inclusionAI/Ring-lite-rl-data huggingface dataset:
Given that $\angle A = 30^\circ$ and $\angle AFG = \angle AGF$, calculate the sum of $\angle B + \angle D$ in the pentagon shown. Assume that $\angle BFD + \angle AFG = 180^\circ$, and that $\angle AFG + \angle AGF + \angle GAF = 180^\circ$ as in a triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75^\circ}
On a $5 \times 5$ board, pieces made up of $4$ squares are placed, as seen in the figure, each covering exactly $4$ squares of the board. The pieces can be rotated or turned over. They can also overlap, but they cannot protrude from the board. Suppose that each square on the board is covered by at most two pieces. Find the maximum number of squares on the board that can be covered (by one or two pieces).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A hyperbola has one of its foci at $(3, 2),$ and the vertex of the hyperbola closer to this focus is at $(4, 2).$ One of the asymptotes of the hyperbola has slope $\frac{\sqrt2}{2}.$ Find the $x-$coordinate of the center of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6+\sqrt6}
The radius of a semi-circle is increased by 20% from its original length of 6.4 cm. What would be the new perimeter of the semi-circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39.50}
Given a set of $n$ positive integers in which the difference between any two elements is either divisible by 5 or divisible by 25, find the maximum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
How many different (positive) prime numbers can be given such that the sum of any three of them is also a prime number? (Provide such primes!)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from twenty consecutive natural numbers to some number \( a \) is 4460, and the sum of the distances from these same twenty numbers to the number \( a^2 \) is 2755. Find all possible values of \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\{-\frac{37}{2}, \frac{39}{2}\right\}}
设正整数 $n\geqslant3,(a_1,a_2,\cdots,a_n)$是任意$n$元不同的实数，且其和是正数，若它的一个排列($b_1,b_2,\cdotp\cdotp\cdotp\cdotp,b_n$)满足对于任意的$k=1,2,\cdotp\cdotp\cdotp,n$,均有 $b_1+b_2+\cdotp\cdotp\cdotp+b_k>0$,则这个排列称为好的，求好的排列至少有多少个？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{好的排列至少有$( n - 1 ) !$个.}
For a positive integer  $n$ , let  $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$ . Find the smallest  $n$  such that  $\left|f_n^{\prime \prime}(0)\right|>2023$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
The numbers from 1 to 9 are placed in the cells of a 3x3 table so that the sum of the numbers on one diagonal is 7 and the sum on the other diagonal is 21. What is the sum of the numbers in the five shaded cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Fill the numbers $1, 2, 3, \ldots, 9, 10$ into 10 circles that form a rectangle. The sum of the numbers on each side of the rectangle must be equal. What is the maximum possible sum for each side?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $X_{1}, \dots, X_{n}$ be a random sample from $f(x \mid \theta) = \theta\cdot x^{\theta-1}$ for $0 < x< 1 ; \theta > 0$. Find a function of $\theta, g(\theta)$ for which there exists an unbiased estimator of $\theta$ whose variance attains the Cramèr-Rao lower bound.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\theta^{-1}}
We have a  $6 \times 6$  square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let $N = 30^{2015}$. Find the number of ordered 4-tuples of integers $(A, B, C, D) \in \{1, 2, \ldots, N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3 + Bn^2 + 2Cn + D$ is divisible by $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that one root of the equation $x^2 - 4ax + 3a^2 = 0$ is less than 1 and the other root is greater than 1, find the range of values for the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3} < a < 1}
Find the number of moles of HCl formed on combining 2 moles of CH4 and 4 moles of Cl2. How many moles of HCl are formed in the reaction?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
To obtain the graph of the function $y = \sqrt{2}\sin(2x - \frac{\pi}{4})$, determine the horizontal shift required to transform the graph of the function $y = \sin x + \cos x$. Express your answer in radians, specifying the direction of the shift (left or right).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4} \text{ units to the right}}
如果正整数n使得$2 n + 1$和$3 n + 1$都是完全平方数，试问$5 n + 3$能否是一个素数？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$5 n + 3 $是合数.}
In the polar coordinate system, find the equation of the circle with center at (1, 0) and radius 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \cos \theta}
