inclusionAI/Ring-lite-rl-data huggingface dataset:
A shopkeeper fixes the marked price of an item at a certain percentage above its cost price. The percentage of discount allowed to gain 8% is 16.92307692307692%. What is the percentage above the cost price at which the marked price is fixed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30\%}
Given a sequence $\{a_{n}\}$ that satisfies the equation: ${a_{n+1}}+{({-1})^n}{a_n}=3n-1$ ($n∈{N^*}$), calculate the sum of the first $60$ terms of the sequence $\{a_{n}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2760}
In a 10×10 grid square, the centers of all unit squares are marked (a total of 100 points). What is the minimum number of lines, not parallel to the sides of the square, needed to go through all the marked points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
The flag of the United Nations consists of a polar map of the world, with the North Pole as center, extending approximately to $45^{\circ}$ South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude $30^{\circ} \mathrm{S}$. In the very close vicinity of this parallel, how much are East and West distances exaggerated as compared to North and South distances? The original answer is in the format $m\pi/n\sqrt{j}\sim2.42$, please provide the value of m + n + j.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In the spring round of the 2000 City Tournament, high school students in country $N$ were presented with six problems. Each problem was solved by exactly 1000 students, and no two students solved all six problems together. What is the smallest possible number of high school students in country $N$ who participated in the spring round?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
设实数 $x ,y$ 满足 $\frac{x^{2}}{5}+\frac{y^{2}}{4}=1$, 求 $\sqrt{x^{2}+y^{2}-2 y+1}+\sqrt{x^{2}+y^{2}-2 x+1}$ 的最小值。答案的形式为 $k\sqrt{m} - \sqrt{n}$，请给出k+m+n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given an integer  $n\ge 2$ , compute  $\sum_{\sigma} \textrm{sgn}(\sigma) n^{\ell(\sigma)}$ , where all  $n$ -element permutations are considered, and where  $\ell(\sigma)$  is the number of disjoint cycles in the standard decomposition of  $\sigma$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
In the finals of a beauty contest among giraffes, there were two finalists: the Tall one and the Spotted one. There are 135 voters divided into 5 districts, each district is divided into 9 precincts, and each precinct has 3 voters. Voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts is the winner; finally, the giraffe that wins in the majority of districts is declared the winner of the final. The Tall giraffe won. What is the minimum number of voters who could have voted for the Tall giraffe?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
The number of distinct even numbers that can be formed using the digits 0, 1, 2, and 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
# Task 4.

Over two days, 50 financiers raised funds to combat a new virus. Each of them made a one-time contribution of a whole number of thousands of rubles, not exceeding 100. Each contribution on the first day did not exceed 50 thousand, while on the second day, it was more than this amount; and no pair of the 50 contributions differed by exactly 50 thousand. What amount was raised?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2525}
Using the digits 0, 1, 2, 3, and 4, how many even numbers can be formed without repeating any digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{163}
A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to twice the radius of the circle. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{\pi}}
Let $g$ be a function satisfying $g(x^2y) = g(x)/y^2$ for all positive real numbers $x$ and $y$. If $g(800) = 4$, what is the value of $g(7200)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{81}}
An isosceles trapezoid \(ABCD\) is circumscribed around a circle. The lateral sides \(AB\) and \(CD\) are tangent to the circle at points \(M\) and \(N\), respectively, and \(K\) is the midpoint of \(AD\). In what ratio does the line \(BK\) divide the segment \(MN\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:3}
Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC = 2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle ABC$ at a point $D \neq A$. Find the length of segment $CD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3521}
John and Roger can finish the work in 24 days if they work together. They worked together for 16 days and then Roger left. After Roger left, a third person, Anna, helped John for 8 days before leaving as well. John finished the remaining work in another 8 days. In how many days can John alone finish the work?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
The sequence \(\left\{a_{n}\right\}_{n \geq 1}\) is defined by \(a_{n+2}=7 a_{n+1}-a_{n}\) for positive integers \(n\) with initial values \(a_{1}=1\) and \(a_{2}=8\). Another sequence, \(\left\{b_{n}\right\}\), is defined by the rule \(b_{n+2}=3 b_{n+1}-b_{n}\) for positive integers \(n\) together with the values \(b_{1}=1\) and \(b_{2}=2\). Find \(\operatorname{gcd}\left(a_{5000}, b_{501}\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
Trevor is a comic book illustrator. In the past three months, he has drawn 220 pages of the comic. The comic he illustrates releases a new issue once a month. The first month, Trevor drew at a speed of 5 pages per day, while in the second and third months, he drew at a speed of 4 pages per day. The first and second months' issues were the same number of pages, but the third month's issue was four pages longer than either of the others. Additionally, each month's issue contained 3 bonus pages created by another artist. How many pages was the first issue of the comic, excluding the bonus pages?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{69}
Let $N = 99999$. Then $N^3 = \ $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999970000299999}
Ten football teams played each other exactly once. As a result, each team ended up with exactly $x$ points.

