inclusionAI/Ring-lite-rl-data huggingface dataset:
A traffic light runs repeatedly through the following cycle: green for 30 seconds, then yellow for 3 seconds, and then red for 30 seconds.  Leah picks a random three-second time interval to watch the light.  What is the probability that the color changes while she is watching?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{7}}
In tetrahedron $ABCD$, $AD > AB$, $AD \perp AB$, $AD \perp AC$, $\angle BAC = \frac{\pi}{3}$. Let the areas of triangles $ADB$, $ADC$, $ABC$, and $BCD$ be $S_{1}$, $S_{2}$, $S_{3}$, and $S_{4}$, respectively. It is known that $S_{1} + S_{2} = S_{3} + S_{4}$. Find the value of $\frac{S_{3}}{S_{1}} + \frac{S_{3}}{S_{2}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/2}
On an $xy$ plane, how many $2n$-move paths can be made that begin and end at $(0,0)$ but pass through $(k,n-k)$ with $0\leq k\leq n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\binom{n}{k}^2\)}
Given that the solution set of the inequality $(ax-a^{2}-4)(x-4) \gt 0$ with respect to $x$ is $A$, and $A$ contains a total of $n$ integers, the minimum value of the real number $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
The graph of the power function $y=f(x)$ passes through the point $(-2, -\frac{1}{8})$. Find the value of $x$ that satisfies $f(x)=27$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
In the rhombus \(ABCD\), point \(Q\) divides side \(BC\) in the ratio \(1:3\) starting from vertex \(B\), and point \(E\) is the midpoint of side \(AB\). It is known that the median \(CF\) of triangle \(CEQ\) is equal to \(2\sqrt{2}\), and \(EQ = \sqrt{2}\). Find the radius of the circle inscribed in rhombus \(ABCD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{7}}{2}}
A sequence $\{a_n\}$ is defined such that $a_i = i$ for $i = 1, 2, 3, \ldots, 2020$ and for $i > 2020$, $a_i$ is the average of the previous 2020 terms. What is the largest integer less than or equal to $\displaystyle\lim_{n \to \infty} a_n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1347}
Given natural numbers \( k \) and \( n \) such that \( 1 < k \leq n \). What is the smallest \( m \) for which the following statement holds: for any placement of \( m \) rooks on an \( n \times n \) board, it is always possible to select \( k \) rooks from these \( m \) such that no two of the selected rooks can attack each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(k-1) + 1}
At the base of the pyramid $S A B C$ lies an equilateral triangle $A B C$, and all lateral faces have equal areas. The edge $S A$ is equal to 2, and the edge $S B$ is equal to $\sqrt{2}$. A section of the pyramid is made through vertex $B$ perpendicular to the edge $S C$. Find the area of this section.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{6}}{40}}
The sequence \(\{a_n\}\) is defined as follows: \[ a_n = \sqrt{1 + \left(1 + \frac{1}{n} \right)^2} + \sqrt{1 + \left(1 - \frac{1}{n} \right)^2}. \] Calculate the value of the expression given below: \[ \frac{4}{a_1} + \frac{4}{a_2} + \dots + \frac{4}{a_{96}}. \] The original answer is in the form \(\sqrt{m} - 1\). Please provide the sum of the integer part of \(\sqrt{m}\) and 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{137}
In a country with 15 cities, some of which are connected by airlines belonging to three different companies, it is known that even if any one of the airlines ceases operations, it will still be possible to travel between any two cities (possibly with transfers) using the remaining two companies' flights. What is the minimum number of airline routes in the country?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Natural numbers $1, 2, 3, \cdots, 2014, 2015, 2016$ are arranged in a circle in clockwise order.
Starting from number 1, we perform the following operations in clockwise order.

Step 1: Cross out 1 and keep 2.

Step 2: Cross out 3 and 4, and keep 5.

Step 3: Cross out 6, 7, and 8, and keep 9.

Step 4: Cross out 10, 11, 12, and 13, and keep 14;

and so on: in the nth step, first cross out n numbers, then keep 1 number. Repeat these operations until all numbers are crossed out.

