inclusionAI/Ring-lite-rl-data huggingface dataset:
Triangle A's base is 20% greater than the base of Triangle B, and A's height is some percent less than the height of Triangle B. The area of Triangle A is 0.25% less than the area of Triangle B. What is the percentage by which Triangle A's height is less than Triangle B's height?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.875\%}
There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.

All the students in this class sat at a round table.

- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers next to me."

What is the minimum number of troublemakers that can be in the class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A boy named Vasya wrote down the nonzero coefficients of a polynomial \( P(x) \) of degree seven in his notebook. He then calculated the derivative of the resulting polynomial and wrote down its nonzero coefficients, continuing this process until he arrived at a constant, which he also wrote down.

What is the smallest number of distinct numbers he might have obtained?

The coefficients are recorded with their signs; if there are free terms, they are also written down. If there's a monomial of the form \( \pm x^{n} \), \( \pm 1 \) is written.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\sin x = \sin (n^2 x)$ on the interval $[0, 2\pi]$. What is $\sum_{n=2}^{100} G(n)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{676797}
During the unloading of a barge, initially, four cranes with equal power operated for 2 hours. Then, two additional cranes with lesser, but equal power were brought into action. After this, it took another 3 hours to complete the unloading. If all these cranes had started working simultaneously, the unloading would have been completed in 4.5 hours. If one crane of greater power and one crane of lesser power worked together, how long would it take to unload the barge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.4 \text{ hours}}
The light glows for some seconds. It glowed 331.27 times between 1:57:58 and 3:20:47 am. What is the frequency of the light glow in seconds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Find the number of positive divisors $d$ of $15! = 15 \cdot 14 \cdot \cdots \cdot 2 \cdot 1$ such that $\gcd(d, 60) = 5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Let $f=y^2-x^3$ and $g=y^3-x^7$. Calculate the intersection multiplicity of $f$ and $g$ at $(0,0)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
On each side of an equilateral triangle, a point is taken. The sides of the triangle with vertices at these points are perpendicular to the sides of the original triangle. In what ratio does each of these points divide the side of the original triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
For each positive integer \( n \), define \( s(n) \) to be the sum of the digits of \( n \). For example, \( s(2014) = 2+0+1+4 = 7 \). Determine all positive integers \( n \) with \( 1000 \leq n \leq 9999 \) for which \( \frac{n}{s(n)} = 112 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1008, 1344, 1680, 2688}
Given a geometric progression \( b_1, b_2, \ldots, b_{3000} \) with all positive terms and a total sum \( S \). It is known that if every term with an index that is a multiple of 3 (i.e., \( b_3, b_6, \ldots, b_{3000} \)) is increased by 50 times, the sum \( S \) increases by 10 times. How will \( S \) change if every term in an even position (i.e., \( b_2, b_4, \ldots, b_{3000} \)) is increased by 2 times?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{8}}
Let $ABCD$ be a trapezoid with $AB \parallel CD$, $AB = 5$, $BC = 9$, $CD = 10$, and $DA = 7$. Lines $BC$ and $DA$ intersect at point $E$. Let $M$ be the midpoint of $CD$, and let $N$ be the intersection of the circumcircles of $\triangle BMC$ and $\triangle DMA$ (other than $M$). If $EN^2 = \frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90011}
The number of games won by five basketball teams is shown in a bar chart. The teams' names are not displayed. The following clues provide information about the teams:

1. The Hawks won more games than the Falcons.
2. The Warriors won more games than the Knights, but fewer games than the Royals.
3. The Knights won more than 22 games.

How many games did the Warriors win? The win numbers given in the bar chart are 23, 28, 33, 38, and 43 games respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Given that \( g \) is a twice differentiable function over the positive real numbers such that
\[ g(x) + 2x^3 g'(x) + x^4 g''(x) = 0 \quad \text{for all positive reals } x \]
and
\[ \lim_{x \to \infty} x g(x) = 1, \]

find the real number \(\alpha > 1\) such that \( g(\alpha) = \frac{1}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{\pi}}
If $(1+x+x^2)^6 = a_0 + a_1x + a_2x^2 + \ldots + a_{12}x^{12}$, then find the value of $a_2 + a_4 + \ldots + a_{12}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{364}
There was an equation of the form \(x^{2} + px + q = 0\) written on the board, with non-zero integer coefficients \(p\) and \(q\).

