inclusionAI/Ring-lite-rl-data huggingface dataset:
Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\frac{n}{d}$ where $n$ and $d$ are integers with $1 \le d \le 5$. What is the probability that
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath>
is a real number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{25}}
43. (FIN 1) Evaluate  
$$ S=\sum_{k=1}^{n} k(k+1) \cdots(k+p), $$  
where $n$ and $p$ are positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(n+1) \cdots(n+p+1)}{p+2}}
A parking lot has a row of eight parking spaces numbered from 1 to 8. Now, 2 trucks and 2 cars are to be parked at the same time, with each parking space accommodating at most one vehicle. If vehicles of the same type must be parked in adjacent parking spaces, there are a total of ______ parking arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Let \( a \) be a given complex number satisfying \( 0 < \left\vert a\right\vert < 2 \) and let \( \gamma \) be the unit circle of radius 3 oriented in the positive sense. Determine the value of the contour integral \( \oint_{\gamma}{\Re\left(z\right) \over z - a}\,\mathrm{d}z \) in terms of \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi i a}
In the convex pentagon $ABCDE$, the angles $\angle A$ and $\angle C$ are both $90^{\circ}$. The sides $AB$ and $AE$ are equal, as well as the sides $BC$ and $CD$. The length of the diagonal $AC$ is $1$. Find the area of the pentagon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
[ Triangle inequality (miscellaneous). ] [ Minimum or maximum distance (length).]

Petya bought a "Constructor" set, which contained 100 sticks of different lengths. The instructions for the "Constructor" state that any three sticks from the set can form a triangle. Petya decided to test this statement by forming triangles with the sticks. The sticks are arranged in the constructor in ascending order of length. What is the
minimum number of checks (in the worst case) that Petya needs to make to prove or disprove the statement in the instructions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Arrange the 5 natural numbers from 1 to 5 in a row from left to right, in such a way that starting from the third number, each number is either the sum or difference of the previous two numbers. How many valid arrangements are there? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Circles with radii \( r \) and \( R \) touch each other externally at point \( K \). A line touches these circles at different points \( A \) and \( B \).

Find the area of triangle \( A K B \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 r R \sqrt{r R}}{r+R}}
There are $2012$ backgammon checkers (stones, pieces) with one side black and the other side white. These checkers are arranged in a line such that no two consecutive checkers are the same color. At each move, we choose two checkers and flip them along with all the checkers between them. What is the minimum number of moves required to make all checkers the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006}
Given that the sides of $\triangle ABC$ are $13$, $14$, and $15$. There are 4 circles $O$, $O_1$, $O_2$, and $O_3$ with the same radius $r$ placed inside $\triangle ABC$. Circle $O_1$ is tangent to sides $AB$ and $AC$, circle $O_2$ is tangent to sides $BA$ and $BC$, circle $O_3$ is tangent to sides $CB$ and $CA$, and circle $O$ is tangent to circles $O_1$, $O_2$, and $O_3$. Find $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{260}{129}}
定义在$R$上的函数$y=f\left( x-1 \right)$的图像关于$\left( 1,0 \right)$对称，且当$x\in \left( -\infty ,0 \right)$时，$f\left( x \right)+x{f}'\left( x \right)<0$（其中${f}'\left( x \right)$是$f\left( x \right)$的导函数），若$a=\left( {{3}^{0.3}} \right)\cdot f\left( {{30.3}} \right),b=\left( \text{lo}{{\text{g}}_{\pi }}3 \right)\cdot f\left( \text{lo}{{\text{g}\pi }}3 \right),$ $c=\left( \text{lo}{{\text{g}}_{3}}\frac{1}{9} \right)\cdot f\left( \text{lo}{{\text{g}3}}\frac{1}{9} \right)$，
则$a,b,c$的大小关系是________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$c>a>b$}
Place eight spheres, each with a radius of 1, in two layers inside a cylinder. Ensure that each sphere is tangent to its four neighboring spheres, as well as tangent to one base and the lateral surface of the cylinder. Determine the height of the cylinder.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[4]{8} + 2}
A group of \( n \) friends participated in a math contest consisting of eight short-answer problems \( S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8 \) and four full-solution problems \( F_1, F_2, F_3, F_4 \). Each person in the group correctly solved exactly 11 out of the 12 problems.