What is the last number to be crossed out?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
Given the universal set $U=\{a^{2}-2,2,1\}$ and set $A=\{a,1\}$, find the complement of $A$ with respect to $U$, denoted as $∁\_U A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{2\}}
Find the anti-derivative of $\frac{e^{2x}-1}{e^{2x}+1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln \cosh x+C}
Let point \( P \) lie on the face \( ABC \) of a tetrahedron \( ABCD \) with edge length 2. The distances from \( P \) to the planes \( DAB \), \( DBC \), and \( DCA \) form an arithmetic sequence. Find the distance from \( P \) to the plane \( DBC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{6}}{9}}
Given a function $$ f(n) = f(5n/13) + f(12n/13) + n \;\;\;\;∀n \geq 0 $$ I would like to find a function $g(n)$ such that $f ∈ Ө(g(n))$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$, and $C$ are sets for which 
\[n(A) + n(B) + n(C) = n(A \cup B \cup C)\] 
and 
\[|A| = |B| = 100,\] 
then find the minimum possible value of $|A \cap B \cap C|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{97}
Find the smallest positive integer \( n \) that satisfies the following condition: For any finite set of points \( A \) in the plane, if any \( n \) points in \( A \) can always be covered by two lines, then there exist two lines that can cover all the points in \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate $$\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
5. In a convex quadrilateral $A B C D$, the areas of triangles $A B D$ and $B C D$ are equal, and the area of $A C D$ is half the area of $A B D$. Find the length of the segment $C M$, where $M$ is the midpoint of side $A B$, if it is known that $A D=12$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
The supermarket sold two types of goods, both for a total of 660 yuan. One item made a profit of 10%, while the other suffered a loss of 10%. Express the original total price of these two items using a formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1333\frac{1}{3}}
7.1. Two wheels of radii $r_{1}$ and $r_{2}$ roll along a straight line $l$. Find the set of points of intersection $M$ of their common internal tangents.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2r_{1}r_{2}}{r_{1}+r_{2}}}
Find the smallest positive integer  $n$  for which one can select  $n$  distinct real numbers such that each of them is equal to the sum of some two other selected numbers.

*Proposed by Anton Trygub*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
On Thursday, January 1, 2015, Anna buys one book and one shelf. For the next two years, she buys one book every day and one shelf on alternate Thursdays, so she next buys a shelf on January 15th. On how many days in the period from Thursday, January 1, 2015, until (and including) Saturday, December 31, 2016, is it possible for Anna to arrange all her books on all her shelves such that there is an equal number of books on each shelf?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
There are 4 soap bars shaped like rectangular prisms, with lengths of 10 cm, widths of 5 cm, and heights of 3 cm. If these 4 soap bars are placed in a packaging box, what is the minimum amount of material in square centimeters needed for the box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{460}
In a triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpendicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the largest positive integer solution of the equation $\left\lfloor\frac{N}{3}\right\rfloor=\left\lfloor\frac{N}{5}\right\rfloor+\left\lfloor\frac{N}{7}\right\rfloor-\left\lfloor\frac{N}{35}\right\rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
A chemical plant plans to improve the technology of a chemical product and now decides to optimize the processing temperature. The experimental range is set to 60～81℃, with an accuracy requirement of ±1℃. If the technicians are preparing to use the fractional method for optimization, the maximum number of experiments needed to find the optimal temperature is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Vasya has $n$ candies of several types, where $n > 145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Embed $R^2$ in the projective plane $RP^2$ by the map $(x, y) → [1, x, y]$. Find the point of intersection in $RP^2$ of the projective lines corresponding to the parallel lines $y = mx$ and $y = mx + c$ in $R^2$ (where $c \neq 0$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,1,m]}
How many Carmichael numbers are there less than 100,000?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Mouse Jerry decided to give cat Tom a birthday pie in the shape of an $8 \times 8$ square. He put fish in three pieces marked with the letter "P", sausage in two pieces marked with the letter "K", and in one piece added both fish and sausage, but did not mark this piece (all other pieces are without filling). Jerry also told Tom that in any $6 \times 6$ square there are at least 2 pieces with fish, and in any $3 \times 3$ square at most one piece with sausage.

What is the minimum number of pieces of pie Tom needs to eat to ensure he gets a piece with both fish and sausage?

