inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $ABCD$ be an isosceles trapezoid with $\overline{AD} \parallel \overline{BC}$. The incircle of $\triangle ABC$ has center $I$ and is tangent to $\overline{BC}$ at $P$. The incircle of $\triangle ABD$ has center $J$ and is tangent to $\overline{AD}$ at $Q$. If $PI = 8$, $IJ = 25$, and $JQ = 15$, compute the greatest integer less than or equal to the area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1728}
If $(x-1)^{2011} = a + a_1x + a_2x^2 + \ldots + a_{2010}x^{2010} + a_{2011}x^{2011}$, then find the value of $a_1 + a_2 + a_3 + \ldots + a_{2010} + a_{2011}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
There are 198 positive integers between a certain number and 2000 (both inclusive) that are multiples of 10. What is the lower limit of this range?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Jon’s textbooks weigh three times as much as Brandon’s textbooks. Jon has four textbooks with certain weights. Brandon's textbooks weigh 8 pounds. What are the weights of Jon's textbooks if their total weight is three times the weight of Brandon's textbooks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Determine the smallest positive integer \( n \) such that any coloring of the \( n \) vertices of a regular \( n \)-gon with three colors (red, yellow, and blue) results in four vertices of the same color forming an isosceles trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Given: Three vertices \( A, B, \) and \( C \) of a square are on the parabola \( y = x^2 \). 
Find: The minimum possible area of such a square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Raashan, Sylvia, and Ted play the following game. Each starts with $\textdollar1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $\textdollar1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $\textdollar1$? (For example, Raashan and Ted may each decide to give $\textdollar1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $\textdollar0$, Sylvia will have $\textdollar2$, and Ted will have $\textdollar1$, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $\textdollar1$ to, and the holdings will be the same at the end of the second round.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{4}$}
There are 3012 positive numbers with both their sum and the sum of their reciprocals equal to 3013. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{12052}{3013}}
On a straight line $\ell$ there are four points, $A$, $B$, $C$, and $D$ in that order, such that $AB = BC = CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is formed. Segments $EA$ and $ED$ are drawn, and a point $F$ is chosen so that when drawing the segments $FA$ and $FE$, an equilateral triangle $FAE$ is formed outside the triangle $EAD$. Finally, the lines $EB$ and $FA$ are drawn, which intersect at the point $G$. If the area of triangle $EBD$ is $10$, calculate the area of triangle $EFG$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given the inequality about $x$, $$\frac {x-a}{x^{2}-3x+2}$$≥0, the solution set is {x|1＜x≤a or x＞2}. Determine the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,2)}
Let $S$ be the set of integers of the form $2^x + 2^y + 2^z$, where $x$, $y$, $z$ are pairwise distinct non-negative integers. Determine the 100th smallest element of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{577}
The charge for a single room at hotel P for a 5-day stay is 20 percent less than the charge for a single room at hotel R and 10 percent less than the charge for a single room at hotel G for the same duration. Additionally, the price per night at each hotel varies based on the number of rooms booked. If a group books 5 rooms at hotel R, their price per room is 15 percent less than when they book only 1 room. The charge for a single room at hotel R, when 5 rooms are booked, is what percent greater than the charge for a single room at hotel G for a 5-day stay, when no discounts are applied for the number of rooms booked?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.375\%}
Inside the triangle $ABC$, a point $D$ is chosen such that $\angle BAD = 60^\circ$ and $\angle ABC = \angle BCD = 30^\circ$. It is given that $AB = 15$ and $CD = 8$. Find the length of segment $AD$. If necessary, round the answer to 0.01 or write the answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.5}
In triangle \( KLM \), with all sides distinct, the bisector of angle \( KLM \) intersects side \( KM \) at point \( N \). A line through point \( N \) intersects side \( LM \) at point \( A \), such that \( MN = AM \). Given that \( LN = a \) and \( KL + KN = b \), find \( AL \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^2}{b}}
Let $A=\{0,1\}$ and $B=\{0,1,2\}$. Calculate the total number of possible mappings from set A to set B, where each element in A corresponds to a unique element in B. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The sum of five consecutive numbers is 225 and the smallest of these numbers is a multiple of 7. What is the smallest number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Given that \(p, q, \frac{2q-1}{p}, \frac{2p-1}{q} \in \mathbf{Z}\), and \(p > 1\), \(q > 1\), find the value of \(p + q\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