We create an \( 8 \times 4 \) table. Inside the square located in the \( i \)-th row and \( j \)-th column, we write down the number of people who correctly solved both problem \( S_i \) and \( F_j \). If the 32 entries in the table sum to 256, what is the value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A positive integer $n$ is loose if it has six positive divisors and satisfies the property that any two positive divisors $a<b$ of $n$ satisfy $b \geq 2 a$. Compute the sum of all loose positive integers less than 100.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{512}
If there is a 95% confidence that events A and B are related, determine the condition that the test statistic K^2 must satisfy. Express your answer as an inequality involving K^2 and the specific numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K^2 > 3.841}
Find the value of \(2 - 13 \cos 2\alpha + \sin^{-1}(2\alpha)\), given that \(\operatorname{ctg} \alpha = -\frac{1}{5}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{57}{5}}
A triangle with vertices $(6, 5)$, $(8, -3)$, and $(9, 1)$ is reflected about the line $x=8$ to create a second triangle.  What is the area of the union of the two triangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{32}{3}}
Find the sum of the infinite series $\frac{1}{2\cdot 4}+\frac{1\cdot3}{2\cdot4\cdot6}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8}+\dots$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
Find the smallest four-digit number SEEM for which there is a solution to the puzzle MY + ROZH = SEEM. (The same letters correspond to the same digits, different letters - different.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2003}
Consider the graph in 3-space of
\[0 = xyz(x + y)(y + z)(z + x)(x - y)(y - z)(z - x).\]
This graph divides 3-space into \(N\) connected regions. What is \(N\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
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. The area of the rhombus is in the form \(k\sqrt{3} - m\). Please find the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given an infinite geometric sequence \{a\_n}(n∈N*) with the first term a\_1=1 and common ratio q=13\*, find the limit of the sum of the first n terms S\_n, i.e., limn→∞Sn=\_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Randomly select two points on a line segment of length 3 to divide it into three segments. Calculate the probability that exactly two of these segments are longer than 1. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
How many distinct ways can we color the faces of a tetrahedron using four different colors, taking into account the symmetries of the shape (orientation-preserving only)? Apply Burnside's orbit-counting theorem, where $G$ is the symmetry group of the tetrahedron, $X_g$ is the number of colorings fixed by element $g \in G$, and $r$ is the number of orbits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Let \( A, B, C, \) and \( D \) be four points lying in this order on a circle. Assume that there is a point \( K \) on the line \( AB \) such that \( BD \) bisects the line \( KC \) and \( AC \) bisects the line \( KD \). Determine the smallest possible value that \( \left|\frac{AB}{CD}\right| \) can take.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
已知复数$z _ { 1 } , z _ { 2 }$满足$\vert z _ { 1 } \vert \geq 1 ,$,$\vert z _ { 2 } \vert \geq \frac { 3 } { 2 }$.则复数$i ^ { 2 0 0 1 } z _ { 1 } + i ^ { 1 9 9 9 } z _ { 2 } + 2 z _ { 1 } z _ { 2 }$的模长的最小值是.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac { 1 } { 2 }$}
A small sphere with a radius of 1 is placed inside a closed cubical box with an edge length of 6. If the box is arbitrarily turned, find the volume of the space that the sphere cannot reach.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56 - \frac{40\pi}{3}}
已知 a_k 为 \log_2k, \log_3k, \cdots, \log_{2018}k 中整数项的个数，则 \ds{k=1}{2018}a_k=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4102}
A kite has sides $15$ units and $20$ units meeting at a right angle, and its diagonals are $24$ units and $x$ units. Find the area of the kite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
Given a hyperbola E with vertices A and B, and a point M on E forming an isosceles triangle ABM with a 120° vertex angle, calculate the eccentricity of E. Express your answer as a numerical value, including the square root symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Problem 6. A row of new recruits stood facing the sergeant. On the command "left," some turned left, while the rest turned right. It turned out that six times more soldiers were looking at the back of their neighbor than in the face. Then, on the command "about face," everyone turned in the opposite direction. Now, seven times more soldiers were looking at the back of their neighbor than in the face. How many soldiers are in the row?

$[8$ points] (A. V. Shipovalov)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
1. (YUG) Find all real numbers $x \in[0,2 \pi]$ such that
$$
2 \cos x \leq|\sqrt{1+\sin 2 x}-\sqrt{1-\sin 2 x}| \leq \sqrt{2} \text {. }
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi/4\leqx\leq7\pi/4}
How much work is required to slam a solid wooden door with a mass of $24 \mathrm{~kg}$ so that its edge hits the door frame at a speed of $5 \mathrm{~m} / \mathrm{s}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100 \, \mathrm{J}}
What is the minimum value of the expression $$\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{a+b+d}{c}\right\rfloor+\left\lfloor\frac{a+d+c}{b}\right\rfloor+\left\lfloor\frac{d+b+c}{a}\right\rfloor$$ where $a, b, c,$ and $d$ are positive real numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
已知集合$A = \{ x \vert x = a _ { 0 } + a _ { 1 } \times 8 + a _ { 2 } \times 8 ^ { 2 } + a _ { 3 } \times 8 ^ { 3 } \}$，其中


