inclusionAI/Ring-lite-rl-data huggingface dataset:
A man buys a house for $10,000 and rents it. He puts $12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep; pays  $325 a year taxes and realizes $5\frac{1}{2}\%$ on his investment. The monthly rent (in dollars) is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83.33}
Given \( x_{i}=\frac{i}{101} \), find the value of \( S=\sum_{i=1}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Mark had a box of chocolates. He consumed $\frac{1}{4}$ of them and then gave $\frac{1}{3}$ of what remained to his friend Lucy. Mark and his father then each ate 20 chocolates from what Mark had left. Finally, Mark's sister took between five and ten chocolates, leaving Mark with four chocolates. How many chocolates did Mark start with?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{104}
A company is lining a 900-foot path with a fence. Fence poles are placed at a certain distance apart. The entire path is lined with a fence except for a 42-foot bridge. There are 286 fence poles placed along the whole path on both sides. What is the distance between each fence pole?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Santa Claus has 36 identical gifts distributed among 8 bags. Each bag contains at least 1 gift, and the number of gifts in each of the 8 bags is unique. From these bags, select some bags such that the total number of gifts in the selected bags can be evenly divided among 8 children, with each child receiving at least one gift. Determine how many different ways the selection can be made.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of a square is 24, and the perimeter of a small rectangle is 16. What is the perimeter of the large rectangle?

The perimeter of a shape is the sum of its side lengths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
If x is greater than 3 and less than 6, and y is greater than 6 and less than a certain number, then the greatest possible positive integer difference of x and y is 4. What is the upper bound for y?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
John chops down 30 trees. He can make 20 planks from the first 10 trees, 25 planks from the next 10 trees, and 30 planks from the last 10 trees. The labor cost for the first 10 trees is $120 per tree, $80 per tree for the next 10 trees, and $60 per tree for the last 10 trees. A table takes 15 planks to make, and the selling price of the tables varies; the first 10 tables sell for $350 each, the next 20 tables for $325 each, and every table after that for $300 each. How much profit did John make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$13400}
Given the function $f(x) = x^2 + 4mx + n$ is decreasing on the interval $[2, 6]$, find the range of real numbers $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -3]}
Given the parabola $C: y^2 = 2px \ (0 < p < 4)$ with focus $F$, and a moving point $P$ on $C$. Let $A(4, 0)$ and $B(p, \sqrt{2}p)$ be such that the minimum value of $|PA|$ is $\sqrt{15}$. Find the value of $|BF|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{2}}
What is the value of \[(\log 5)^{3}+(\log 20)^{3}+(\log 8)(\log 0.25)\] where $\log$ denotes the base-ten logarithm?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2$}
Ephram is growing $3$ different variants of radishes in a row of $13$ radishes total, but he forgot where he planted each radish variant and he can't tell what variant a radish is before he picks it. Ephram knows that he planted at least one of each radish variant, and all radishes of one variant will form a consecutive string, with all such possibilities having an equal chance of occurring. He wants to pick three radishes to bring to the farmers market, and wants them to all be of different variants. Given that he uses optimal strategy, the probability that he achieves this can be expressed as $\frac{m}{n}$, 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{17}
In the complex plane, let $z_1, z_2, z_3$ be the roots of the polynomial $p(x) = x^3- ax^2 + bx - ab$. Find the number of integers $n$ between $1$ and $500$ inclusive that are expressible as $z^4_1 +z^4_2 +z^4_3$ for some choice of positive integers $a, b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
In a 5x7 grid with 1 gray square, how many rectangles with an area of 15 can be drawn such that they do not contain or intersect the gray square? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Carol spends ten hours filling up her pool. During the first hour, the pool fills at a rate of 8 gallons of water per hour. For the next two hours, the pool fills at 10 gallons of water per hour. From the fourth hour to the fifth hour, the pool fills at a rate of 14 gallons of water per hour. During the sixth hour, the pool starts filling at a rate of 12 gallons of water per hour. 

At the start of the seventh hour, the pool develops a leak, losing 8 gallons of water per hour. From the seventh hour to the ninth hour, two more leaks occur. The first leak starts during the eighth hour, and the pool loses 10 gallons of water per hour. The second leak starts during the ninth hour, and the pool loses 6 gallons of water per hour. 

