inclusionAI/Ring-lite-rl-data huggingface dataset:
In the diagram, \(ABCD\) is a rectangle with \(AD = 13\), \(DE = 5\), and \(EA = 12\). Calculate the area of \(ABCD\). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Rhombus $ABCD$ is similar to rhombus $BFDE$. The area of rhombus $ABCD$ is $24$, and $\angle BAD = 60^\circ$. What is the area of rhombus $BFDE$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given a right-angled triangle, one of whose acute angles is $\alpha$. Find the ratio of the radii of the circumscribed and inscribed circles and determine for which value of $\alpha$ this ratio will be the smallest.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2} + 1}
Sean enters a classroom in the Memorial Hall and sees a 1 followed by 2020 0's on the blackboard. As he is early for class, he decides to go through the digits from right to left and independently erase the $n$th digit from the left with probability $\frac{n-1}{n}$. (In particular, the 1 is never erased.) Compute the expected value of the number formed from the remaining digits when viewed as a base-3 number. (For example, if the remaining number on the board is 1000 , then its value is 27 .)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{681751}
There are 48 students participating in three sports activities. Each student only participates in one activity, and the number of participants in each activity is different. Each number contains the digit "6". How many students are participating in each of the three sports activities?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6, 16, 26}
Six rhombi of side length 1 are arranged as shown. What is the perimeter of this figure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
As shown in Figure 1, in triangle \( \triangle ABC \), the circumcenter is \( O \) and the incenter is \( I \). It is given that \( OI \perp AI \) and \( AB = 10 \), \( AC = 18 \). Find the length of \( BC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
A segment of length $1$ is divided into four segments. Then there exists a quadrilateral with the four segments as sides if and only if each segment is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\text{less than} \frac{1}{2}$}
All positive integers whose binary representations (excluding leading zeroes) have at least as many 1's as 0's are put in increasing order. Compute the number of digits in the binary representation of the 200th number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
There are three bags. One bag contains three green candies and one red candy. One bag contains two green candies and two red candies. One bag contains one green candy and three red candies. A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. If the first candy had been green, the child randomly chooses one of the other two bags and randomly selects a second candy from that bag. If the first candy had been red, the child randomly selects a second candy from the same bag as the first candy. If the probability that the second candy is green is given by the fraction  $m/n$  in lowest terms, find  $m + n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{217}
Find all numbers  $x \in \mathbb Z$  for which the number
\[x^4 + x^3 + x^2 + x + 1\]
is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1, 0, 3}
Several letters A and several letters B were sitting on a pipe. After some A's fell and some B's disappeared, only three letters remained on the pipe, and the following dialogue occurred between them:

First letter: "I am the only letter like me here."

Second letter: "There are fewer than two A's here."

Third letter: "There is one B among us."

It turned out that each letter told the truth if it was an A, and lied if it was a B. Determine the identity of each letter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{B, A, B}
There are 800 marbles in a bag. Each marble is colored with one of 100 colors, and there are eight marbles of each color. Anna draws one marble at a time from the bag, without replacement, until she gets eight marbles of the same color, and then she immediately stops. Suppose Anna has not stopped after drawing 699 marbles. Compute the probability that she stops immediately after drawing the 700th marble.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{99}{101}}
The integers $1,2,4,5,6,9,10,11,13$ are to be placed in the circles and squares below with one number in each shape. Each integer must be used exactly once and the integer in each circle must be equal to the sum of the integers in the two neighbouring squares. If the integer $x$ is placed in the leftmost square and the integer $y$ is placed in the rightmost square, what is the largest possible value of $x+y$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In a sequence, there are several minus signs. Two players take turns converting one or two adjacent minus signs to plus signs. The player who converts the last minus sign wins. Who will win with optimal play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{First Player}
The school principal, the facilities manager, and the parent committee, not agreeing with each other, each bought a rug for the school auditorium measuring $10 \times 10$ meters. Upon reflection, they decided to place all three rugs as shown in the picture: the first rug $6 \times 8$ meters in one corner, the second rug $6 \times 6$ meters in the opposite corner, and the third rug $5 \times 7$ meters in one of the remaining corners. 

