inclusionAI/Ring-lite-rl-data huggingface dataset:
A rectangle with perimeter $176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Among the $n$ angles of a convex $n$-gon, $n-1$ angles are equal to $150^\circ$, and the remaining angle is less than $150^\circ$. For which $n$ is this possible? List all possible answers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8, 9, 10, 11}
5. Given the number $800 \ldots 008$ (80 zeros). It is required to replace some two zeros with non-zero digits so that after the replacement, the resulting number is divisible by 198. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14080}
An ideal gas is used as the working substance of a heat engine operating cyclically. The cycle consists of three stages: isochoric pressure reduction from $3 P_{0}$ to $P_{0}$, isobaric density increase from $\rho_{0}$ to $3 \rho_{0}$, and a return to the initial state, represented as a quarter circle in the $P / P_{0}, \rho / \rho_{0}$ coordinates with the center at point $(1,1)$. Determine the efficiency of this cycle, knowing that it is 8 times less than the maximum possible efficiency for the same minimum and maximum gas temperatures as in the given cycle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{9}}
There are four boxes of apples. By weighing three of the boxes at a time, the following four different weights are obtained: $70 \mathrm{~kg}, 80 \mathrm{~kg}, 73 \mathrm{~kg}, 77 \mathrm{~kg}$. The weights of the four boxes of apples are $\qquad$ kg, $\qquad$ kg, $\qquad$ kg, $\qquad$ kg.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23, 27, 20, 30}
平面上的n个圆满足：存在平面上的6个点，任一点都至少在这n个圆中的3个圆上．求n的最小值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given an isosceles triangle with an internal angle of 40°, determine the possible degrees of its apex angle. Express your answer in degrees (°), including all possible angles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40°, 100°}
A prime number is divided by 21 with a remainder. Find all values of the remainder that are composite numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4, 8, 10, 16, 20}
In five pots arranged in a row, Rabbit poured three kilograms of honey (not necessarily into each pot and not necessarily evenly). Winnie-the-Pooh can take any two pots that are next to each other. What is the maximum amount of honey that Winnie-the-Pooh can be guaranteed to eat?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 \text{ kg}}
The original price of a saree is Rs. 600 after successive discounts of 20%, 15%, and 10% are applied. If a new saree with the same original price goes through successive discounts of 20%, 15%, 10%, and an additional discount of 'x%', what is the final sale price of the new saree in terms of 'x'?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{367.2}
空间中有9个点，其中任意4个点不共面．在这9点之间连线，要求不产生四面体.最多有几个三角形？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
When conducting market research, what statistical measure of shoe models should a shoe store focus on in order to increase sales? Provide your answer as a single word or brief phrase.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Mode}
For how many positive integers $n \le 1000$ is
$\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$
not divisible by $3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
$2019$  circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the value of \[3+\frac{1}{3+\frac{1}{3+\frac{1}{3}}}?\] The original answer is in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{142}
29 函数f定义在有序正整数对的集合上，且满足下列性质：

(1)f(x,x)=x,

(2)f(x,y)=f(y,x)-

(3)(x+y)f(x,y)=yf(x,x+y).