From time to time, different students would come to the board, erase the equation, and then create and write an equation of the same form, the roots of which were the coefficients of the erased equation. At some point, the newly created equation matched what was originally written on the board. What equation was originally written on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + x - 2}
Let set $A=\{(x,y)|y=a|x|\}$, and set $B=\{(x,y)|y=x+a\}$. If $A\cap B$ is a singleton set, the range of the real number $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1,1]}
410 deputies sat around a round table, each of whom was either a knight who always tells the truth or a liar who always lies. Each of the deputies said: "Among my twenty neighbors to the left and twenty neighbors to the right, there are exactly 20 liars in total." It is known that at least half of the people at the table are liars. How many knights are at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
在凸四边形 $A B C D$ 中, $\overrightarrow{B C}=2 \overrightarrow{A D}$. 点 $P$ 是四边形所在平面上一点, 满足 $\overrightarrow{P A}+2020 \overrightarrow{P B}+\overrightarrow{P C}+2020 \overrightarrow{P D}=\mathbf{0}$. 设 $s, t$ 分别为四边形 $A B C D$ 与 $\triangle P A B$的面积，则 $\frac{t}{s}=$ $\qquad$.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{2358}
If $w,w^2$ are non-real cube roots of unity and $a,b,c \in R$ such that $$\frac{1}{a+w}+\frac{1}{b+w}+\frac{1}{c+w}=2 w^2$$ and $$\frac{1}{a+w^2}+\frac{1}{b+w^2}+\frac{1}{c+w^2}=2 w$$ then find $$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Over all real numbers $x$ and $y$ such that $$x^{3}=3 x+y \quad \text { and } \quad y^{3}=3 y+x$$ compute the sum of all possible values of $x^{2}+y^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $x, y, z$ be real numbers satisfying $$\frac{1}{x}+y+z=x+\frac{1}{y}+z=x+y+\frac{1}{z}=3$$ The sum of all possible values of $x+y+z$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6106}
Simplify \[\frac{1}{\dfrac{2}{\sqrt{5}+2} + \dfrac{3}{\sqrt{7}-2}}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{5} + \sqrt{7} + 2}{23 + 4\sqrt{35}}}
A is half as efficient as B, and B is twice as efficient as C. Together, A, B, and C can finish a job in 26 days. When they work in pairs, A and B can finish the job in 13 days, while A and C can finish it in 39 days. How many days would it take for B and C to finish the job together, assuming they all work at a constant rate?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34\frac{2}{3}}
实系数方程 $x^{4}+a x^{3}+b x^{2}+c x+d=0$ 的根都不是实数, 其中两个根的和为 $2+i$, 另两根的积为 $5+6 i$,则 $b$ 等于多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Two candles of the same thickness and material have lengths in the ratio \(21:16\). They start burning at the same time. After 18 minutes, the length ratio of the longer candle to the shorter one is \(15:11\). How many more minutes can the longer candle continue to burn?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
What is the sum of the digits of the smallest possible value of n for which there are more than 1,000,000 ways to arrange n identical black socks and 2n identical gold socks in a row so that there are at least 2 gold socks between any 2 black socks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In how many ways can the letters in $BEEKEEPER$ be rearranged so that two or more $E$s do not appear together? Provide the number of such arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
How many 6-tuples $ (a_1,a_2,a_3,a_4,a_5,a_6)$ are there such that each of $ a_1,a_2,a_3,a_4,a_5,a_6$ is from the set $ \{1,2,3,4\}$ and the six expressions
\[ a_j^2 \minus{} a_ja_{j \plus{} 1} \plus{} a_{j \plus{} 1}^2\]
for $ j \equal{} 1,2,3,4,5,6$ (where $ a_7$ is to be taken as $ a_1$) are all equal to one another?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
The foci of the ellipse \(\frac{x^2}{25} + \frac{y^2}{b^2} = 1\) and the foci of the hyperbola
\[\frac{x^2}{225} - \frac{y^2}{144} = \frac{1}{36}\]
coincide. Find \(b^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.75}
In a field arranged in 10 rows side by side, choose 2 rows to plant two different crops, A and B, with one row for each crop. To facilitate the growth of the crops, it is required that the interval between crops A and B is not less than 6 rows. How many different planting methods are there in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Four red candies and three green candies can be combined to make many different flavors. Flavors are different if the percent red is different, so 3 red / 0 green is the same flavor as 2 red / 0 green; and likewise 4 red / 2 green is the same flavor as 2 red / 1 green. If a flavor is to be made using some or all of the seven candies, how many different flavors are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