Find the area of the part of the auditorium covered with rugs in three layers (give the answer in square meters).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
菱形（大小形状不固定）的3个顶点依次在边长为1的正方形的边AB，BC，AD上，求菱形第4个顶点的轨迹所成图形之面积．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Two right triangles, $ABC$ and $ACD$, are joined as shown. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. What is the number of square units in the area of the fourth square?

Note that the diagram is not provided, but imagine it similarly to the reference where:
- $AB$ and $CD$ have squares on them (inside the triangles $ABC$ and $ACD$ respectively),
- $AC$ is a common hypotenuse for triangles $ABC$ and $ACD$,
- A square is drawn on $AD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let  $n_0$  be the product of the first  $25$  primes. Now, choose a random divisor  $n_1$  of  $n_0$ , where a choice  $n_1$  is taken with probability proportional to  $\phi(n_1)$ . ( $\phi(m)$  is the number of integers less than  $m$  which are relatively prime to  $m$ .) Given this  $n_1$ , we let  $n_2$  be a random divisor of  $n_1$ , again chosen with probability proportional to  $\phi(n_2)$ . Compute the probability that  $n_2\equiv0\pmod{2310}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{256}{5929}}
Let  $n$  points with integer coordinates be given in the  $xy$ -plane. What is the minimum value of  $n$  which will ensure that three of the points are the vertices of a triangel with integer (possibly, 0) area?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Consider all the real sequences  $x_0,x_1,\cdots,x_{100}$  satisfying the following two requirements:
(1) $x_0=0$ ;
(2)For any integer  $i,1\leq i\leq 100$ ,we have  $1\leq x_i-x_{i-1}\leq 2$ .
Find the greatest positive integer  $k\leq 100$ ,so that for any sequence  $x_0,x_1,\cdots,x_{100}$  like this,we have
\[x_k+x_{k+1}+\cdots+x_{100}\geq x_0+x_1+\cdots+x_{k-1}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
The rectangle $ABCD^{}_{}$ below has dimensions $AB^{}_{} = 12 \sqrt{3}$ and $BC^{}_{} = 13 \sqrt{3}$.  Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P^{}_{}$.  If triangle $ABP^{}_{}$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles.  Find the volume of this pyramid.

