). It turns out that \( BD = ED \), and \( \angle BDC = \angle DEB \). Find the length of the segment \( AC \) given that \( AB = 7 \) and \( BE = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
What is the area of the region inside the quadrilateral \(P Q R S\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
In the Cartesian coordinate system $xOy$, given points $A(0, -2)$ and $B(1, -1)$, and $P$ is a moving point on the circle $x^2+y^2=2$, the maximum value of $\frac{PB}{PA}$ is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( \int_{0}^{\frac{\pi}{2}} M(a) \sin (2a) \ da \), where \( M(a) \) is the maximum value of the function \( f(x) = \int_{0}^{\pi} \sin (x-t) \sin (2t-a) \ dt \) for \( a \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{28}{27}}
Rita sells three items: Item A, Item B, and Item C. She makes a profit of 60% on Item A by selling it for Rs. 600, a profit of 40% on Item B by selling it for Rs. 800, and a profit of 50% on Item C by selling it for Rs. 900. Find the overall average cost price of these 3 items.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{515.48}
Let $M$ be the set of natural numbers such that there exists a group of this order which is not solvable. Determine the minimal distance $D$ between two numbers in $M$. Given examples show $D \leq 12$, and the Feit-Thompson theorem implies $D > 1$. What is the value of $D$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
What is the minimum number of acute-angled triangles into which a square can be divided?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A circle is constructed on the side $BC$ of triangle $ABC$ as its diameter, and it intersects segment $AB$ at point $D$. Find the ratio of the areas of triangles $ABC$ and $BCD$, given that $AC = 15$, $BC = 20$, and $\angle ABC = \angle ACD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25/16}
If a certain digit and y are the two digits of the number 653xy such that this number is divisible by 80, then the sum of the two digits equals 2. What is the value of the first digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Simplify the expression $E[E(ZX \mid Y) \mid Z,Y]$. Provide the simplified form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E(ZX \mid Y)}
Evaluate the line integral of the conservative vector field \( F(x,y,z) = 3x^2zi + z^2j + (x^3 + 2yz)k \) along the curve \( C \) defined by the parametric equations \( r(t) = \left(\frac{\ln t}{\ln 2}\right)i + z^{3/2}j + t\cos(\pi t)k \), where \( 1 \leq t \leq 4 \), using the fundamental theorem of line integrals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{159}
Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Basil could have called out?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In how many ways can a lottery ticket be filled out so that the sum of the 5 marked numbers is at least 228?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21974634}
Let \( G \) be a cubic graph with \( q \) edges and girth at least 5. Determine the number of paths of length 4 in \( G \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8q}