|  |  |  |  |  |  |  |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
|  | P |  |  |  |  |  |  |
|  | K |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
|  |  |  |  | P | K |  |  |
|  |  |  |  |  |  |  |  |
|  | P |  |  |  |  |  |  |
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given $m\in R$, find the range of values for $m$ such that the maximum value of the function $f(x)=|x^{2}-4x+9-2m|+2m$ on the interval $[0,4]$ is $9$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, \frac{7}{2}]}
Rectangle $ABCD_{}^{}$ has sides $\overline {AB}$ of length $4$ and $\overline {CB}$ of length $3.$ Divide $\overline {AB}$ into $168$ congruent segments with points $A=P_0, P_1, \cdots, P_{168}=B$ , and divide $\overline {CB}$ into $168$ congruent segments with points $C=Q_0, Q_1, \cdots, Q_{168}=B$ . For $1 \le k \le 167$ , draw the segments $\overline {P_kQ_k}$ . Repeat this construction on the sides $\overline {AD}$ and $\overline {CD}$ , and then draw the diagonal $\overline {AC}$ . Find the sum of the lengths of the $335$ parallel segments drawn.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{840}
Evaluate the integral $\int_0^\infty J_n(bx)\,dx$, where $J_n(x)$ is the $n$th order Bessel function of the first kind.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{b}$}
Given a hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0, b>0)$ with left and right foci $F_{1}$, $F_{2}$, and a point $P$ on $C$, where $\angle F_{1}PF_{2}=60^{\circ}$, find the eccentricity $e$ of $C$ when $\sin \angle PF_{2}F_{1}=3\sin \angle PF_{1}F_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{7}}{2}}
A teacher is making packed lunches for a field trip. Each student needs 2 sandwiches, and will travel in a group with 5 other students. There are a total of 5 groups. How many pieces of bread will the teacher need to make enough sandwiches for all the students?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given that the sequences $\{a_{n}\}$ and $\{b_{n}\}$ satisfy ${b_{n+1}}{a_n}+{b_n}{a_{n+1}}={({-2})^n}+1,{b_n}=\frac{{3+{{({-1})}^{n-1}}}}{2}({n∈{N^*}})$, and $a_{1}=2$, find $a_{2n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1 - {4}^{n}}{2}}
In the diagram, $\triangle ABC$ is right-angled. Side $AB$ is extended in each direction to points $D$ and $G$ such that $DA = AB = BG$. Similarly, $BC$ is extended to points $F$ and $K$ so that $FB = BC = CK$, and $AC$ is extended to points $E$ and $H$ so that $EA = AC = CH$. Find the ratio of the area of the hexagon $DEFGHK$ to the area of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13:1}
2. Let $x, y, z$ be positive real numbers, and $x y z + x + z = y$, find the maximum value of $P = \frac{2}{x^{2}+1} - \frac{2}{y^{2}+1} + \frac{3}{z^{2}+1}$. (1999 Vietnam Mathematical Olympiad)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{3}}
Ittymangnark and Kingnook are an Eskimo couple living in the most northern region of the Alaskan wilderness. Together, they live with their two children, Oomyapeck and Yurraknalik, and their two dogs, Ankaq and Nanuq. Every day Ittymangnark catches enough fish for the entire family and the pets to eat for the day, and they need to split the fish in a fair manner: Ittymangnark gets 3 fish, Kingnook gets 4 fish, Oomyapeck gets 1 fish, Yurraknalik gets 2 fish, Ankaq gets 1 fish, and Nanuq gets 3 fish. After they have split the fish, they give Oomyapeck all of the eyes, who eats 24 out of the eyes himself and gives 4 eyes to Yurraknelik. Additionally, Oomyapeck gives Ankaq 6 eyes and Nanuq 8 eyes. How many fish did Ittymangnark catch in total for the whole family and pets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
What is the molecular weight of a compound having 2 Al, 3 C, and 9 O
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{233.99}
Diana can either invest $20,\!000$ dollars for $4$ years with a simple interest rate of $6\%$ or an interest rate of $7\%$ which compounds quarterly. How many more dollars, rounded to the nearest dollar, would she get with the better interest rate than with the worse one?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1599 \text{ dollars}}
Acute-angled triangle \(ABC\) is inscribed in a circle with center at \(O\). The arcs are given as follows: \(\overarc{AB} = 120^\circ\) and \(\overarc{BC} = 72^\circ\). A point \(E\) is located on the minor arc \(AC\) such that \(OE\) is perpendicular to \(AC\). Determine the ratio of the magnitudes of angles \(\angle OBE\) and \(\angle BAC\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The volume of the triangular pyramid P-ABC is $8 \sqrt {3}$. All the vertices of the pyramid are on the surface of the sphere O. PC is perpendicular to the plane ABC, PC=4, and $∠CAB= \frac {π}{3}$. What is the minimum surface area of sphere O?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48π}
Warehouse A and Warehouse B originally stored whole bags of grain. If 90 bags are transferred from Warehouse A to Warehouse B, then the grain in Warehouse B will be twice that in Warehouse A. If a certain number of bags are transferred from Warehouse B to Warehouse A, then the grain in Warehouse A will be six times that in Warehouse B. What is the minimum number of bags originally stored in Warehouse A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153}
Bald Qiang said, "I will take a 45-minute nap, and when I wake up, it will be just time for afternoon work." However, he slept 20 minutes longer and woke up at 1:30 PM. Calculate the time at which Bald Qiang started his nap. Express your answer in 12-hour time format, including AM/PM designation (e.g., 12:00 PM).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12:25 PM}