[asy] pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6); draw(B--C--P--D--C^^D--A); filldraw(A--P--B--cycle, gray, black); label("$A$", A, SE); label("$B$", B, NE); label("$C$", C, NW); label("$D$", D, SW); label("$P$", P, N); label("$13\sqrt{3}$", A--D, S); label("$12\sqrt{3}$", A--B, E);[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{594}
5. A semicircle of radius 1 is drawn inside a semicircle of radius 2, as shown in the diagram, where $O A=O B=2$.
A circle is drawn so that it touches each of the semicircles and their common diameter, as shown.
What is the radius of the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{9}}
Find all natural numbers such that if you add their smallest divisor greater than one to them, you get 30.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25, 27, 28}
The real numbers  $a_1,a_2,a_3$  and  $b{}$  are given. The equation \[(x-a_1)(x-a_2)(x-a_3)=b\]has three distinct real roots,  $c_1,c_2,c_3.$  Determine the roots of the equation \[(x+c_1)(x+c_2)(x+c_3)=b.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-a_1, -a_2, -a_3}
From the edge of a large square sheet, a small square was cut off as shown in the figure, resulting in the perimeter of the sheet increasing by 10%. By what percentage did the area of the sheet decrease?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\%}
As shown in the figure, there is a square wall with a side length of 50 meters. Two people, A and B, start from points A and C respectively, moving along the wall in a clockwise direction. It is known that A walks 5 meters per second and B walks 3 meters per second. Determine the minimum number of seconds it will take for both A and B to be on the same side of the square at the same time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Let  $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$ . Compute the following expression in terms of  $k$ : 
\[ E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}.  \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(k-2)^{2}(k+2)^{2}}{4k(k^{2}+4)}}
Find the largest positive integer  $n$  for which the inequality 
\[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\]
holds true for all  $a, b, c \in [0,1]$ . Here we make the convention  $\sqrt[1]{abc}=abc$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given is a positive integer  $k$ . There are  $n$  points chosen on a line, such the distance between any two adjacent points is the same. The points are colored in  $k$  colors. For each pair of monochromatic points such that there are no points of the same color between them, we record the distance between these two points. If all distances are distinct, find the largest possible  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3k - 1}
My club has 15 members.  In how many ways can we choose a president, vice-president, secretary, and treasurer, if either the secretary or the treasurer must be elected vice-president and no other member can hold more than one office?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5,\!460}
Evaluate the limit: $$\lim_{n\to\infty}\int^{\infty}_0n\ln\Big(1+\frac{e^{-x}}{n}\Big)dx.$$ Find the value of this limit using appropriate techniques.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A string consisting of letters A, C, G, and U is untranslatable if and only if it has no AUG as a consecutive substring. For example, ACUGG is untranslatable. Let \(a_{n}\) denote the number of untranslatable strings of length \(n\). It is given that there exists a unique triple of real numbers \((x, y, z)\) such that \(a_{n}=x a_{n-1}+y a_{n-2}+z a_{n-3}\) for all integers \(n \geq 100\). Compute \((x, y, z)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(4,0,-1)}
A bowl contained $10\%$ blue candies and $25\%$ red candies.  A bag containing three quarters red candies and one quarter blue candies was added to the bowl.  Now the bowl is $16\%$ blue candies.  What percentage of the candies in the bowl are now red?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45\%}
Maria must work 8 hours each day, excluding her 45-minute lunch break. If she begins working at 7:25 A.M. and takes her lunch break at noon, what time will her working day end? Express your answer in 12-hour clock format (e.g., 4:10 P.M.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:10 P.M.}
Sophie wants to complete the grid shown so that each row and each column of the grid contains the digits 1, 2, and 3 exactly once. Calculate the sum of the digits she will write in the shaded cells. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a class, there are \( n \) boys and \( n \) girls (\( n \geq 3 \)). They sat around a round table such that no two boys and no two girls sit next to each other. The teacher has \( 2n \) cards with the numbers \( 1, 2, 3, \ldots, 2n \), each appearing exactly once. He distributes one card to each student in such a way that the number on any girl's card is greater than the number on any boy's card. Then each boy writes down the sum of the numbers on three cards: his own and those of the girls sitting next to him. For which values of \( n \) could all the \( n \) sums obtained be equal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \text{ is odd}}
Knot is on an epic quest to save the land of Hyruler from the evil Gammadorf. To do this, he must collect the two pieces of the Lineforce, then go to the Temple of Lime. As shown on the figure, Knot starts on point $K$, and must travel to point $T$, where $O K=2$ and $O T=4$. However, he must first reach both solid lines in the figure below to collect the pieces of the Lineforce. What is the minimal distance Knot must travel to do so?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{5}}
an article is sold at a certain price . by selling it at 2 / 3 of that price one loses 10 % . find the gain percent at original price .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35\%}
Find the minimum area of the part bounded by the parabola  $ y\equal{}a^3x^2\minus{}a^4x\ (a>0)$  and the line  $ y\equal{}x$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3}}
Given the sets $P={x|x^{2}+x-6=0}$ and $Q={x|ax+1=0}$ where $Q \subset P$, find all possible values of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \in {0, \frac{1}{3}, -\frac{1}{2}}}
Calculate the molecular weight of 4 moles of Calcium hydroxide, 2 moles of aluminum sulfate, and 3 moles of potassium nitrate combined in a single compound.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1284.07}