求最小正整数n，使得任取n个不同的正整数中必存在8个不同的正整数，其和被8整除．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Find the range of negative values for \( a \) such that the inequality \( \sin^2 x + a \cos x + a^2 \geq 1 + \cos x \) holds for all \( x \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \leq -2}
A person plans to process 408 parts in 15 days. In the first three days, he processes 24 parts each day. Calculate the minimum number of parts he needs to process each day for the remaining days to complete the task ahead of schedule. Express your answer as an integer number of parts per day.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
A square with sides of length $1$ is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points on three of the sides, as shown. Find $x$, the length of the longer parallel side of each trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{6}}
A sequence of positive integers $a_1,a_2,\ldots $ is such that for each $m$ and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$, then $a_m$ is a divisor of $a_n$ and $a_m<a_n$. Find the least possible value of $a_{2000}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
Square  $ABCD$  has center  $O$ ,  $AB=900$ ,  $E$  and  $F$  are on  $AB$  with  $AE<BF$  and  $E$  between  $A$  and  $F$ ,  $m\angle EOF =45^\circ$ , and  $EF=400$ . Given that  $BF=p+q\sqrt{r}$ , wherer  $p,q,$  and  $r$  are positive integers and  $r$  is not divisible by the square of any prime, find  $p+q+r$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{307}
An elephant and a lion are currently 1 mile apart. The elephant runs directly away from the lion at 19 miles per hour, while the lion runs directly towards the elephant at 24 miles per hour.  How many minutes will it take for the lion to catch the elephant?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
There are 2001 coins on the table. Two players take turns playing the following game: on their turn, the first player can take any odd number of coins from 1 to 99, and the second player can take any even number of coins from 2 to 100, and so on. The player who cannot make a move loses. Who wins with optimal play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{First player wins}
A city's population of 1.75 million is projected to increase by 325% over the next 10 years. What is the expected population after this period?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.4375}
If M = {(x, y) | tan(πy) + sin^2(πx) = 0} and N = {(x, y) | x^2 + y^2 ≤ 2}, find the number of elements in the intersection of sets M and N, denoted as |M ∩ N|. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
As shown in the diagram, square ABCD and square EFGH have their corresponding sides parallel to each other. Line CG is extended to intersect with line BD at point I. Given that BD = 10, the area of triangle BFC is 3, and the area of triangle CHD is 5, what is the length of BI?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["15/4"]}
In the figure, \(ABCD\) is a square of side length \(1\). The rectangles \(JKHG\) and \(EBCF\) are congruent. The original answer is in the form \(k - \sqrt{m}\). Find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In the calculations shown, each letter stands for a digit. They are used to make some two-digit numbers. The two numbers on the left have a total of 79. What is the total of the four numbers on the right?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{158}
Team A was tasked with installing 66 air conditioners in Community A, and Team B was tasked with installing 60 air conditioners in Community B. Both teams started and finished their tasks at the same time. If Team A installs 2 more air conditioners per day than Team B, calculate the daily installation numbers of Team A and Team B. Express your answer as two whole numbers, representing the rates for Team A and Team B, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22, 20}
Given $$(x+a)^{2}(2x- \frac {1}{x})^{5}$$, if the expansion does not contain the term $x^3$, then $a= \_\_\_\_\_\_ ．$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm1}
The teacher fills some numbers into the circles in the diagram (each circle can and must contain only one number). The sum of the three numbers in each of the left and right closed loops is 30, and the sum of the four numbers in each of the top and bottom closed loops is 40. If the number in circle \(X\) is 9, what is the number in circle \(Y\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Returning from the Palace of Sports, Nina and her coach had the following dialogue:

- Do you have a brother?
- I have three brothers, - Nina replied.
- What are their ages?
- The product of the numbers representing their ages is 36, and the sum equals the number on that bus over there at the stop.

After looking at the bus number, the coach said that this information was not enough for him.

- The eldest one also loves sports, - added Nina.
- Now I know how old each of them is, - said the coach and accurately named each boy's age.

What are the ages of Nina's brothers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2, \; 2, \; 9}
In 2006, the revenues of an insurance company increased by 25% and the expenses increased by 15% compared to the previous year. The company's profit (revenue - expenses) increased by 40%. What percentage of the revenues were the expenses in 2006?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55.2}
Two trains of different lengths are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. The trains will be clear of each other in 21.998240140788738 seconds from the moment they meet. If the second train is 320 m long, how long is the first train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{119.96}
A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
A gambler plays the following coin-tossing game. He can bet an arbitrary positive amount of money. Then a fair coin is tossed, and the gambler wins or loses the amount he bet depending on the outcome. Our gambler, who starts playing with   $ x$  forints, where  $ 0<x<2C$ , uses the following strategy: if at a given time his capital is  $ y<C$ , he risks all of it; and if he has  $ y>C$ , he only bets  $ 2C\minus{}y$ . If he has exactly  $ 2C$  forints, he stops playing. Let  $ f(x)$  be the probability that he reaches  $ 2C$  (before going bankrupt). Determine the value of  $ f(x)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x}{2C}}
As shown in the figure, given that \( BE = 3EC \) and \( ED = 3AD \), and the area of triangle \( BDE \) is 18 square centimeters, find the area of triangle \( ABC \) in square centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Find the $z^{-2},z^{-1},..., z^2$ term of the Laurent series expansion for $f(z)=\frac{1}{\cos z}+\frac{1}{z^2}$ at $z=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 0, 1, 0, \(\frac{1}{2}\)}
Let $Q:=[0,2] \times [0,2]$ ; then how to evaluate ${\int\int}_Q\lfloor x+y\rfloor dxdy$ ? where $\lfloor . \rfloor$ denotes the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
从银行中取出伍角、壹元、贰元、伍元、拾元、伍拾元、壹百元的纸币共10张，共有多少种不同的取法？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8008}
Point  $A$  lies on the circumference of a circle  $\Omega$  with radius  $78$ .  Point  $B$  is placed such that  $AB$  is tangent to the circle and  $AB=65$ , while point  $C$  is located on  $\Omega$  such that  $BC=25$ .  Compute the length of  $\overline{AC}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
What is the greatest positive integer  $m$  such that  $ n^2(1+n^2-n^4)\equiv 1\pmod{2^m} $  for all odd integers  $n$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ m = 7 }
Can we find the expected value of the squared Euclidean distance of a point sampled uniformly from the standard simplex in $n$ dimensions using an alternative method that does not involve the Dirichlet distribution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{n+1}}
A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{2 \sqrt{3}}{3}$}
For the reaction: $2X + 3Y \rightarrow 3Z$, the combination of 2.00 moles of X with 2.00 moles of Y produces 1.75 moles of Z. Calculate the percent yield of this reaction. Express your answer as a percentage with one decimal place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{87.5%}
Suppose $A=\{1,2,\dots,2002\}$ and $M=\{1001,2003,3005\}$. $B$ is an non-empty subset of $A$. $B$ is called a $M$-free set if the sum of any two numbers in $B$ does not belong to $M$. If $A=A_1\cup A_2$, $A_1\cap A_2=\emptyset$ and $A_1,A_2$ are $M$-free sets, we call the ordered pair $(A_1,A_2)$ a $M$-partition of $A$. Find the number of $M$-partitions of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{501}}
For each  even positive integer $x$, let $g(x)$ denote the greatest power of 2 that  divides $x.$ For example, $g(20)=4$ and $g(16)=16.$ For each positive integer $n,$ let $S_n=\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than 1000 such that $S_n$ is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{899}
Anna and Brigitte, two little girls, are twins; they look identical. At school, they sit together at one desk; one always sits on the left, and the other on the right. Each of them claims to be Brigitte. The teacher knows from experience that, generally speaking, the girl on the right lies one time out of four, and the girl on the left lies one time out of five. But the teacher wants to know the probability that Anna is sitting on the left. Help her.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{7}}
As shown in the figure, the side length of the square $ABCD$ is $18$. Point $E$ is the midpoint of $CD$, and $ABFH$ is a rectangle. The areas of the two shaded triangles are equal. What is the area of the quadrilateral $AEFB$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
How many ways are there to make two  $3$ -digit numbers  $m$  and  $n$  such that  $n=3m$  and each of six digits  $1$ ,  $2$ ,  $3$ ,  $6$ ,  $7$ ,  $8$  are used exactly once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Moor made a lopsided ice cream cone. It turned out to be an oblique circular cone with the vertex directly above the perimeter of the base (see diagram below). The height and base radius are both of length  $1$ . Compute the radius of the largest spherical scoop of ice cream that it can hold such that at least  $50\%$  of the scoop’s volume lies inside the cone.