What is the largest possible value of $x$? (A win earns 3 points, a draw earns 1 point, and a loss earns 0 points.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
What is the limit of the following expression as $r$ approaches infinity?

$$\lim_{r \to \infty} \frac{\int_0^{\pi/2}\left(\frac{x}{\pi/2}\right)^{r-1}\cos x\,dx}{\int_0^{\pi/2}\left(\frac{x}{\pi/2}\right)^r\cos{x}\,dx}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that the sum of the squares of two numbers $x$ and $y$ is 7, and their sum of cubes is 10, find the maximum value of $x+y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Sixteen dots are arranged in a 4x4 grid. The distance between any two dots is defined as the minimum number of horizontal and vertical steps along the grid lines needed to travel from one dot to the other. For instance, two adjacent dots are a distance of 1 apart, and two dots at opposite corners of the grid are a distance of 6 apart.

The problem is to find the mean distance between two distinct dots in the grid, expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine the value of $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Find the least upper bound for the set of values \((x_1 x_2 + 2x_2 x_3 + x_3 x_4) / (x_1^2 + x_2^2 + x_3^2 + x_4^2)\), where \(x_i\) are real numbers, not all zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}+1}{2}}
Given the sample 7, 8, 9, x, y has an average of 8 and a standard deviation of $\sqrt{2}$, the value of xy is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
In the city park, there are various attractions for kids and their parents. The entrance ticket to the park is $5, but for each attraction, you have to pay separately - such ticket costs $2 for kids and $4 for parents. A family with a certain number of children, their parents, and her grandmother paid $55 for visiting the park and one attraction inside. How many children were in the family?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Meiyangyang is using two types of decorations to decorate a "Christmas tree". The decorations are hung on each vertex of small triangles. Meiyangyang can decorate in any order. If two vertices of a small triangle already have the same decoration, then the third vertex will have the other decoration; otherwise, the third vertex will have the same decoration. How many different designs of the "Christmas tree" are there? (No rotation or flipping is allowed)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
If the real part and the imaginary part of both $\frac{z}{20}$ and $\frac{20}{z}$ are between 0 and 1, what is the area of the geometric figure formed by such complex numbers $z$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300 - 50\pi}
Let $ABCD$ be a parallelogram with an area of 160. The diagonals $AC$ and $BD$ intersect at point $E$. Point $P$ is on the line segment $\overline{AE}$ such that $EC = 4EP$. If the line $DP$ intersects $AB$ at point $F$, find the area of quadrilateral $BFPC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
将一个正方形分割成有限个白色和红色矩形，其边均与正方形的边平行. 在每个白色矩形内，写下它的宽与高的比值；在每个红色矩形内，写下它的高与宽的比值. 我们计算出所有矩形内写的数的总和 x. 如果两种颜色矩形的总面积相同，求 x 的最小可能值.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{7}
Define a sequence $a_i$ as follows: \( a_1 = 181 \) and for \( i \ge 2 \), \( a_i = a_{i-1}^2 - 1 \) if \( a_{i-1} \) is odd and \( a_i = \frac{a_{i-1}}{2} \) if \( a_{i-1} \) is even. Find the least \( i \) such that \( a_i = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Four, (50 points) Let $X$ be a 56-element set. Find the smallest positive integer $n$, such that for any 15 subsets of set $X$, if the union of any seven of them has at least $n$ elements, then there must exist three subsets among these 15 such that their intersection is non-empty.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
Given the function $f(x)=\sin (ωx+ \frac {π}{3})(ω > 0)$, the graph of the function $y=f(x)$ is shifted to the right by $\frac {2}{3}π$ units and overlaps with the original graph. What is the minimum value of $ω$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A sequence is defined recursively as follows: \( t_{1} = 1 \), and for \( n > 1 \):

- If \( n \) is even, \( t_{n} = 1 + t_{\frac{n}{2}} \).
- If \( n \) is odd, \( t_{n} = \frac{1}{t_{n-1}} \).

Given that \( t_{n} = \frac{19}{87} \), find the sum of the digits of \( n \).