Jamie, Linda, and Don bought bundles of roses at a ﬂower shop, each paying the same price for each
bundle. Then Jamie, Linda, and Don took their bundles of roses to a fair where they tried selling their
bundles for a ﬁxed price which was higher than the price that the ﬂower shop charged. At the end of the
fair, Jamie, Linda, and Don donated their unsold bundles of roses to the fair organizers. Jamie had bought
20 bundles of roses, sold 15 bundles of roses, and made  $60$  proﬁt. Linda had bought 34 bundles of roses,
sold 24 bundles of roses, and made $69 proﬁt. Don had bought 40 bundles of roses and sold 36 bundles of
roses. How many dollars proﬁt did Don make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares. The measure of $\angle GDA$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^{\circ}}
There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let  $ N$  be the total number of intersection points of these circles. Determine the smallest possible values of  $ N$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3012}
Using only the paths and the directions shown, how many different routes are there from $\text{M}$ to $\text{N}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
If the two roots of the function $f(x)=x^{2}-ax-b$ are $2$ and $3$, then the roots of the function $g(x)=bx^{2}-ax-1$ are ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}, -\dfrac{1}{3}}
Given that \(A\) is a prime number less than 100, and the results of \(A+10\), \(A-20\), \(A+30\), \(A+60\), \(A+70\) are all prime numbers, find all possible values of \(A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37, 43, 79}
There is a polynomial $P$ such that for every real number $x$,  
\[  
  x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x).  
\]  
When $P$ is written in standard polynomial form, how many of its coefficients are nonzero?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{341}
Let $f_0(x)=x+|x-100|-|x+100|$, and for $n\geq 1$, let $f_n(x)=|f_{n-1}(x)|-1$. For how many values of $x$ is $f_{100}(x)=0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{301}
Tina is a professional boxer. She wins her first 10 fights of her career. She then goes on to win 5 more before losing her first fight, and then doubles her number of wins before losing again. She then retires. How many more wins than losses does she have at the end of her career?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Water boils at 212 °F or 100 °C and ice melts at 32 °F or 0 °C at standard atmospheric pressure (1 atm). The boiling and melting points of water can change depending on the atmospheric pressure. If the temperature of a pot of water is 55 °C and the atmospheric pressure is 0.8 atm, what is the temperature of the pot of water in °F, taking into account the effects of the modified atmospheric pressure on the boiling and melting points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{131}
Petya and Vasya are playing a game. Initially, there are 11 piles of 10 stones each on the table. The players take turns, starting with Petya. On each turn, a player takes 1, 2, or 3 stones. Petya always chooses to take all the stones from any one pile, while Vasya always takes all the stones from different piles (if there is more than one). The player who cannot make a move loses. Which player can ensure a win, regardless of how the opponent plays?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Vasya}
A point \( D \) is taken on the extension of side \( BC \) of triangle \( ABC \) such that line \( AD \) is tangent to the circumcircle \( \omega \) of triangle \( ABC \). Line \( AC \) intersects the circumcircle of triangle \( ABD \) at point \( E \), with \( AC:CE = 1:2 \). It turns out that the angle bisector of \( \angle ADE \) is tangent to circle \( \omega \). Find the angles of triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^{\circ}, 60^{\circ}, 90^{\circ}}
On the coordinate plane, find the area of the part enclosed by the curve  $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$  for  $a>0$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^2}{2} (4 - \pi)}
Determine the number of integers $2 \le n \le 2016$ such that $n^n-1$ is divisible by $2$, $3$, $5$, $7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
求函数$f ( a , b ) = ( a - b ) ^ { 2 } + ( \sqrt { 2 - a ^ { 2 } } - \frac { 9 } { b } ) ^ { 2 }$的最小值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$f(a, b)$ 有最小值 8 .}
Ava and Tiffany participate in a knockout tournament consisting of a total of 32 players. In each of 5 rounds, the remaining players are paired uniformly at random. In each pair, both players are equally likely to win, and the loser is knocked out of the tournament. The probability that Ava and Tiffany play each other during the tournament is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{116}
Three sectors of a circle are removed from a regular hexagon to form the shaded shape shown. Each sector has a perimeter of 18 mm. What is the perimeter, in mm, of the shaded shape formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 2013 such steps, let $D$ be the number of dice on the table. What is the expected value (average value) of $6^D$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10071}
The segment $AB$ with a length of $16 \mathrm{~cm}$ is the diameter of a circle with center $O$. A secant line intersects the circle at points $C$ and $D$ and the line $AB$ at point $P$, as shown in the figure. If $OD = DP$ and $\angle APC = 18^\circ$, what is the value of the angle $\angle AOC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54^\circ}
Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the diagram which is the same as the previous one. If $AB = 36$ units and $CD = 72$ units, what is the area of square $BCFE$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2592}
A non-negative integer  $n$  is said to be *squaredigital* if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, 1, 81}
Let $f(x) = (x - 5)(x - 12)$ and $g(x) = (x - 6)(x - 10)$. 
Find the sum of all integers $n$ such that $\frac{f(g(n))}{f(n)^2}$ is defined and an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Let  $A$  be the set  $\{k^{19}-k: 1<k<20, k\in N\}$ . Let  $G$  be the GCD of all elements of  $A$ .
Then the value of  $G$  is?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{798}