In the tenth hour, all leaks have been fixed, but the filling rate decreases to 6 gallons of water per hour. At the end of the ten hours, how many gallons of water are still left in the pool?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
James and his younger sibling, Sarah, both have growth spurts. Initially, James was 2/3s as tall as his 72-inch tall uncle, and Sarah was only 3/4 as tall as James. During the growth spurt, James gains 10 inches, while Sarah gains 12 inches. What is the combined height difference between the uncle and both James and Sarah now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Scott, Mary, Ken, and Amy committed to raising $15,000 for their local children's cancer hospital. Currently, Mary's collection is five times what Ken has and three times Scott’s funds, while Amy's collection is two-thirds of the total raised by Ken and Scott together. They also agreed that for every $100 collected, they will each contribute another 10% of their funds as a bonus. If Ken has raised $900, by how much have the four of them exceeded or fallen short of their goal, including the bonus contribution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$5650}
Lil writes one of the letters \( \text{P}, \text{Q}, \text{R}, \text{S} \) in each cell of a \( 2 \times 4 \) table. She does this in such a way that, in each row and in each \( 2 \times 2 \) square, all four letters appear. In how many ways can she do this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Find (in terms of \( n \geq 1 \)) the number of terms with odd coefficients after expanding the product:
\[ \prod_{1 \leq i<j \leq n}(x_{i} + x_{j}) \]

For example, for \( n = 3 \), the expanded product is given by \( x_{1}^{2} x_{2} + x_{1}^{2} x_{3} + x_{2}^{2} x_{3} + x_{2}^{2} x_{1} + x_{3}^{2} x_{1} + x_{3}^{2} x_{2} + 2 x_{1} x_{2} x_{3} \), resulting in 6 terms with odd coefficients.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
Two trains, 200 m and 160 m long, run at the speeds of 68 kmph and 40 kmph, in opposite directions on parallel tracks. However, the tracks are not on the same level: one track is at ground level and the other is elevated. The angle of elevation between the tracks is 15 degrees. What is the time they take to cross each other, taking into account the difference in heights between the tracks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Calculate the L.C.M of 4/9, 5/7, 7/15, 9/13, 13/21, 16/35, and 19/45.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1244880}
In an opaque bag, there are a total of 50 glass balls in red, black, and white colors. Except for the color, everything else is the same. After several trials of drawing balls, Xiaogang found that the probability of drawing a red or black ball is stable at 15% and 45%, respectively. What could be the possible number of white balls in the bag?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the number of ordered triples $(x,y,z)$ of real numbers such that
\[x^4 + y^4 + z^4 - 4xyz = -1.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A cistern is two-thirds full of water. There are three pipes: Pipe A can fill the remaining part in 12 minutes, Pipe B in 8 minutes, and Pipe C can empty the full cistern in 24 minutes. If Pipes A and B fill while Pipe C is open, how much time will they take to fill the cistern completely?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $f(x)$ be a real function with smallest period $5$, and $g(x)$ be defined as $f(2^x)$. Given that $g(x)$ has $25$ zeros in the range $[0,5)$, find the number of zeros that $g(x)$ has in the range $[1,4)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Find the number of ordered pairs of positive integers $(x, y)$ that satisfy the equation
$$
x \sqrt{y} + y \sqrt{x} + \sqrt{2009 x y} - \sqrt{2009 x} - \sqrt{2009 y} - 2009 = 0.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Gold is 19 times as heavy as water and copper is 9 times as heavy as water. These should be mixed in a ratio of 9 to get an alloy that is a certain times as heavy as water. How heavy is the alloy compared to water?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given the sequence $\left\{a_{n}\right\}$ defined by
$$
a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right] \quad (n \in \mathbf{Z}_{+}),
$$
where $[x]$ denotes the greatest integer not exceeding the real number $x$, determine the minimum value of the constant $C$ such that for any positive integer $n$, the following inequality holds:
$$
\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{288}}
Let $ABC$ be a scalene triangle whose side lengths are positive integers. It is called [i]stable[/i] if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?