求f(14,52)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{364}
Determine the greatest positive integer  $ n$  such that in three-dimensional space, there exist n points  $ P_{1},P_{2},\cdots,P_{n},$  among  $ n$  points no three points are collinear, and for arbitary  $ 1\leq i < j < k\leq n$ ,  $ P_{i}P_{j}P_{k}$  isn't obtuse triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A paper cup has a base that is a circle with radius $r$, a top that is a circle with radius $2r$, and sides that connect the two circles with straight line segments. This cup has height $h$ and volume $V$. A second cup, exactly the same shape as the first, is held upright inside the first cup so that its base is a distance of $\frac{h}{2}$ from the base of the first cup. The volume of liquid that will fit inside the first cup and outside the second cup can be written as $\frac{m}{n} \cdot V$, 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{93}
Using equal-length toothpicks to form a rectangular diagram as shown, if the length of the rectangle is 20 toothpicks long and the width is 10 toothpicks long, how many toothpicks are used?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{430}
一个不均匀的骰子，掷出 $1,2,3,4,5,6$ 点的概率依次成等差数列.独立地先后掷该骰子两次，所得的点数分别记为 $a, b$ .若事件 " $a+b=7$ " 发生的概率为 $\frac{1}{7}$ ，则事件 " $a=b$ "发生的概率为 $\qquad$.  原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
In the diagram, \(P Q R S\) is a rectangle. Also, \(\triangle S T U\), \(\triangle U V W\), and \(\triangle W X R\) are congruent. Calculate the fraction of the area of rectangle \(P Q R S\) that is shaded. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
The polar curve $r = 1 + \cos \theta$ is rotated once around the point with polar coordinates $(2,0).$  What is the area of the region it sweeps over?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{16 \pi}{3}}
A social survey agency conducted a monthly income survey on 10,000 residents of a certain area and drew a frequency distribution histogram of the sample based on the collected data (as shown in the diagram). To analyze the relationship between residents' income and various factors such as age, education, and occupation, 100 people need to be further surveyed using the stratified sampling method from these 10,000 individuals. The question is: how many people should be sampled from the monthly income bracket of [¥2500, ¥3000)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Find the dihedral angles of a trihedral angle, whose plane angles are $90^{\circ}, 90^{\circ}$, and $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ, 90^\circ, \alpha}
Five square sheets with a side length of 10 centimeters are placed in such a manner that each small square's covered part is a smaller square whose side length is half of the original square's side length. Determine the perimeter of the outer contour (thick lines) in the figure in centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120 \, \text{cm}}
Given the set M={α, β, γ}, where α, β, and γ are complex numbers, if the product of any two numbers in set M and the square of any one number in set M are still elements in M, then set M is equal to \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1, 0, 1}
Find all ordered triples of primes  $(p, q, r)$  such that \[ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1.  \] *Reid Barton*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(p, q, r) \in \{(2, 5, 3), (3, 2, 5), (5, 3, 2)\}}
In the diagram, the side lengths of four squares are shown. The area of the fifth square is denoted by k. Using the given relationships between the heights of the squares on the left and right sides, find the value of k. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Given an inscribed quadrilateral $A B C D$. The rays $A B$ and $D C$ intersect at point $K$. It turns out that points $B, D$, as well as the midpoints of segments $A C$ and $K C$, lie on the same circle. What values can the angle $A D C$ take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^
}
A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{3\sqrt{7}-\sqrt{3}}{2}$}
The graph of $y=f(x)$ is obtained by shifting the graph of $y=\cos \left(2x+\frac{\pi}{6}\right)$ to the left $\frac{\pi}{6}$ units. How many intersection points does the graph of $y=f(x)$ have with line $y=\frac{1}{2}x-\frac{1}{2}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3$}