[center]<see attached>[/center]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5} - 1}{2}}
In the numbers between 10 and 40, exactly one number has two of the digits as a specific digit. What is that digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A cylindrical tank has a radius of 10 meters and a length of 30 meters. The tank is lying on its side on a flat surface and is filled with water to a depth of 5 meters. Water is added to the tank and the depth of the water increases from 5 meters to \(10 + 5\sqrt{2}\) meters. If the volume of water added to the tank, in cubic meters, can be written as \(a \pi + b + c \sqrt{p}\) for some integers \(a, b, c\) and prime number \(p\), determine the quadruple \((a, b, c, p)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1250, 1500, 750, 3)}
On a board, there are 10 natural numbers written, among which there may be equal ones, and the square of each of them divides the sum of all the others. What is the greatest number of different numbers that can be among the written ones?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given  $a$ ,  $b$ , and  $c$  are complex numbers satisfying

\[ a^2+ab+b^2=1+i \]
\[ b^2+bc+c^2=-2 \]
\[ c^2+ca+a^2=1, \]

compute  $(ab+bc+ca)^2$ . (Here,  $i=\sqrt{-1}$ )
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{-11 - 4i}{3}}
What are the last 8 digits of $$11 \times 101 \times 1001 \times 10001 \times 100001 \times 1000001 \times 111 ?$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19754321}
How many ordered pairs of positive real numbers (a,b) satisfy the equation $(1+2a)(2+2b)(2a+b)=32ab$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1$}
In triangle $ABC,$ $AB = 3$ and $AC = 5.$  Let $O$ be the circumcenter of triangle $ABC.$  Find $\overrightarrow{OA} \cdot \overrightarrow{BC}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8}
In the diagram, two pairs of identical isosceles triangles are cut off from square ABCD, leaving rectangle PQRS. The total area cut off is 200 m^2. Find the length of PR in meters. Express your answer as a whole number or exact decimal value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
8.4. Solve the inequality $\sqrt{x^{2}+x-56}-\sqrt{x^{2}+25 x+136}<8 \sqrt{\frac{x-7}{x+8}}$, and find the sum of its integer solutions that belong to the interval $[-25 ; 25]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{285}
The pattern in the series of figures consists of black, white, and gray squares. In which figure will there be 2112 gray squares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
In $\triangle{ABC}$ medians $\overline{AD}$ and $\overline{BE}$ intersect at $G$ and $\triangle{AGE}$ is equilateral. Then $\cos(C)$ can be written as $\frac{m\sqrt p}n$, where $m$ and $n$ are relatively prime positive integers and $p$ is a positive integer not divisible by the square of any prime. What is $m+n+p?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$44$}
Squirrels $A$, $B$, and $C$ have several pine cones in total. Initially, squirrel $A$ has 26 pine cones, and it takes 10 pine cones to evenly divide between $B$ and $C$. Then, squirrel $B$ takes 18 pine cones and evenly divides them between $A$ and $C$. Finally, squirrel $C$ divides half of its current pine cones evenly between $A$ and $B$. At this point, all three squirrels have the same number of pine cones. How many pine cones did squirrel $C$ originally have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86}
Suppose that  $S$  is a series of real numbers between  $2$  and  $8$  inclusive, and that for any two elements  $y > x$  in  $S,$    $$ 98y - 102x - xy \ge 4. $$  What is the maximum possible size for the set  $S?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
The diagram shows a rectangle ABCD in which AB = 1 metre and AD = 4 metres. The points E and G are the midpoints of AD and AB respectively, and the points F and H are the midpoints of AE and AG respectively. Find the area of the shaded rectangle. Express your answer in square metres (m^2) as a fraction or decimal value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Five cards labeled $1,3,5,7,9$ are laid in a row in that order, forming the five-digit number 13579 when read from left to right. A swap consists of picking two distinct cards, and then swapping them. After three swaps, the cards form a new five-digit number $n$ when read from left to right. Compute the expected value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50308}
Consider an  $n\times{n}$  grid formed by  $n^2$  unit squares. We define the centre of a unit square as the intersection of its diagonals.
Find the smallest integer  $m$  such that, choosing any  $m$  unit squares in the grid, we always get four unit squares among them whose centres are vertices of a parallelogram.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n}