[i]Proposed by Evan Chen[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
How many injective functions are there from an uncountable set to the set of bounded functions from the natural numbers to the natural numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2^{\aleph_0}$}
The squares of a $3\times3$ grid are filled with positive integers such that $1$ is the label of the upper- leftmost square, $2009$ is the label of the lower-rightmost square, and the label of each square divides the  ne directly to the right of it and the one directly below it. How many such labelings are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2448}
29. Find the largest natural number in which each non-edge digit is less than the arithmetic mean of the adjacent digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96433469}
A frog is at the center of a circular shaped island with radius $r$. The frog jumps $\frac{1}{2}$ meters at first. After the first jump, it turns right or left at exactly $90^\circ$, and it always jumps one half of its previous jump. After a finite number of jumps, what is the least $r$ that ensures the frog can never fall into the water? The original answer is in \frac{\sqrt{m}}{n} format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that Jones is waiting for one of two telephone booths occupied by A and B, where the duration of their calls follows an exponential distribution with λ = 1/8, what is the probability that Jones is not the last to finish their call?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
Compare the following expressions: $-|-3|\_\_\_\_\_\_-\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{>}
55. Three faces of an $8 \times 8 \times 8$ cube are painted blue, and the other three faces are painted red, such that no vertex has three faces of the same color. How many small cubes from this large cube have both a blue and a red face?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Given a real number $a$, the floor of $a$, written $\lfloor a \rfloor$, is the greatest integer less than or equal to $a$. For how many real numbers $x$ such that $1 \le x \le 20$ is \[ x^2 + \lfloor 2x \rfloor = \lfloor x^2 \rfloor + 2x \? \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{362}
Given a right triangular pyramid with pairwise perpendicular edges, its three side faces have areas of 4, 6, and 12 square units. Calculate the surface area of the circumscribing sphere. Express your answer in terms of pi (π) and include the unit'square units'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56\pi}
Determine the probability that both banknotes are counterfeit given that one of the two randomly selected 100 yuan banknotes from a stack of 20 (which contains 5 counterfeit notes) is verified to be counterfeit upon examination with a money detector.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{17}}
Let $a, b, c, x, y,$ and $z$ be complex numbers such that:
\[ a = \frac{b+c}{x-2}, \quad b = \frac{c+a}{y-2}, \quad c = \frac{a+b}{z-2}. \]
Given that:
\[ xy + yz + zx = 1000 \quad \text{and} \quad x + y + z = 2016, \]
find the value of $xyz$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4044}
Let \( F_{1}, F_{2}, F_{3}, \ldots \) be a sequence of convex quadrilaterals where \( F_{k+1} \) (for \( k=1, 2, 3, \ldots \)) is obtained as follows: \( F_{k} \) is cut along a diagonal, one of the parts is flipped over, and then glued along the cut line with the other part. What is the maximum number of distinct quadrilaterals that can be contained in this sequence? (Polygons are considered distinct if they cannot be made to coincide by motion.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
设$Z = \{ 1 , 2 , \cdots , n \}$，k为正整数，$\frac { n } { 2 } \leq k \leq n$，求满足下列条件的映射$f : Z \rightarrow Z$的个数：$( i ) f ^ { 2 } = f ;( i i ) f ( x )$的值域$f ( Z )$、中恰有k个不同元素；（iii）对每个$y \inf ( Z ) .$，使$f ( x ) = y$的$x \in Z$至多两个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ \frac { n ! } { ( n - k ) ! ( 2 k - n ) ! } .$}
How many positive integers divide $5n^{11}-2n^5-3n$ for all positive integers $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Calculate: $\int_{-1}^{1}\left(2 \sqrt{1-{x}^{2}-\sin x}\right)dx = $______．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Let the function $f(x) = x^3 - 3ax + b$ $(a \neq 0)$. If the curve $y = f(x)$ is tangent to the line $y = 8$ at the point $(2, f(2))$, find the value of the product $ab$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Let $ABCD$ be a convex quadrilateral with $AC=20$, $BC=12$ and $BD=17$. If $\angle{CAB}=80^{\circ}$ and $\angle{DBA}=70^{\circ}$, then find the area of $ABCD$.

[i]2017 CCA Math Bonanza Team Round #7[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
Given a positive integer \( k \), a sequence \( \{a_n\} \) satisfies the following conditions:
\[ a_1 = 3, \quad a_{n+1} = \left(3^{\frac{2}{2x-1}} - 1\right) S_n + 3 \quad (n \in \mathbb{Z}_+) \]
where \( S_n \) is the sum of the first \( n \) terms of the sequence \( \{a_n\} \).