Given four positive numbers, the first three form an arithmetic sequence with a sum of 48. The last three form a geometric sequence, with the last number being the maximum value of the function $y=21-4x-x^2$. Find these four numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12, 16, 20, 25}
The odd numbers from 5 to 21 are used to build a 3 by 3 magic square. If 5, 9 and 17 are placed as shown, what is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given a positive integer  $n$ , determine the maximum number of lattice points in the plane a square of side length  $n +\frac{1}{2n+1}$  may cover.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n+1)^2}
Two bicycle enthusiasts are leaving Cincinnati at the same time. One is biking 840 miles to Boston, and the other is biking 440 miles to Atlanta. They plan to travel for seven days, covering equal distances each day. However, the terrain and weather conditions vary along their routes. The biker going to Boston will face 2 days of hilly terrain (decreasing daily mileage by 20%), 2 days of rain (decreasing daily mileage by 10%), and 1 rest day (no mileage). The biker going to Atlanta will face 1 day of hilly terrain (decreasing daily mileage by 20%), 1 day of rain (decreasing daily mileage by 10%), and 1 rest day (no mileage). What is the greatest number of miles a day each can bike considering these factors, if they want to cover equal distances each day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77.19}
Of the students attending a school athletic event, $80\%$ of the boys were dressed in the school colors, $60\%$ of the girls were dressed in the school colors, and $45\% $ of the students were girls. Find the percentage of students attending the event who were wearing the school colors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
Let $n>3$ be a positive integer. Equilateral triangle $ABC$ is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$. Please provide the value of $m-d$ when $n = 5$. The answer should be an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In the diagram, \(A B C D\) is a rectangle, \(P\) is on \(B C\), \(Q\) is on \(C D\), and \(R\) is inside \(A B C D\). Also, \(\angle P R Q = 30^\circ\), \(\angle R Q D = w^\circ\), \(\angle P Q C = x^\circ\), \(\angle C P Q = y^\circ\), and \(\angle B P R = z^\circ\). What is the value of \(w + x + y + z\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Find all numbers between  $100$  and  $999$  which equal the sum of the cubes of their digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153, 370, 371, 407}
求出具有下述性质的正整数 $n$ : 它被 $\leqslant \sqrt{n}$ 的所有正整数整除.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1,2,3,4,6,8,12,24$}
Find all solutions in integers for the equation: \( x^{3} + y^{3} = 2^{30} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(0, 2^{10}\right), \left(2^{10}, 0\right)}
the cricket team of 11 members is 29 yrs old & the wicket keeper is 3 yrs older . if the ages ofthese 2 are excluded , the average age of theremaining players is 1 year less than the average age of the whole team . what is the average age of the team ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
In a 4 by 4 grid, each of the 16 small squares measures 3 cm by 3 cm and is shaded. Four unshaded circles are then placed on top of the grid as shown. The area of the visible shaded region can be written in the form $A-B\pi$ square cm. What is the value $A+B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Find the amount of Water formed on combining some moles of Sodium hydroxide and 1 mole of Perchloric acid if 18 grams of Water are formed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Four identical isosceles trapezoids are placed such that their lower bases form the two diagonals of a square. If $PX = 3XQ$, find the ratio of the area of the shaded regions to the area of the entire square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.375}
The gardener Fedya has a miracle tree with seven branches in his garden. On each branch, there can either grow 6 apples, 5 pears, or 3 oranges. Fedya discovered that the tree has fruit of all types, with the most pears and the fewest apples.

