inclusionAI/Ring-lite-rl-data huggingface dataset:
the least number which when divided by 5 , 6 , 7 and 8 leaves a remainder 3 , but when divided by 9 leaves no remainder , is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1683}
Given that $b_1, b_2, b_3, b_4$ are positive integers, the polynomial $g(z)=(1-z)^{b_1}\left(1-z^2\right)^{b_2}\left(1-z^3\right)^{b_3}\left(1-z^4\right)^{b_4}$ simplifies to $1-2z$ when terms higher than the 4th degree are ignored. Additionally, $\alpha$ is the largest root of the polynomial $f(x)=x^3-b_4 x^2+b_2$. Find the remainder when $\left[\alpha^{1995}\right]$ is divided by 9, where $[x]$ denotes the greatest integer less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Andrew wants to write the numbers 1, 2, 3, 4, 5, 6, and 7 in the circles in the diagram so that the sum of the three numbers joined by each straight line is the same. What number should he write in the top circle? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A cat ate half of the smallest fish in 30 seconds. In total, it ate 5 fish and took a whole number of minutes to do so (the cat eats fish at a constant 'cell' rate). The picture shows all the fish the cat had. Which fish did the cat not eat?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The figure consists of five squares and two right-angled triangles. The areas of three of the squares are 5, 8, and 32, as shown. Find the area of the shaded square. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
A parallelogram has three of its vertices at $(-1,0)$, $(2,4)$ and $(2,-4)$.  What is the positive difference between the greatest possible perimeter and the least possible perimeter of the parallelogram?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A cross composed of two identical large squares and two identical small squares is placed inside an even larger square. Calculate the side length of the largest square in centimeters if the area of the cross is \( 810 \mathrm{~cm}^{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36} \, \text{cm}
In triangle \( \triangle ABC \), \( AM \) and \( CN \) are the medians to sides \( BC \) and \( AB \), respectively, intersecting at point \( O \). \( P \) is the midpoint of \( AC \), and \( MP \) intersects \( CV \) at \( Q \). If the area of \( \triangle OMQ \) is \( n \), calculate the area of \( \triangle ABC \) in terms of \( n \). Express your answer as a multiplier of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24n}
Given the odd function $f(x)$ is increasing on the interval $[3, 7]$ and its minimum value is 5, determine the nature of $f(x)$ and its minimum value on the interval $[-7, -3]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
Given that α is an angle in the second quadrant and |cos(α/3)| = -cos(α/3), determine the quadrant of α/3. Express your answer as the name of the quadrant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{second quadrant}
The vertex $A$ of parallelogram $A B C D$ is connected by segments with the midpoints of the sides $B C$ and $C D$. One of these segments turned out to be twice as long as the other. Determine whether angle $B A D$ is acute, right, or obtuse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Obtuse}}
It is easy to place the complete set of ships for the game "Battleship" on a $10 \times 10$ board (see illustration). What is the smallest square board on which this set can be placed? (Recall that according to the rules, ships must not touch each other, even at the corners.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7 \times 7}
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? Write your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Rectangle \(ABCD\) has length 9 and width 5. Diagonal \(AC\) is divided into 5 equal parts at \(W, X, Y\), and \(Z\). Determine the area of the shaded region.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let  $m\geq 4$  and  $n\geq 4$ . An integer is written on each cell of a  $m \times n$  board. If each cell has a number equal to the arithmetic mean of some pair of numbers written on its neighbouring cells, determine the maximum amount of distinct numbers that the board may have. 