求具有下述性质的最小正数 $c$ ：对任意整数 $n \geq 4$,以及集合 $A \subseteq\{1,2, \cdots, n\}$, 若 $|A|>c n$, 则存在函数 $f: A \rightarrow\{1,-1\}$, 满足  
$$  
\left|\sum_{a \in A} f(a) \cdot a\right| \leq 1.  
$$
The original answer is in \frac{m}{n} format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $\triangle ABC$ be a triangle, and let $l$ be the line passing through its incenter and centroid. Assume that $B$ and $C$ lie on the same side of $l$, and that the distance from $B$ to $l$ is twice the distance from $C$ to $l$. Suppose also that the length $BA$ is twice that of $CA$. If $\triangle ABC$ has integer side lengths and is as small as possible, what is $AB^2+BC^2+CA^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
For real numbers $x$ and $y$, let $M$ be the maximum value of the expression $x^4y + x^3y + x^2y + xy + xy^2 + xy^3 + xy^4$, subject to $x + y = 3$. Find $\lfloor M \rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
A circle passes through the vertices $A$ and $C$ of triangle $ABC$, intersecting side $AB$ at point $E$ and side $BC$ at point $F$. The angle $AEC$ is 5 times the angle $BAF$, and the angle $ABC$ is $72^{\circ}$. Find the radius of the circle if $AC=6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given a triangle with  $ \triangle ABC $ , with:  $ \angle C = 36^o$  and   $ \angle A = \angle B $ . Consider the points  $ D $  on  $ BC $ ,  $ E $  on  $ AD $ ,   $ F $  on  $ BE $ ,   $ G $  on  $ DF $  and   $ H $  on   $ EG $ , so that the rays  $ AD, BE, DF, EG, FH $  bisect the angles   $ A, B, D, E, F $  respectively. It is known that  $ FH = 1 $ . Calculate   $ AC$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11 + 5\sqrt{5}}{2}}
Given an arithmetic sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $2\vec{OC} = a_4\vec{OA} + a_8\vec{OB}$, and points A, B, C are not collinear (the line does not pass through point O), find $S_{11}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let \( a_{i} = 0,1 \) for \( i = 1, 2, \cdots, n \). Find the number of sequences \(\left(a_{1}, a_{2}, \cdots, a_{n}\right)\) that satisfy \( a_{1} \leqslant a_{2}, a_{2} \geqslant a_{3}, a_{3} \leqslant a_{4}, a_{4} \geqslant a_{5}, \cdots \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_{n+2}}
A fair six-sided die has faces numbered 1, 2, 3, 4, 5, 6. The die is rolled four times, and the results are a, b, c, d. What is the probability that one of the numbers in the set {a, a+b, a+b+c, a+b+c+d} equals 4? Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{343}{1296}}
Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have:
\[
  (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]
Compute the number of distinct possible values of $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
在锐角$\triangle ABC$中，三边a，b，c满足$a>b>c$，则$\sin B$的取值范围为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$( \frac { \sqrt { 2 } } { 2 } , 1 )$}
Suppose there are 100 cookies arranged in a circle, and 53 of them are chocolate chip, with the remainder being oatmeal. Pearl wants to choose a contiguous subsegment of exactly 67 cookies and wants this subsegment to have exactly \(k\) chocolate chip cookies. Find the sum of the \(k\) for which Pearl is guaranteed to succeed regardless of how the cookies are arranged.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
Let \( S = \{1, 2, 3, 4\} \), and consider a sequence \( a_1, a_2, \cdots, a_n \) of \( n \) terms where \( a_i \in S \) such that for any non-empty subset \( B \) of \( S \), there exists a subsequence of \( n \) adjacent terms that forms the set \( B \). Find the minimum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let a positive integer \( n \) be called a cubic square if there exist positive integers \( a, b \) with \( n = \gcd(a^2, b^3) \). Count the number of cubic squares between 1 and 100 inclusive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
The distances between the points on a line are given as $2, 4, 5, 7, 8, k, 13, 15, 17, 19$. Determine the value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $f(X, Y, Z) = X^5Y - XY^5 + Y^5Z - YZ^5 + Z^5X - ZX^5$.

Find \( \frac{f(2009, 2010, 2011) + f(2010, 2011, 2009) - f(2011, 2010, 2009)}{f(2009, 2010, 2011)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A circle of radius $t$ is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius $r=1+\sin \frac{\pi}{8}$.  Find $rt$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos^2 \frac{\pi}{8}}
Maddie has a paper ribbon of length \(36 \text{ cm}\). She divides it into four rectangles of different lengths. She draws two lines joining the centers of two adjacent rectangles. What is the sum of the lengths of the lines that she draws?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18 \text{ cm}}
In the "Turn Back Difference" game, two polynomials $-m$ and $-n$ are operated on in sequence as follows: after the first operation, three polynomials $-m$, $-n$, $-n+m$ are obtained; after the second operation, four polynomials $-m$, $-n$, $-n+m$, $m$ are obtained, and so on. The operation rule is: the polynomial added in each operation is the difference between the last polynomial obtained in the previous operation and the one before it. Calculate the sum of the polynomials obtained after the $2023$rd operation of this game. Express your answer as a simplified polynomial expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2n}