For a sequence of distinct real numbers $b_1, b_2, \ldots, b_k$ in the interval $[1, 2023]$, what is the minimum value of $k$ such that there must exist $b_i$ and $b_j$ satisfying $0 < b_i - b_j < 1 + 3\sqrt[3]{b_i \cdot b_j}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
The Razorback shop makes $210 off each jersey and a certain amount off each t-shirt. During the Arkansas and Texas Tech game, they sold 177 t-shirts and 23 jerseys. A t-shirt costs $30 more than a jersey. How much does the shop make off each t-shirt?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
A 6-liter solution is 40% alcohol. How many liters of pure alcohol must be added to produce a solution that is 50% alcohol?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.2}
Sasha places rooks on an empty chessboard: the first one anywhere he wants, and each subsequent rook in such a way that it attacks an odd number of previously placed rooks. What is the maximum number of rooks he can place in this manner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
设集合 $A$ 中元素全为正整数, 且对任意 $x, y \in A(x \neq y)$, 有 $|x-y| \geqslant \frac{1}{25} x y$, 问 $A$ 中最多有几个数?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
I have four distinct rings that I want to wear on my right hand (five distinct fingers). One of these rings is a Canadian ring that must be worn on a finger by itself, and the rest can be arranged however I want. If I have two or more rings on the same finger, then I consider different orders of rings along the same finger to be different arrangements. How many different ways can I wear the rings on my fingers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
Hugo can fold a small box in 3 seconds and a medium one in twice that time. Tom can fold both the small and medium boxes in 4 seconds. They want to leave as early as possible, and it takes them 7200 seconds to fold a certain number of small boxes and 1800 medium boxes together. How many small boxes do they fold?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4200}
An eight-sided die, with faces numbered from 1 to 8, is tossed three times. Given that the sum of the first two tosses equals the third, calculate the probability that at least one "2" is tossed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{28}}
5% of people in a village in Sri Lanka died by bombardment, 15% of the remainder left the village on account of fear. The population is now reduced to 3294. What was the initial population of the village?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4080}
In a ten-mile race $\textit{First}$ beats $\textit{Second}$ by $2$ miles and $\textit{First}$ beats $\textit{Third}$ by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does $\textit{Second}$ beat $\textit{Third}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2\frac{1}{2}$}
A TV station once continuously played 5 different commercial advertisements during a certain time period. Now, it plans to only keep 2 of these commercial advertisements, add one new commercial advertisement, and two different public service announcements (PSAs). The two PSAs cannot be played consecutively nor can they be placed at the beginning or end of the sequence. Calculate the total number of different sequences that can be arranged.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Our third and final item comes to us from Germany, I mean Geometry. It is known that a regular $n$-gon can be constructed with straightedge and compass if $n$ is a prime that is 1 plus a power of 2. It is also possible to construct a $2 n$-gon whenever an $n$-gon is constructible, or a $p_{1} p_{2} \cdots p_{m}$-gon where the $p_{i}$ 's are distinct primes of the above form. What is really interesting is that these conditions, together with the fact that we can construct a square, is that they give us all constructible regular $n$-gons. What is the largest $n$ less than $4,300,000,000$ such that a regular $n$-gon is constructible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{32}}
The eight vertices of a cube are randomly labelled with the integers from 1 to 8 inclusive. Judith looks at the labels of the four vertices of one of the faces of the cube. She lists these four labels in increasing order. After doing this for all six faces, she gets the following six lists: $(1,2,5,8),(3,4,6,7),(2,4,5,7),(1,3,6,8),(2,3,7,8)$, and $(1,4,5,6)$. Identify the label of the vertex of the cube that is farthest away from the vertex labelled 2. Express your answer as a single integer representing the vertex label.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
It is given that $x = -2272$, $y = 10^3 + 10^2c + 10b + a$, and $z = 1$ satisfy the equation $ax + by + cz = 1$, where $a, b, c$ are positive integers with $a < b < c$. Find $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1987}
A retired worker receives an annual pension that is proportional to the square root of the number of years they served. If they serve $a$ additional years, their pension increases by $p$ dollars. If they serve $b$ additional years $(b \neq a)$, their pension increases by $q$ dollars. What is their annual pension (expressed in terms of $a$, $b$, $p$, and $q$)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a q^{2}-b p^{2}}{2(b p-a q)}}
