inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $A$ be a set with $\text{card}(A) = a$. Determine the cardinal number of the set of countably infinite subsets of $A$. Express your answer in terms of $a$. \( \text{Hint: Consider the set of injective functions from } \mathbb{N} \rightarrow A. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^{\aleph_0}}
A circular sector with radius \( R \) and an opening angle of \( \frac{\pi}{2} \) is rotated around its axis of symmetry (the x-axis) to form a homogeneous body. Determine the position of the center of gravity of this body.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3R}{8}}
Let $ S(N) $ be the number of 1's in the binary representation of an integer $ N $, and let $ D(N) = S(N + 1) - S(N) $. Compute the sum of $ D(N) $ over all $ N $ such that $ 1 \leq N \leq 2017 $ and $ D(N) < 0 $.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
-1002
}
Determine the set of discontinuities of the function:
\[ f(x) = \lim_{n \to \infty} \left( \lim_{t \to 0} \frac{(\sin(n!\pi x))^2}{(\sin(n!\pi x))^2 + t^2} \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Find the asymptotic approximation for the expression \( \left(1 - \sqrt{ \frac{2}{n-1} } \cdot \frac{ \Gamma(n/2) }{ \Gamma( \frac{n-1}{2} ) } \right) \) as \( n \to \infty \), where \( n \) is a positive integer, using the Taylor series expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4n}}
Let \( f \) be a function such that \( f'(x) \) is continuous on \([0,1]\) and satisfies the following conditions: \( f(1) = 0 \), \( \int_0^1 [f'(x)]^2 \, dx = 7 \), and \( \int_0^1 x^2 f(x) \, dx = \frac{1}{3} \). Calculate \( \int_0^1 f(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{5}}
Determine the values of $a$ for which the given population model exhibits cyclical behavior.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 3}
Calculate the sum of the series $\sum\limits_{-\infty}^{\infty} \frac{\sin^3 3^k}{3^k}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The sum of the series is \( \frac{3}{4} \).

}
Determine the probability that a random polynomial of degree \( n \), with integer coefficients between \(-L\) and \(L\), is irreducible over the rationals and has the symmetric group \( S(n) \) as its Galois group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the values of $p$ for which the Fourier transform $\mathcal{F}: L^p(\mathbb{T}) \rightarrow l^{p'}(\mathbb{Z})$ is a continuous and surjective mapping.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of zeros of the entire analytic function $g(z)$ inside the unit square in the first quadrant, given the conditions $g(z+1) = g(z)$, $g(z+i) = e^{\pi}e^{-2\pi iz}g(z)$, and that $g(z)$ has no zeros on the boundary of the square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of partitions of the set \( \{0, 1, 2, 3, 4, 5, 6, 7\} \) that have an even number of parts. What is the cardinality of this set of partitions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2095}
Determine the number of Hausdorff topologies on a set with 5 elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider a compact connected Riemannian 2-manifold $M$ with positive curvature and diameter 1. Determine the supremum of the average distance between two randomly chosen points on $M$, distributed according to the volume form on $M$. Assume $M$ is diffeomorphic to a 2-sphere. What is this supremum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the integral \( \int_{\mathbb{R}} (F(x+a) - F(x)) \, dx \) for any real number \( a \), where \( F \) is a distribution function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a}
8. Let $A B C$ be an equilateral triangle with side length 8 . Let $X$ be on side $A B$ so that $A X=5$ and $Y$ be on side $A C$ so that $A Y=3$. Let $Z$ be on side $B C$ so that $A Z, B Y, C X$ are concurrent. Let $Z X, Z Y$ intersect the circumcircle of $A X Y$ again at $P, Q$ respectively. Let $X Q$ and $Y P$ intersect at $K$. Compute $K X \cdot K Q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{304}
Sari and Ken climb up a mountain. On the day they are supposed to reach the summit, Sari wakes up at 08:00 and starts climbing at a constant pace. Ken starts climbing only at 10:00, when Sari is already 700 meters ahead of him. Nevertheless, Ken climbs at a constant pace of 500 meters per hour, and reaches the summit before Sari. If Sari is 50 meters behind Ken when he reaches the summit, at what time did Ken reach the summit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15:00}
Evaluate the integral \( A = \int_0^1 \{\operatorname{arccosh}(x)\} \, dx \), where \( \{x\} \) denotes the fractional part of \( x \) and for complex \( x, \{x\} = \{\Re x\} + i\{\Im x\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i(1 - \cos 1)}
Find the supremum of the set of maximum eigenvalues of correlation matrices, where a correlation matrix is defined as a positive semidefinite matrix $B \in M_{n}(\mathbb{C})$ with the identity matrix $I$ as its diagonal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Given the function $f(x)=\frac{1}{9}x^3+x^2-2ax+1$, if the function $f(x)$ has an extremum on the interval $(1,2)$, then the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{7}{6}, \frac{8}{3}\right)}