Let 
\[ b_n = \frac{1}{n} \log_3 (a_1 a_2 \ldots a_n) \quad (n \in \mathbb{Z}_+) \]
and define
\[ T_k = \sum_{i=1}^{2k} \left| b_i - \frac{3}{2} \right|. \]

If \( T_k \in \mathbb{Z}_+ \), find all possible values of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ shown in Figure 2 is cut into two parts by a plane passing through point $A$, the trisection point $H$ on the edge $C C^{\prime}$ closer to $C$, and the midpoint $F$ on the edge $D D^{\prime}$. Determine the volume ratio of the two parts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{19}{89}}
Given the number of elements of a list ($n$) and the quadratic mean ($QM$) of that list, can we find a lower bound for the arithmetic mean ($AM$) of that list? If so, what is the greatest lower bound for the $AM$ that we can find?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{QM}{\sqrt{n}}\)}
Evaluate the expression: $\left(2\left(3\left(2\left(3\left(2\left(3 \times (2+1) \times 2\right)+2\right)\times 2\right)+2\right)\times 2\right)+2\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5498}
A train running at a speed of 36 km/h passes an electric pole in 14 seconds. How long is the platform if it takes the train 48.997200223982084 seconds to pass it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{349.97}
A couple agreed to split their expenses equally, including the salary of their house help. While doing her job, the house help had an accident; thus, she underwent three medical procedures which cost her $128, $256, and $64 respectively. The couple decided to pay half of the total medical expenses and the other half will be deducted from her $160 salary. Additionally, they also deducted a 5% tax from her salary. If the husband paid for the entire half of the hospital bill, how much does he still need to pay so that they split the expenses for the house help equally?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$76}
Let $M$ be the set of all polynomials $P(x)$ with pairwise distinct integer roots, integer coefficients, and all absolute values of the coefficients less than $2007$. What is the highest degree among all the polynomials in the set $M$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A solid in the shape of a right circular cone is $4$ inches tall and its base has a $3$-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum -shaped solid $F,$ in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$ . Given that $k=\frac m n,$ 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{512}
According to the relevant tax laws of the country, the author's remuneration should be subject to personal income tax. The specific regulations are as follows: For each individual remuneration income, the balance after a fixed amount or fixed rate minus the specified expenses is the taxable income. For each income not exceeding 4,000 yuan, first deduct an expense of 800 yuan per remuneration income; for each income exceeding 4,000 yuan, first deduct 20% of the expenses, and in both cases, a tax rate of 20% is applied, and according to the regulations, a 30% tax is levied on the taxable amount. It is known that a person has published a manuscript and paid a tax of 280 yuan, the pre-tax remuneration this person should receive is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2800 \text{ yuan}}
Given the first term, the last term, and the number of terms in a geometric sequence, determine the product of the terms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(a_1 a_n)^{\frac{n}{2}}}
Given that all faces of the tetrahedron P-ABC are right triangles, and the longest edge PC equals $2\sqrt{3}$, the surface area of the circumscribed sphere of this tetrahedron is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\pi}
A technician makes a round-trip to and from a certain service center by the same route that consists of three stops. If the technician completes the drive to the first stop and then completes 75% of the drive from the first stop to the second stop, 50% from the second stop to the third stop, and 25% from the third stop to the service center, what percent of the complete round-trip journey has the technician completed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31.25\%}
Compute the expected value $E[{X \choose 3}]$ for a geometrically distributed random variable $X$ with parameter $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{(1-p)^2}{p^3}\)}
如果三角形 $\triangle A B C$ 的三个内角 $A, B, C$ 的余切 $\cot A, \cot B, \cot C$ 依次成等差数列, 则角 $B$ 的最大值是以 $\frac{m\pi}{n}$ 表示，求 $m + n$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Troy's and Emily's homes are 75 meters and 98 meters away from school, respectively. However, there is a park and a river along their routes, which force them to take different detours each day.

On Monday, Troy takes a detour adding 15 meters to his route, while Emily takes a detour adding 10 meters. On Tuesday, Troy takes another detor adding 20 meters, while Emily takes one adding 25 meters. On Wednesday, both take a detour adding 10 meters each. On Thursday, Troy has no detours while Emily takes one adding 15 meters. On Friday, Troy takes a detour adding 5 meters, and Emily takes one adding 10 meters.

Given the varying detours and distances added on each day, find out how much farther Emily walks to school and back in five days.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{270}
Given that the domain of the function f(x) is \u211d, and the following conditions hold:
- When x < 0, f(x) = x^3 - 1
- When -1 \u2264 x \u2264 1, f(-x) = -f(x)
- When x > 1/2, f(x + 1/2) = f(x - 1/2)
Find the value of f(6). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In an isosceles triangle \(ABC\) with base \(AC\), the side \(AB\) is equal to 2, segment \(AD\) is the angle bisector. A tangent \(DH\) to the circumcircle of triangle \(ADB\) passes through point \(D\) and point \(H\) lies on side \(AC\). Find the area of triangle \(ABC\) if \(CD = \sqrt{2} \cdot CH\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{4 \sqrt{2}-5}}