In January, an air-conditioning unit costs $470. In February, there was a 12% discount on the unit due to a clearance sale. Following that, in March, after restocking, the price increased by 8% due to production cost adjustments. In April, a Spring promotion offered a 10% discount on all air-conditioning units. In June, another 5% price increase was applied due to increased demand during the summer months. In August, the company reduced the price by 7% during the mid-year sales event. In October, prices increased again by 6% in preparation for the holiday season. In November, the company offers a seasonal promotion with a 15% discount on the final price. How much will an air-conditioning unit cost in November considering all these changes throughout the year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$353.71}
What is the minimum width that an infinite strip of paper must have in order to cut out any triangle with an area of 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[4]{3}}
In the triangle $A_{1} A_{2} A_{3}$, the point $B_{i}$ is the quarter point closest to $A_{i}$ on the side $A_{i} A_{i+1}$, and the point $C_{i}$ is the quarter point closest to $A_{i+1}$ on the same side $\left(i=1,2,3, A_{4} \equiv A_{1}\right)$. What fraction of the area of triangle $A_{1} A_{2} A_{3}$ is the area of the common part of triangles $B_{1} B_{2} B_{3}$ and $C_{1} C_{2} C_{3}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{49}{160}}
In a candy store, a salesperson placed 91 candies of several kinds in a row on the counter. It turned out that between any two candies of the same kind, there is an even number of candies. What is the minimum number of kinds of candies that could be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
In a right-angled triangle $LMN$, suppose $\sin N = \frac{5}{13}$ with $LM = 10$. Calculate the length of $LN$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
After giving a discount of 15% on the initial selling price and then offering a further discount of rs. 55, the shopkeeper still gets a profit of 25%. If the cost price is rs. 220, find the markup percentage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76.47\%}
Find the Taylor series expansion of the function $x(t) = \frac{1}{1+\frac{1}{2}\Omega^2t^2}$ up to the term $t^4$, given the differential equation $\ddot x(t)=3\Omega^2 x^2(t) -4\Omega^2 x^3(t)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-\frac{\Omega^2}{2}\,t^2+\frac{\Omega^4}{8}\,t^4}
21. (1999 National Team Selection Test) For non-negative real numbers $x_{1}, x_{2}$, $\cdots, x_{n}$ satisfying the condition $x_{1}+x_{2}+\cdots+x_{n}=1$, find the maximum value of $\sum_{j=1}^{n}\left(x_{j}^{4}-x_{j}^{5}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{12}}
\frac{3}{2} + \frac{5}{4} + \frac{9}{8} + \frac{17}{16} + \frac{33}{32} + \frac{65}{64} - 7 =
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{64}}
I'm stuck with this expression while solving. Can you help me solve it?

$$\lim_{n \to \infty}n(\frac{\cos(\frac{2n+1}{n^2+n})\sin(\frac{1}{n^2+n})}{\cos(\frac{2n+3}{(n+1)^2+n+1})\sin(\frac{1}{(n+1)^2+n+1})}-1)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider a large square divided into a grid of \(5 \times 5\) smaller squares, each with side length \(1\) unit. A shaded region within the large square is formed by connecting the centers of four smaller squares, creating a smaller square inside. Calculate the ratio of the area of the shaded smaller square to the area of the large square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{25}}
A game show offers a contestant three prizes $A, B$ and $C$, each of which is worth a whole number of dollars from $＄$ $1$ to $＄$ $9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order $A, B, C$. As a hint, the digits of the three prices are given. On a particular day, the digits given were $1, 1, 1, 1, 3, 3, 3$ . Find the total number of possible guesses for all three prizes consistent with the hint.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
不等式$({{m}^{2}}-2m-3){{x2}}-(m-3)x-1<0$的解集为$R$，求实数$m$的取值范围
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\left( -\frac{1}{5} \right.,\left. 3 \right]$}
Given the line $x-my-1=0$ intersects the circle $(x-1)^{2}+(y-2)^{2}=4$ at points $A$ and $B$, and the length of chord $AB$ is $2\sqrt{3}$, find the value of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±\frac{\sqrt{3}}{3}}
10.301. Find the ratio of the area of a square inscribed in a segment with an arc of $180^{\circ}$ to the area of a square inscribed in a segment of the same circle with an arc of $90^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10:1}
Find the probability that the minimum of three independent exponential random variables with the same rate is less than or equal to the sum of the other two.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/4}
For any $x \in \mathbb{R}$, the inequality $|x+5| \geq m+2$ always holds. Find the range of values for the real number $m$ that satisfy this inequality. Express your answer as an inequality involving $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \leq -2}