已知实数 $x, y$ 满足 $x|x|+\frac{y|y|}{3}=1$, 则 $|\sqrt{3} x+y-4|$ 的取值范围为 [p, q)。请给出 p + q 的整数部分。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The base radius of a cone is 6 cm, and the central angle of the sector in the unfolded lateral surface of the cone is 240°. Find the slant height of the cone. Express your answer in centimeters as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given $N=2^n$ (where $n\in\mathbb{N}^*, n\geq 2$), place $N$ numbers $x_1, x_2, \ldots, x_N$ sequentially into $N$ positions labeled from 1 to $N$, resulting in the permutation $P_0=x_1x_2\ldots x_N$. Extract the numbers located in the odd and even positions of this permutation and place them in order into the corresponding first $\frac{N}{2}$ and last $\frac{N}{2}$ positions, forming permutation $P_1=x_1x_3\ldots x_{N-1}x_2x_4\ldots x_N$. Let's call this operation a C transformation. Divide $P_1$ into two segments, each containing $\frac{N}{2}$ numbers, and perform a C transformation on each segment to obtain $P_2$. For $2\leq i\leq n-2$, divide $P_i$ into $2^i$ segments, each with $\frac{N}{2^i}$ numbers, and apply C transformations to each segment to get $P_{i+1}$. For example, when $N=8$, $P_2=x_1x_5x_3x_7x_2x_6x_4x_8$, and $x_7$ is in the fourth position in $P_2$. When $N=32$, find the position of $x_{21}$ in $P_3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A train travelling at 80 kmph passes through a tunnel of 70 km at 5:12 am. The train leaves the tunnel at a certain time. The length of the train is 1 km. At what time did the train leave the tunnel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6:05:15}
A class of 30 students wrote a history test. Of these students, 25 achieved an average of 75%. The other 5 students achieved an average of 40%. Calculate the overall class average percentage, rounded to the nearest whole number. Express your answer as a percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{69%}
Rachel starts with the number $1000$ in her hands. Each minute, she flips a fair coin. If it lands heads, she adds 1 to the number by placing it in her left pocket, transforming it to $x+1$. If it lands tails, she inverts the number by placing it in her right pocket, transforming it to $x^{-1}$. After eight minutes, the expected value of the number in Rachel's hands is $E$. Compute $\left\lfloor\frac{E}{10}\right\rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Emily wants to know how much it rained last week. She sees that it rained 2 inches on Monday morning and 1 more inch later that day. It rained some amount on Tuesday compared to Monday. It did not rain on Wednesday but on Thursday it rained 1 inch. The biggest storm was on Friday when the total was equal to Monday through Thursday combined. The daily average rain total for the week was 4 inches. What is the ratio of the rain on Tuesday to the total rain on Monday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
Given two geometric sequences $\{a_n\}$ and $\{b_n\}$, satisfying $a_1=m$, $b_1-a_1=1$, $b_2-a_2=2$, $b_3-a_3=3$. If the general term formula of the sequence $\{a_n\}$ is unique, then the value of the positive real number $m$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
A $39$-tuple of real numbers $(x_1,x_2,\ldots, x_{39})$ satisfies the following conditions:
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\]
Determine the ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1,x_2,\ldots, x_{39})$ satisfying the given conditions. This ratio is expressed as $\frac{a}{b}$ for coprime positive integers $a$ and $b$. Find the sum $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
There are 60 three-digit numbers between 100 and 500, where the sum of the first two digits is equal to a certain number. What is that number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
For \(0 \leq x \leq 1\) and positive integer \(n\), let \(f_0(x) = |1 - 2x|\) and \(f_n(x) = f_0(f_{n-1}(x))\). How many solutions are there to the equation \(f_{10}(x) = x\) in the range \(0 \leq x \leq 1\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2048}
Find the value of $k$ for the ellipse $\frac{x^2}{k+8} + \frac{y^2}{9} = 1$ with an eccentricity of $\frac{1}{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{5}{4}}
Sabrina has a fair tetrahedral die with faces numbered 1, 2, 3, and 4. She rolls the die to create a sequence by recording the number on its bottom face. However, she discards any roll if appending its number to the sequence results in two consecutive terms that sum to 5. Sabrina stops once all four numbers appear in the sequence. Find the expected (average) number of terms in Sabrina's sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
若函数$f ( x ) = \lg [ ( k ^ { 2 } - 1 ) x ^ { 2 } - ( k + 1 ) x + 1 ]$的值域是R，则实数k的取值范围是
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$[ 1 , \frac { 5 } { 3 } ]$}
Compare the sizes: $-\frac{3}{5}$ ____ $-\frac{1}{5}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{<}
Let point $P$ be on the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$. A line through $P$ intersects the asymptotes at $P_{1}$ and $P_{2}$, and $\overrightarrow{P_{1} P} \overrightarrow{P P_{2}}$ $=3$. Let $O$ be the origin. Find the area of $\triangle O P_{1} P_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
How many distinct four-digit positive integers are there such that the product of their digits equals 18?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Let $a > 1$, and $m = \log_a(a^2+1)$, $n = \log_a(a-1)$, $p = \log_a(2a)$. Determine the relationship among $m$, $n$, and $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m > p > n}
a volunteer organization is recruiting new members . in the fall they manage to increase their number by 9 % . by the spring however membership falls by 19 % . what is the total change in percentage from fall to spring ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11.71\%}
Determine the coefficient of the fourth term in the expansion of $( 3x - \frac{1}{2 \cdot 3x} )^{8}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-63}
## Task 4 - 261244