(From the 38th American High School Mathematics Examination, 1987)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Among all positive integers \( n \) less than 2012, how many \( n \) are there such that \( 2^n - n^2 \) is divisible by 7?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{576}
A positive integer has exactly $81$ divisors, which are arranged in a $9 \times 9$ table. In this table, for any two numbers in the same row or column, one of them divides the other. Find the maximum possible number of distinct prime divisors of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $a$, $b \in \mathbb{R}$, $c \in [0,2\pi)$. If for any real number $x$, the equation $2\sin (3x- \frac {\pi}{3})=a\sin (bx+c)$ holds, determine the number of sets of parameters $(a, b, c)$ that satisfy this condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
At the beginning of the week, a toy car factory, Factory A, made 60 cars on Monday. Factory B, which is twice as efficient as Factory A, also started producing toy cars that day. From Tuesday to Friday, each factory doubled its production rate from the previous day, but on some days, defective products were found. 

On Tuesday, Factory A had a 5% defect rate, and on Thursday, Factory B had a 3% defect rate. There were no defective items on the other days. How many non-defective toy cars did the factories make in total during the week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5545}
Given the sequence ${a_n}$ satisfying: $a_1=1$, $a_n=a_{n-1}^2+2a_{n-1}$ for $n\geqslant 2$, and let $b_n= \frac {1}{a_{n+1}}+ \frac {1}{a_{n}+2}$ for $n\in\mathbb{N^*}$, find the sum of the first $n$ terms of the sequence ${b_n}$, $S_n=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1- \frac {1}{2^{2^n}-1}}
In the equilateral triangle \(ABC\), a circle is drawn passing through the center of the triangle and touching side \(BC\) at its midpoint \(D\). A line is drawn from point \(A\) tangent to the circle at point \(E\), such that \(\angle BAE < 30^{\circ}\). Find the area of triangle \(ABE\) if the area of triangle \(ABC\) is \(\frac{10}{4-\sqrt{2}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the maximum number of colors used in coloring integers  $n$  from  $49$  to  $94$  such that if  $a, b$  (not necessarily different) have the same color but  $c$  has a different color, then  $c$  does not divide  $a+b$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A rising number, such as $34689$, is a positive integer each digit of which is larger than each of the digits to its left. There are $\binom{9}{5} = 126$ five-digit rising numbers. When these numbers are arranged from smallest to largest, the $97^{\text{th}}$ number in the list does not contain the digit
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
If $f(n)$ is the sum of the digits of $n^2+1$ (where $n \in \mathbb{N}^*$). For example, since $14^2+1=197$, and $1+9+7=17$, thus $f(14)=17$. Let $f_1(n)=f(n)$, $f_2(n)=f(f_1(n))$, ..., $f_{k+1}(n)=f(f_k(n))$, where $k \in \mathbb{N}^*$, then $f_{2005}(8)=$  ．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let \( A_{1}B_{1}C_{1}, A_{2}B_{2}C_{2} \), and \( A_{3}B_{3}C_{3} \) be three triangles in the plane. For \( 1 \leq i \leq 3 \), let \( D_{i}, E_{i} \), and \( F_{i} \) be the midpoints of \( B_{i}C_{i}, A_{i}C_{i} \), and \( A_{i}B_{i} \), respectively. Furthermore, for \( 1 \leq i \leq 3 \) let \( G_{i} \) be the centroid of \( A_{i}B_{i}C_{i} \). Suppose that the areas of the triangles \( A_{1}A_{2}A_{3}, B_{1}B_{2}B_{3}, C_{1}C_{2}C_{3}, D_{1}D_{2}D_{3}, E_{1}E_{2}E_{3} \), and \( F_{1}F_{2}F_{3} \) are 2, 3, 4, 20, 21, and 2020, respectively. Compute the largest possible area of \( G_{1}G_{2}G_{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{917}
Consider functions $f$ that satisfy \[|f(x)-f(y)|\leq \frac{1}{2}|x-y|\] for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, find the greatest possible value of
\[f(f(800))-f(f(400))?\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
## Task 3 - 100733

Of the students in an 8th grade class, exactly 35 belong to the school choir and exactly 710 belong to the school sports club (SSG). Exactly 25 of the total number of students in this class are members of both the choir and the SSG.