Cards numbered 1 and 2 must be placed into the same envelope, and three cards are left to be placed into two remaining envelopes, find the total number of different methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let \( A = \{1, 2, \ldots, 2002\} \) and \( M = \{1001, 2003, 3005\} \). For any non-empty subset \( B \) of \( A \), \( B \) is called an \( M \)-free set if the sum of any two elements in \( B \) does not belong to \( M \). If \( A = A_1 \cup A_2 \), \( A_1 \cap A_2 = \varnothing \), and both \( A_1 \) and \( A_2 \) are \( M \)-free sets, then the ordered pair \(\left(A_{1}, A_{2}\right)\) is called an \( M \)-partition of \( A \). Find the number of all \( M \)-partitions of \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{501}}
A parallelogram $ABCD$ is inscribed in an ellipse $\frac{x^2}{4}+\frac{y^2}{2}=1$. The slope of line $AB$ is $k_1=1$. Calculate the slope of line $AD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
The bases \( AB \) and \( CD \) of trapezoid \( ABCD \) are 65 and 31, respectively, and its diagonals are mutually perpendicular. Find the dot product of vectors \( \overrightarrow{AD} \) and \( \overrightarrow{BC} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors and satisfy $\overrightarrow{a} \cdot \overrightarrow{b} = 0$, find the maximum value of $(\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}) \cdot (\overrightarrow{a} + \overrightarrow{c})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + \sqrt{5}}
In triangle $ABC$, the angle bisectors $AA_{1}$ and $BB_{1}$ intersect at point $O$. Find the ratio $AA_{1} : OA_{1}$ given $AB=6, BC=5$, and $CA=4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 : 1}
Sheena is sewing the bridesmaid's dresses for her sister's wedding. She can sew one dress in 15 hours. There are 7 bridesmaids in the wedding. Each bridesmaid's dress takes a different number of hours to sew due to different styles and sizes. The hours needed to sew the bridesmaid's dresses are as follows: 15 hours, 18 hours, 20 hours, 22 hours, 24 hours, 26 hours, and 28 hours. If Sheena sews the dresses for various hours each week as follows: 5 hours in the first week, 3 hours in the second week, 6 hours in the third week, and 4 hours in the fourth week, and then continues by repeating this cycle. Additionally, she needs to spend 10 hours to finalize the dresses by adding accessories and decorations. How many weeks will it take her to complete all the bridesmaid's dresses, including the additional tasks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Find an asymptotic approximation for the number of ways to partition a set of $2N$ distinct elements into $N$ pairs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt 2 (2N/e)^N}
Five children (Tian Tian, Shi Tou, Kimi, Cindy, Angela) need to switch fathers, with each child able to choose any one of the other four fathers except their own. Find the total number of different combinations of choices for the five children. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Define $K(n,0)=\varnothing $ and, for all nonnegative integers m and n, $K(n,m+1)=\left\{ \left. k \right|\text{ }1\le k\le n\text{ and }K(k,m)\cap K(n-k,m)=\varnothing  \right\}$.  Find the number of elements of $K(2004,2004)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{127}
设函数 \varphi_1(x),\varphi_2(x) 满足\varphi_i''(x)-\ln(x+1)\varphi_i'(x)+e^x\varphi_i(x)=0,i=1,2,x>-1.定义 w(x)=\varphi_1'(x)\varphi_2(x)-\varphi_1(x)\varphi_2'(x)，若 w(0)=1，则 [w(2)]=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
5. The volume of tetrahedron $ABCD$ is $\frac{1}{6}$, and it satisfies $\angle ACB=45^{\circ}, AD+BC+\frac{AC}{\sqrt{2}}=3$, then $CD=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Let \( N \) be the positive integers. The function \( f : N \to N \) satisfies \( f(1) = 5 \), \( f(f(n)) = 4n + 9 \), and \( f(2n) = 2n+1 + 3 \) for all \( n \). Find \( f(1789) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3581}
How many ways are there to fill in a $3 \times 3$ grid of cells with $0$'s and $2$'s, one number in each cell, such that each $2 \times 2$ contiguous subgrid contains exactly three $2$'s and one $0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
There are 99 children standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If a child receives two balls, one of the balls is irrevocably lost. What is the minimum amount of time after which only one ball can remain with the children?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
The principal of a certain school decided to take a photo of the graduating class of 2008. He arranged the students in parallel rows, all with the same number of students, but this arrangement was too wide for the field of view of his camera. To solve this problem, the principal decided to take one student from each row and place them in a new row. This arrangement displeased the principal because the new row had four students fewer than the other rows. He then decided to take one more student from each of the original rows and place them in the newly created row, and noticed that now all the rows had the same number of students, and finally took his photo. How many students appeared in the photo?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}