How many fruits in total grew on the miracle tree?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$7$}
Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{1} + \frac{1}{1} + \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{1} + \frac{1}{3}$.

Find the least positive integer $k$ making the number $k! \cdot S_{2016}$ an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine whether the set $$ D:= \bigcap_{n \in \mathbb{N}} ]\frac{1}{n},1]$$ has a minimum or infimum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
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}
Charles has $5q + 1$ quarters and Richard has $q + 5$ quarters. The difference in their money in dimes is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$10(q - 1)$}
Let $P(x)=x^{2}-1$. How many distinct real solutions does the following equation have:  
$$  
\underbrace{P(P(\ldots(P}_{2004}(x)) \ldots))=0?  
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2005}
In a school tennis tournament with $m \ge 2$ participants, each match consists of 4 sets. A player who wins more than half of all sets during a match gets 2 points for this match. A player who wins exactly half of all sets during the match gets 1 point, and a player who wins less than half of all sets gets 0 points. During the tournament, each participant plays exactly one match against each remaining player. Find the least number of participants $m$ for which it is possible that some participant wins more sets than any other participant but obtains less points than any other participant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A person walks on a 400-meter circular track at a speed of 6 kilometers per hour. They start from point A and walk clockwise for 1 minute, then counterclockwise for 3 minutes, and clockwise again for 5 minutes. What is the minimum time they need to return to the starting point A? Express your answer in minutes as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If Bill can buy 3 pairs of jeans, 2 shirts, and 1 hat for $89, or 2 pairs of jeans, 3 shirts, and 2 hats for $102, or 4 pairs of jeans, 1 shirt, and 3 hats for $125, how much does one shirt cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$12.53}
Vasya cut a triangle out of cardboard and numbered its vertices with the numbers 1, 2, and 3. It turns out that if Vasya rotates his triangle 15 times clockwise around the vertex numbered 1 by an angle equal to the angle at this vertex, the triangle returns to its original position. If Vasya rotates his triangle 6 times clockwise around the vertex numbered 2 by an angle equal to the angle at this vertex, the triangle returns to its original position. Vasya claims that if he rotates his triangle \( n \) times around the vertex numbered 3 by an angle equal to the angle at this vertex, the triangle will return to its original position. What is the minimal \( n \) that Vasya could name so that his statement is true for at least some cardboard triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
394. The random variable $X$ is given by the probability density function $f(x)=(1 / 2) \sin x$ in the interval $(0, \pi)$; outside this interval, $f(x)=0$. Find the variance of the function $Y=\varphi(X)=X^{2}$, using the density function $g(y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\pi^{4}-16\pi^{2}+80)/4}
一个三位数等于它的个位数字的阶乘之和，则此三位数的个位数字之和为多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Among the $n$ angles of a convex $n$-gon, $n-1$ angles are equal to $150^{\circ}$, and the remaining angle is less than $150^{\circ}$. For which values of $n$ is this possible? List all possible answers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8, 9, 10, 11}
A  $6$ -inch-wide rectangle is rotated  $90$  degrees about one of its corners, sweeping out an area of  $45\pi$  square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given that ${F_1}$ and ${F_2}$ are the left and right foci of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, $P$ is a point on the line $x = \frac{3a}{2}$, and $\Delta F_2 P F_1$ is an isosceles triangle with a base angle of $30^\circ$, calculate the eccentricity of $E$. Express your answer as a single fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
给定整数 $n \geqslant 2$. 求最小的正实数 $c$, 使得对任意复数 $z_{1}, z_{2}, \cdots, z_{n}$, 均有
$$
\left|\sum_{i=1}^{n} z_{i}\right|+c \sum_{1 \leq i<j \leqslant n}\left|z_{i}-z_{j}\right| \geqslant \sum_{i=1}^{n}\left|z_{i}\right|
$$
若$n=5$，答案写为$\frac{a}{b}$，其中$a,b$互质，求$a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The marked price of a book was 30% less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$35\%$}
Find the mass percentage of O in Chromic acid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54.23\%}
Let  $ f : \mathbf {R} \to \mathbf {R} $  be a smooth function such that  $ f'(x)^2 = f(x) f''(x) $  for all  $x$ . Suppose  $f(0)=1$  and  $f^{(4)} (0) = 9$ . Find all possible values of  $f'(0)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \sqrt{3}}
Jason and Jared take turns placing blocks within a game board with dimensions  $3 \times 300$ , with Jason going first, such that no two blocks can overlap. The player who cannot place a block within the boundaries loses. Jason can only place  $2 \times 100$  blocks, and Jared can only place  $2 \times n$  blocks where  $n$  is some positive integer greater than 3. Find the smallest integer value of  $n$  that still guarantees Jason a win (given both players are playing optimally).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
If there is a positive integer \( m \) such that \( m! \) (m factorial) ends with exactly \( n \) zeros, then the positive integer \( n \) is called a "factorial tail number." How many numbers less than 1992 are not "factorial tail numbers"?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{396}
Let $x_1=1$ and $x_{n+1} =x_n+\left\lfloor \frac{x_n}{n}\right\rfloor +2$, for $n=1,2,3,\ldots $ where $x$ denotes the largest integer not greater than $x$. Determine $x_{1997}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23913}
Determine the smallest natural number that can be represented both as the sum of $2002$ positive integers with the same sum of decimal digits, and as the sum of $2003$ integers with the same sum of decimal digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10010}
In an old script found in the ruins of Persepolis, it is written:

"This script has been finished in a year whose 13th power is 258145266804692077858261512663. If you are skilled in Arithmetic, you will know the year this script is finished easily."

Find the year the script is finished. Provide a reason for your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{183}
Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}