Note: two neighbouring cells share a common side.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
地面上有 10 只小鸟在啄食,其中任意 5 只小鸟中至少有 4 只在同一圆上, 问有鸟最多的圆上最少有几只鸟?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{有鸟最多的圆周上至少有 9 只鸟.}
An electric kettle heats water from room temperature $T_{0}=20^{\circ} \mathrm{C}$ to a temperature of $T_{m}=100^{\circ} \mathrm{C}$ in $t=10$ minutes. How long will it take for all the water to boil away, $t_{1}$, if the kettle is not turned off and the automatic shut-off system is malfunctioning? The specific heat capacity of water is $c=4200$ J/kg·K, and the specific heat of vaporization of water is $L=2.3$ MJ/kg. Round the answer to the nearest whole number of minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68}
In the diagram, triangles \(ABC\) and \(CBD\) are isosceles. The perimeter of \(\triangle CBD\) is 19, the perimeter of \(\triangle ABC\) is 20, and the length of \(BD\) is 7. Find the length of \(AB\). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the largest $r$ such that $4$ balls each of radius $r$ can be packed into a regular tetrahedron with side length $1$. In a packing, each ball lies outside every other ball, and every ball lies inside the boundaries of the tetrahedron. If $r$ can be expressed in the form $\frac{\sqrt{a}+b}{c}$ where $a, b, c$ are integers such that $\gcd(b, c) = 1$, what is $a + b + c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Circle $X$ has a radius of $\pi$. Circle $Y$ has a circumference of $8 \pi$. Circle $Z$ has an area of $9 \pi$. List the circles in order from smallest to the largest radius.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$Z, X, Y$}
Find all integers \( k \) for which the values of the fraction

$$
\frac{2 k^{2}+k-8}{k-1}
$$

are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6, 2, 0, -4}
One notebook, 3 notepads, and 2 pens cost 98 rubles, while 3 notebooks and a notepad cost 36 rubles less than 5 pens. How much does each item cost if the notebook costs an even number of rubles? (Each of these items costs an integer number of rubles.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4, 22, 14}
What is the mass percentage of Al in Aluminum bromide?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10.11\%}
A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of  $50$  meters per second, while each of the spiders has a speed of  $r$  meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum  $c$  so that for any  $r<c,$   the fly can always avoid being caught?

*Author: Anderson Wang*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
On an 8 by 8 Battleship game board, a "patychok" figure is placed. What is the minimum number of shots required to ensure hitting at least one "patychok"? (7 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
For a positive integer  $n$  that is not a power of two, we define  $t(n)$  as the greatest odd divisor of  $n$  and  $r(n)$  as the smallest positive odd divisor of  $n$  unequal to  $1$ . Determine all positive integers  $n$  that are not a power of two and for which we have  $n = 3t(n) + 5r(n)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8p, 60, 100}
设k为给定的正整数，求最小正整数n，使得任意n个整数中总存在两个正整数，它们的和或差被2k整除．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k+2}
The numbers on opposite sides of a die total 7. Two dice are shown with some faces visible. Using the properties of dice, find the sum of the numbers on the unseen faces of the two dice. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In how many different ways can the name "Bolyai János" be read from the following diagram? The reading can only occur continuously, meaning starting from the letter B, moving one letter at a time diagonally to the right or left, until reaching the letter S.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
有20个队参加全国足球冠军决赛，为了使任何三个队中都有两个队相互比赛过，问最少要进行多少场比赛？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
In the diagram, each scale shows the total mass (weight) of the shapes on that scale. Using the information provided by the scales, calculate the mass (weight) of a $\triangle$. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The segment \( PQ \) is divided into several smaller segments. On each of them, a square is constructed (see illustration).

What is the length of the path along the arrows if the length of the segment \( PQ \) is 73? If necessary, round your 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{219}
A triangle is composed of circular cells arranged in  $5784$  rows: the first row has one cell, the second has two cells, and so on (see the picture). The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called **diagonal** if the two cells in it *aren't* in the same row. What is the minimum possible amount of diagonal pairs in the division?
An example division into pairs is depicted in the image.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2892}
Find three numbers in an arithmetic progression with a common difference of 2, such that the sum of their squares is a number consisting of four identical digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41, 43, 45}
What is the minimum number of squares into which a staircase of 15 steps can be cut (see the figure)? Cutting can only be done along the boundaries of the cells.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$ . Determine the maximum value of the sum of the six distances.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[ 5 + \sqrt{3} \]}
The quotient of the division of a three-digit number by the sum of its digits is 13, and the remainder is 15. Find all such three-digit numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106, 145, 184}
Does the following integral converge or diverge? 
$$\int_{0}^{\infty}\frac{e^{-at^2}}{\sqrt{t}}\prod_{i=1}^n\left|\frac{1}{\sqrt{t}}-r_i\right|^2 dt$$
Here $r_i\in\mathbb{R}$ and $a>0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{diverges}
For any finite set $X$, let $| X |$ denote the number of elements in $X$. Define
\[S_n = \sum | A \cap B | ,\]
where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\left\{ 1 , 2 , 3,  \cdots , n \right\}$ with $|A| = |B|$.