$a _ { i } \in \{ 0 ,$,1,2,3,4,5,6,7}, i=0,1,2,3

且$a _ { 3 } \neq 0$，若正整数m,n∈A满足m+n=2018，且$m _ { . } > n$，则符合条件的正整数m有＿个（请用数字作答）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{497}
Two adjacent coins (neighboring by side) weigh \$9\$ grams each, and the remaining coins weigh \$10\$ grams each. What is the minimum number of weighings on a scale, which shows the total weight in grams, needed to identify these two coins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Karcsi has 10 identical balls: 5 are red, 3 are white, and 2 are green. He also has two boxes, one of which can hold 4 balls and the other can hold 6 balls. In how many ways can he distribute the balls between the two boxes? (The arrangement of balls within the boxes does not matter.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Let  $k$  be a positive integer. A sequence of integers  $a_1, a_2, \cdots$  is called  $k$ -pop if the following holds: for every  $n \in \mathbb{N}$ ,  $a_n$  is equal to the number of distinct elements in the set  $\{a_1, \cdots , a_{n+k} \}$ . Determine, as a function of  $k$ , how many  $k$ -pop sequences there are.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k}
There are 8 seats in a row. If each seat can only be occupied by 1 person, and 4 people are seated, the number of different ways to arrange the seating such that exactly two empty seats are adjacent is ______ (provide your answer as a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
In triangle $\triangle ABC$, $AB=AC=10cm$, $\angle A=60^{\circ}$, then $BC=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
An eager young acoustic engineering student is shadowing a seasoned consultant on a project to design a new concert hall. The student is tasked with calculating the optimal number of sound panels needed for a section of the hall to ensure the best acoustics. The hall section is rectangular and measures 60 feet in length and 40 feet in width. Each sound panel covers an area of 5 square feet. If the consultant advises that 20% of the total wall area should be covered with sound panels, how many sound panels does the student need to recommend for this section?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
The teacher invented a puzzle by replacing the example \(a + b = c\) with the addition of two natural numbers with their digits replaced by letters: identical digits are replaced by identical letters, and different digits by different letters. (For example, if \(a = 23\), \(b = 528\), then \(c = 551\), resulting in the puzzle \(A B + V A \Gamma = VVD\) precisely up to the choice of letters). It turned out that the original example can be uniquely restored from the given puzzle. Find the smallest possible value of the sum \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
设平面上有 \(n\) 个点，任取出三点，均能被单位圆覆盖，且存在三点不能被恰好 1 个单位圆覆盖。求可能覆盖这些点的最小圆半径。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The administrator accidentally mixed up the keys for 10 rooms. If each key can only open one room, what is the maximum number of attempts needed to match all keys to their corresponding rooms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Given a parabola $y^2=2px$ ($p>0$) with focus $F$, a circle is drawn with $F$ as the center and $p$ as the radius, intersecting the $y$-axis at points $A$ and $B$. Connect $F$ to $A$, intersecting the parabola at point $D$ (which lies on segment $FA$). Extend $FA$ to intersect the parabola's directrix at point $C$. If $|AD|=m$ and $m\in[1,2]$, determine the maximum value of $|FD|\cdot|CD|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Given a sequence $\{a_n\}$, where $a_1=1$ and $a_{n+1}=3a_n+1$ for $n\in\mathbb{N}^*$, let $b_n= \frac{n}{(3^n-1)\cdot 2^{n-2}}\cdot a_n$, and let $T_n$ be the sum of the first $n$ terms of $\{b_n\}$. If the inequality $2^n\lambda < 2^{n-1}T_n+n$ holds for all $n\in\mathbb{N}_+$, then the range of the real number $\lambda$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty,1)}
To participate in the school club tour event, the public welfare club is preparing to purchase watercolor pens and signature pens for $105$ yuan. It is known that the unit prices of watercolor pens and signature pens are $5$ yuan per box and $7$ yuan per box, respectively. Therefore, the club can buy a maximum of ____ boxes of these two types of pens.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Each edge of a regular tetrahedron is divided into three equal parts. Through each point of division, two planes are drawn, each parallel to one of the two faces of the tetrahedron that do not pass through this point. Into how many parts do the constructed planes divide the tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Find the value of the real number $a$ if the constant term in the expansion of $({(\sqrt[6]{x}-\frac{a}{\sqrt{x}})}^{8})$ is $56$. Express your answer as a real number, including the \pm symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm\sqrt{2}}
He puts them into three boxes, a red one, a white one, and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn. How many ways are there to put the cards in the three boxes so that the trick works?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given two discrete distributions, $X$ with probability mass function $P_X(x) = \frac{1}{3}$ for $x = 1, 2, 3$, and $Y$ with probability mass function $P_Y(y) = \frac{1}{y}$ for $y = 2, 3, 6$, calculate the probability $P(X_1X_2 > Y_1 + Y_2)$, where $X_1$ and $X_2$ are independent samples from $X$, and $Y_1$ and $Y_2$ are independent samples from $Y$. Can this probability be determined algebraically?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{73}{324}}
Evaluate the limit of the double integral as the radius of the circular region approaches 0, given that $f$ is a differentiable function such that $f(0) = 0$:
$$\lim_{r\to 0^{+}} \iint_{x^2 + y^2 \leq r^2} \frac{3}{2\pi r^3} f(\sqrt{x^2+y^2}) \, dx \, dy$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f'(0)}
Determine the number of distinct parse trees for the expression $a+b*c-d$ using the following grammar:

$$\begin{align}
E &\to E + T \mid E - T \mid T \\
T &\to T * F \mid T / F \mid F \\
F &\to (E) \mid id
\end{align}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a regular pentagon \(ABCDE\), point \(K\) is on side \(AE\) such that the ratio \(AK : KE = 3 : 7\), and point \(L\) is on side \(CD\). Given that \(\angle LAE + \angle KCD = 108^\circ\), determine the ratio \(CL : AB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{10}}
Calculate the volume of the solid formed by rotating the region bounded by the graphs of the functions \( y = \arccos \frac{x}{3} \), \( y = \arccos x \), and \( y = 0 \) about the \( y \)-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi^2}
Determine the cardinality of the set of all epsilon numbers less than $\omega_1$. Provide your answer as a cardinal number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
Determine the degree of the line bundle over $\mathbb{P}^1$ with transition function $e^{1/z}$, where $z \neq 0, \infty$. What is the degree of this line bundle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: \[ \lim_{m\to\infty} \frac{\int_0^{\pi/2}(\sin x)^{2m}dx}{\int_0^{\pi/2}(\sin x)^{2m+1}dx} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by $2$ and add $1$ or he may multiply the stored value by $4$ and add $3$. The first player to make the stored value exceed $2^{100}$ wins. The stored value starts at $1$ and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
In the quadrilateral \(ABCD\), it is known that \(AB = BD\), \(\angle ABD = \angle DBC\), and \(\angle BCD = 90^\circ\). On the segment \(BC\), there is a point \(E\) such that \(AD = DE\). What is the length of segment \(BD\) if it is known that \(BE = 7\) and \(EC = 5\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Given the input signal $x(t) = \cos(t)$ and impulse response $h(t) = u(t)$, find the output $y(t)$ using Fourier transforms. Express the final time-domain solution $y(t)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin(t)}
When dividing the polynomial \(x^{1951} - 1\) by \(x^4 + x^3 + 2x^2 + x + 1\), the quotient and remainder are obtained. Find the coefficient of \(x^{14}\) in the quotient.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Find a simple expression or a good upper bound for the expected maximum amount of money found, \(E[\max(x_1, \ldots, x_n)]\), when \(n\) individuals find street money. Each person has a probability of \(\dfrac{1}{2^x}\) to find \(x\) dollars (\(1 \leq x \leq n\)), with probabilities being independent. It is known that the expected value of money found by each person is less than 2 dollars. Initial observations suggest it might be around \(\log(n)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(\log n)}
A PhD entrance question of TIFR 2020 asks to find the integer part of the sum $\sum\limits_{n=1}^{9999}\frac 1{n^{\frac 14}}$. Can you help me find the answer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1332}
In base $-2$ notation, digits are $0$ and $1$ only, and the places increase in powers of $-2$. For example, $11011$ stands for $(-2)^4 + (-2)^3 + (-2)^1 + (-2)^0$ and equals the number $7$ in base $10$.

If the decimal number $2019$ is expressed in base $-2$, how many non-zero digits does it contain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. $M$ is the midpoint of $BC$ with $BM=MC=2$, and $AM=b-c$. Find the maximum area of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
In Mr. Jacob's class, $12$ of the $20$ students received a 'B' on the latest exam. If the same proportion of students received a 'B' in Mrs. Cecilia's latest exam, and Mrs. Cecilia originally had $30$ students, but $6$ were absent during the exam, how many students present for Mrs. Cecilia’s exam received a 'B'?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Find the coefficient of \(x^{33}\) in the expansion of \((x+x^3)(1+5x^6)^{-13}(1-8x^9)^{-37}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83019300}