Determine the smallest positive integer $a$ for which $(a+1)^{5}-a^{5}-1$ is divisible by 18305.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["60"]}
Given a grid of size $m \times n$ (with $m, n > 1$). The centers of all the cells are marked. What is the maximum number of marked centers that can be chosen such that no three of them form the vertices of a right triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n - 2}
If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), where \(a, b\), and \(c\) are positive integers, find the number of different ordered triples \((a, b, c)\) such that \(a+2b+c \leq 40\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
There are \( n \) distinct lines in the plane. One of these lines intersects exactly 5 of the \( n \) lines, another one intersects exactly 9 of the \( n \) lines, and yet another one intersects exactly 11 of them. Which of the following is the smallest possible value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
a, b, and c can do a work in 4 hours, 12 hours, and 6 hours respectively. a starts the work at 6 am, and they work alternately for one hour each in the order of a, b, and c. When will the work be completed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the smallest integer $n > 1$ such that $n^2(n - 1)$ is divisible by 2009.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
A child lines up $2020^2$ pieces of bricks in a row and removes bricks whose positions are square numbers (i.e., the 1st, 4th, 9th, 16th, ... bricks). Then, he lines up the remaining bricks again and removes those that are in a 'square position'. This process is repeated until the number of bricks remaining drops below $250$. How many bricks remain in the end?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Given that \( y = (x - 16)(x - 14)(x + 14)(x + 16) \), find the minimum value of \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-900}
A finite sequence of numbers $(a_1,\cdots,a_n)$ is said to be alternating if $$a_1>a_2~,~a_2<a_3~,~a_3>a_4~,~a_4<a_5~,~\cdots$$  $$\text{or ~}~~a_1<a_2~,~a_2>a_3~,~a_3<a_4~,~a_4>a_5~,~\cdots$$ How many alternating sequences of length $5$ , with distinct numbers $a_1,\cdots,a_5$ can be formed such that $a_i\in\{1,2,\cdots,20\}$ for $i=1,\cdots,5$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32 \times \binom{20}{5}}
Let \( f(x) = \ln x \) and \( g(x) = a x + \frac{b}{x} \). These functions have a common tangent at a common point on the x-axis. When \( x > 1 \), what is the relationship between \( f(x) \) and \( g(x) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) < g(x)}
Find the least positive integer $n$ greater than $1$ such that $n^3 - n^2$ is divisible by $7^2 \times 11$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Marie has 40 notebooks, each containing 150 stamps, 12 binders, each filled with 315 stamps, and 25 folders, each holding 80 stamps. If she decides to keep only 28.5% of her stamps and give away the rest, how many stamps can she give away?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8423}
Let  $\Gamma_1$  be a circle with radius  $\frac{5}{2}$ .  $A$ ,  $B$ , and  $C$  are points on  $\Gamma_1$  such that  $\overline{AB} = 3$  and  $\overline{AC} = 5$ . Let  $\Gamma_2$  be a circle such that  $\Gamma_2$  is tangent to  $AB$  and  $BC$  at  $Q$  and  $R$ , and  $\Gamma_2$  is also internally tangent to  $\Gamma_1$  at  $P$ .  $\Gamma_2$  intersects  $AC$  at  $X$  and  $Y$ .  $[PXY]$  can be expressed as  $\frac{a\sqrt{b}}{c}$ . Find  $a+b+c$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Given the function $f(x)=2\sin(\omega x+\varphi)$, where $(\omega > 0, |\varphi| < \frac{\pi}{2})$, the graph passes through the point $B(0,-1)$, and is monotonically increasing on the interval $\left(\frac{\pi}{18}, \frac{\pi}{3}\right)$. Additionally, the graph of $f(x)$ coincides with its original graph after being shifted to the left by $\pi$ units. If $x_{1}, x_{2} \in \left(-\frac{17\pi}{12}, -\frac{2\pi}{3}\right)$ and $x_{1} \neq x_{2}$, and $f(x_{1}) = f(x_{2})$, calculate $f(x_{1}+x_{2})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
G.H. Hardy once went to visit Srinivasa Ramanujan in the hospital, and he started the conversation with: "I came here in taxi-cab number 1729. That number seems dull to me, which I hope isn't a bad omen." "Nonsense," said Ramanujan. "The number isn't dull at all. It's quite interesting. It's the smallest number that can be expressed as the sum of two cubes in two different ways." Ramanujan had immediately seen that $1729 = 12^{3} + 1^{3} = 10^{3} + 9^{3}$. What is the smallest positive integer representable as the sum of the cubes of three positive integers in two different ways?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{251}