For example, $S_2 = 4$ because the sum is taken over the pairs of subsets
\[(A, B) \in \left\{ (\emptyset, \emptyset) , ( \{1\} , \{1\} ), ( \{1\} , \{2\} ) , ( \{2\} , \{1\} ) , ( \{2\} , \{2\} ) , ( \{1 , 2\} , \{1 , 2\} ) \right\} ,\]
giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4$.
Let $\frac{S_{2022}}{S_{2021}} = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by
1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{245}
The product
$$\left(\frac{1+1}{1^2+1}+\frac{1}{4}\right)\left(\frac{2+1}{2^2+1}+\frac{1}{4}\right)\left(\frac{3+1}{3^2+1}+\frac{1}{4}\right)\cdots\left(\frac{2022+1}{2022^2+1}+\frac{1}{4}\right)$$
can be written as $\frac{q}{2^r\cdot s}$, where $r$ is a positive integer, and $q$ and $s$ are relatively prime odd positive integers. Find $s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the molecular weight of 9 moles of a certain compound if the molecular weight is 972?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8748}
Let $n$ be a positive integer. Each of the numbers $1,2,3,\ldots,100$ is painted with one of $n$ colors in such a way that two distinct numbers with a sum divisible by $4$ are painted with different colors. Determine the smallest value of $n$ for which such a situation is possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
In the star-shaped figure below, if all side lengths are equal to $3$ and the three largest angles of the figure are $210$ degrees, its area can be expressed as $\frac{a \sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is square-free. Compute $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In three out of six circles in a diagram, the numbers 4, 14, and 6 are written. How many ways can natural numbers be placed in the remaining three circles so that the products of the triplets of numbers along each of the three sides of the triangular diagram are the same?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In the game of Guess the Card, two players each have a  $\frac{1}{2}$  chance of winning and there is exactly one winner. Sixteen competitors stand in a circle, numbered  $1,2,\dots,16$  clockwise. They participate in an  $4$ -round single-elimination tournament of Guess the Card. Each round, the referee randomly chooses one of the remaining players, and the players pair off going clockwise, starting from the chosen one; each pair then plays Guess the Card and the losers leave the circle. If the probability that players  $1$  and  $9$  face each other in the last round is  $\frac{m}{n}$  where  $m,n$  are positive integers, find  $100m+n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{164}
Each face of a cube is painted either red or blue, each with probability 1/2. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{5}{16}$}
The product of the proper positive integer factors of $n$ can be written as $n^{(ax+b)/c}$, where $x$ is the number of positive divisors $n$ has, $c$ is a positive integer, and the greatest common factor of the three integers $a$, $b$, and $c$ is $1$. What is $a+b+c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the adjoining figure, \( \triangle CDE \) is an equilateral triangle and \( \square ABCD \) and \( \square DEFG \) are squares. If the measure of \( \angle GDA \) is \( x^\circ \), what is the value of \( x \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Suppose that  $a$  and  $b$  are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of  $p$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
设 $a, b>0$ ，满足：关于 $x$ 的方程 $\sqrt{|x|}+\sqrt{|x+a|}=b$ 恰有三个不同的实数解 $x_{1}, x_{2}, x_{3}$, 且 $x_{1}<x_{2}<x_{3}=b$, 则 $a+b$ 的值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Compute the sum of the series:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{510}
Liquid $X$ does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$cm thick. A rectangular box measuring $6$cm by $3$cm by $12$cm is filled with liquid $X$. Its contents are poured onto a large body of water. What will be the radius, in centimeters, of the resulting circular film?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{2160}{\pi}}}
Determine the expected value ($\mathbb{E}[X]$) and variance of the number of times it is necessary to roll a dice until the result "1" occurs 4 times in a row.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1554}
In the diagram, $\triangle ABC$ is isosceles with $AB = AC$ and $BC = 30 \mathrm{~cm}$. Square $EFGH$, which has a side length of $12 \mathrm{~cm}$, is inscribed in $\triangle ABC$, as shown. The area of $\triangle AEF$, in $\mathrm{cm}^{2}$, is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Wenlu, Xander, Yasser, and Zoe make the following statements:
Wenlu: "Xander is lying."