Calculate what fraction of the total number of students in this class are neither in the choir nor in the SSG!
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{10}}
If the line $2ax - by + 2 = 0$ ($a, b \in \mathbb{R}$) always bisects the circumference of the circle $(x+1)^2 + (y-2)^2 = 4$, what is the maximum value of $ab$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
A rectangular grazing area is to be fenced off on three sides using part of a 100 meter rock wall as the fourth side. The area to be fenced is 36 m by 60 m. Fence posts are to be placed every 12 meters along the fence, including the two posts where the fence meets the rock wall. What is the minimum number of fence posts required to fence this area? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
There are \( N \geq 5 \) natural numbers written on the board. It is known that the sum of all the numbers is 80, and the sum of any five of them is not more than 19. What is the smallest possible value of \( N \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Determine an expression for the area of $\triangle QCA$ in terms of $p$, where the coordinates of point $A$ are now $(3, 12)$. [asy]
size(5cm);defaultpen(fontsize(9));
pair o = (0, 0); pair q = (0, 12); pair b = (12, 0);
pair a = (3, 12); pair t = (3, 0); pair c = (0, 9);

draw((-2, 0)--(15, 0), Arrow);
draw((0, -2)--(0, 15), Arrow);
draw(q--a--b);
draw(a--c--b);

label("$Q(0, 12)$", q, W);
label("$A(3, 12)$", a, NE);
label("$B(12, 0)$", b, S);
label("$O(0, 0)$", o, SW);
label("$x$", (15, 0), E);
label("$y$", (0, 15), N);
label("$C(0, p)$", c, W);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}(12-p)}
Given that the right focus of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ is $F(\sqrt{6},0)$, a line $l$ passing through $F$ intersects the ellipse at points $A$ and $B$. If the midpoint of chord $AB$ has coordinates $(\frac{\sqrt{6}}{3},-1)$, calculate the area of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["12\\sqrt{3}\\pi"]}
Find a formula for the following series:

$$ \sum_{i=1}^m {m \choose i} i! S(n,i)$$

Where $S(n,m)$ is the Stirling numbers of the second kind.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m^n}
Two diameters and one radius are drawn in a circle of radius 1, dividing the circle into 5 sectors. The largest possible area of the smallest sector can be expressed as \(\frac{a}{b} \pi\), where \(a, b\) are relatively prime positive integers. Compute \(100a + b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
A semicircle is inscribed in a triangle, with the semicircle touching the base and the diameter (with its ends on the lateral sides of the triangle) parallel to the base. Find the radius of the semicircle if the base of the triangle is $a$ and the height is $h$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{ah}{a + 2h} }
Solomiya wrote the numbers $1, 2, \ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from the board and write the sum $a+b$ instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
The Gnollish language consists of 3 words, "splargh", "glumph", and "amr". In a sentence, "splargh" cannot come directly before "glumph", and "amr" cannot come directly after "glumph". How many valid 3-word sentences are there in Gnollish?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let f(x) be an odd function defined on R with f(2) = 0, and xf''(x) + f(x) < 0 holds true for all x > 0. Determine the solution set of the inequality xf(x) > 0 in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 0) ∪ (0, 2)}
Find the area of the region enclosed by the curve $y=x^{2}-1$ and the lines $x=2$, $y=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3}}
Let $\mathbf{p}$ be the projection of vector $\mathbf{v}$ onto vector $\mathbf{u},$ and let $\mathbf{q}$ be the projection of $\mathbf{p}$ onto $\mathbf{u}.$ If $\frac{\|\mathbf{p}\|}{\|\mathbf{v}\|} = \frac{3}{4},$ then find $\frac{\|\mathbf{q}\|}{\|\mathbf{u}\|}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{16}}
Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_1}B^{n_2}A^{n_3}\cdots B^{n_k}$, where $n_{1},\cdots, n_k$ are any integers, and $A^0=B^0=1$ as usual. Also, suppose that $A^4=B^7= ABA^{-1}B=1$, $A^2\neq1$, and $B\neq1$. How many elements of $G$ are of the form $C^2$ with $C$ in $G$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The number of edges of a convex polyhedron is 99. What is the maximum number of edges a plane, not passing through its vertices, can intersect?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
510 athletes participate in an arm wrestling tournament. For a victory, 1 point is awarded, for a defeat - 0 points. If a winner initially had fewer points than the opponent, they additionally receive one point from the loser. In each round, participants meet whose number of points differs by no more than 1. The tournament ends as soon as a sole leader is determined. What is the minimum number of rounds that need to be held?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In an exam, there are 4 multiple-choice questions, each with 3 possible answers. A group of students takes the exam, and it is found that for any 3 students, there is at least 1 question for which their answers are all different. What is the maximum number of students that could have taken the exam?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given that the base- $17$  integer  $\overline{8323a02421_{17}}$  (where a is a base- $17$  digit) is divisible by  $\overline{16_{10}}$ , find  $a$ . Express your answer in base  $10$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given the ratio of length $AD$ to width $AB$ of the rectangle is $4:3$ and $AB$ is 40 inches, determine the ratio of the area of the rectangle to the combined area of the semicircles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{3\pi}}