Xander: "Yasser is lying."
Yasser: "Zoe is telling the truth."
Zoe: "Wenlu is telling the truth."
Determine the possible numbers of people telling the truth. Express your answer as a list or range of possible numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 3}
In the diagram, \(\angle AFC = 90^\circ\), \(D\) is on \(AC\), \(\angle EDC = 90^\circ\), \(CF = 21\), \(AF = 20\), and \(ED = 6\). Determine the total area of quadrilateral \(AFCE\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{297}
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}
A single burger is not enough to satisfy a guy's hunger. The five guys go to Five Guys' Restaurant, which has 20 different meals on the menu. Each meal costs a different integer dollar amount between $1 and $20. The five guys have \$20 to split between them, and they want to use all the money to order five different meals. How many sets of five meals can the guys choose?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The product underwent a price reduction from 25 yuan to 16 yuan. Calculate the average percentage reduction for each price reduction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
Given a parabola $y=ax^2+bx+c$ that opens downward and has its vertex at $(2, -3)$, determine the type of value (maximum or minimum) this point represents for the parabola and its numerical value. Express your answer in the format: 'The parabola has a [maximum/minimum] value of [numerical value]'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{The parabola has a maximum value of -3.}
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane that contains this edge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{4}}
A moth starts at vertex $A$ of a certain cube and is trying to get to vertex $B$, which is opposite $A$, in five or fewer "steps," where a step consists of traveling along an edge from one vertex to another. The moth will stop as soon as it reaches $B$. How many ways can the moth achieve its objective?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA=5$ and $CD=2$.  Point $E$ lies on side $CA$ such that $\angle DEF = 60^\circ$.  The area of triangle $DEF$ is $14\sqrt{3}$.  The two possible values of the length of side $AB$ are $p \pm q\sqrt{r}$, where $p$ and $q$ are rational, and $r$ is an integer not divisible by the square of a prime.  Find $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{989}
Point $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$, $P BC$, $P CD$, and $P DA$, respectively. Given that $P A+P B +P C +P D = 23\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$, the second largest of the lengths $O_1O_2$, $O_2O_3$, $O_3O_4$, $O_4O_1$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16902}
As shown in Figure 1, a cross-section of cube \(ABCDEFGH\) passes through vertices \(A\), \(C\), and a point \(K\) on edge \(EF\), and divides the cube into two parts with a volume ratio of 3:1. Find the value of \(\frac{EK}{KF}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
有9本书，其中4本是相同的数学书，3本是相同的语文书，2本是相同的外语书，1本是词典，从中取出4本有多少种不同的取法
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Wendy randomly chooses a positive integer less than or equal to $2020$. The probability that the digits in Wendy's number add up to $10$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{107}
The diagram shows a circle that has been divided into six sectors of different sizes. Two of the sectors are to be painted red, two of them are to be painted blue, and two of them are to be painted yellow. Any two sectors which share an edge are to be painted in different colors. In how many ways can the circle be painted?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Circle $B$ has radius $6\sqrt{7}$.  Circle $A$, centered at point $C$, has radius $\sqrt{7}$ and is contained in $B$.  Let $L$ be the locus of centers $C$ such that there exists a point $D$ on the boundary of $B$ with the following property: if the tangents from $D$ to circle $A$ intersect circle $B$ again at $X$ and $Y$, then $XY$ is also tangent to $A$.  Find the area contained by the boundary of $L$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168\pi}