Two rectangles have their edge lengths chosen uniformly from the interval [0,1]. What is the probability that the second rectangle will fit inside the first one?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Given sets $A=\{x|x\leq-2 \text{ or } x\geq-1\}$, and $B=\{x|m-1<x\leq2m\}$, if $A\cap B=\emptyset$, and $A\cup B=A$, find the range of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m\leq-1}
Find the greatest integer $n$, such that there are $n+4$ points $A$, $B$, $C$, $D$, $X_1, \dots, X_n$ in the plane with $AB \neq CD$ that satisfy the following condition: for each $i=1,2,\dots,n$, the triangles $\triangle ABX_i$ and $\triangle CDX_i$ are equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the complex integral \( \int_{\gamma} \frac{-\cos(1/z)}{\sin(1/z)z^2} \, dz \), where \( \gamma \) is the positively oriented circle \( \partial B(0,2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi i}
In the two regular tetrahedra \(A-OBC\) and \(D-OBC\) with coinciding bases, \(M\) and \(N\) are the centroids of \(\triangle ADC\) and \(\triangle BDC\) respectively. Let \(\overrightarrow{OA}=\boldsymbol{a}, \overrightarrow{OB}=\boldsymbol{b}, \overrightarrow{OC}=\boldsymbol{c}\). If point \(P\) satisfies \(\overrightarrow{OP}=x\boldsymbol{a}+y\boldsymbol{b}+z\boldsymbol{c}\) and \(\overrightarrow{MP}=2\overrightarrow{PN}\), then the real number \(9x+81y+729z\) equals \(\qquad\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{439}
How many positive integers $n$ satisfy the inequality
\[\left\lceil \frac{n}{101} \right\rceil + 1 > \frac{n}{100} \ ?\]
Recall that $\lceil a \rceil$ is the least integer that is greater than or equal to $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15049}
A circular metal wire $S$ in 3-space is described by $S=\{x^2+y^2+z^2=4\}$ and has density $\sigma (x,y,z) = \sqrt{x^2+y^2}$ at the point $(x,y,z)$. Find the mass of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\pi}
Determine the value of the parameter \(a\) such that the substitution \(y(x) = z(x)^a\) transforms the differential equation \(2xy' + y = y^2\sqrt{x - x^2y^2}\) into a homogeneous equation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
A rhombus  $\mathcal R$  has short diagonal of length  $1$  and long diagonal of length  $2023$ . Let  $\mathcal R'$  be the rotation of  $\mathcal R$  by  $90^\circ$  about its center. If  $\mathcal U$  is the set of all points contained in either  $\mathcal R$  or  $\mathcal R'$  (or both; this is known as the *union* of  $\mathcal R$  and  $\mathcal R'$ ) and  $\mathcal I$  is the set of all points contained in both  $\mathcal R$  and  $\mathcal R'$  (this is known as the *intersection* of  $\mathcal R$  and  $\mathcal R'$ , compute the ratio of the area of  $\mathcal I$  to the area of  $\mathcal U$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2023}}
The legs of a trapezoid are \( p \) and \( q \) (\( p < q \)), and the larger base is \( a \). The angles at the larger base are in the ratio \( 2:1 \). Find the length of the smaller base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{p^2 + ap - q^2}{p}}
A cube with edge length 6 units is painted with a blue stripe going down the center of one face, an orange stripe going horizontally across another face, and a green stripe going diagonally across the third face. It is then cut into 216 equal smaller cubes, each with side length of 1 unit. How many smaller cubes have exactly 2 faces painted with different colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Compute the value of the integral \( \int_0^{\infty}{xI\left( 2x \right) \mathrm{e}^{-x^2}\mathrm{d}x} \), where the Bessel function is given by \( I\left( x \right) =\frac{1}{\pi}\int_0^{\infty}{\mathrm{e}^{x\cos \theta}\mathrm{d}\theta} \) for all real \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{e}{2}}
Each week, Paul has a certain amount of homework on weeknights and 5 hours for the entire weekend. This week Paul has practice 2 nights out of the week and can't do any homework those nights. He has to average 3 hours of homework for the other nights to get his week's homework done. How many hours of homework does he have on weeknights?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let's call a natural number "remarkable" if it is the smallest among natural numbers with the same sum of digits. What is the sum of the digits of the two-thousand-and-first remarkable number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2001}
in a class , if 50 % of the boys were girls , then there would be 50 % more girls than boys . what percentage of the overall class is girls ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
Viswam walks half a mile to get to school each day. His route consists of $10$ city blocks of equal length and he takes one minute to walk each block. Today, after walking $5$ blocks, Viswam discovers he has to take a detour, walking $3$ blocks of equal length instead of one block to reach the next corner. From the time he starts his detour what speed, in mph, must he walk in order to get to school at his usual time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4.2
}
A contractor undertakes to do a job within 150 days and hires 25 people to do it. The workers have varying work efficiencies - 15 people have an efficiency of 1 unit per day, while the other 10 have an efficiency of 1.5 units per day. After 40 days, the contractor realizes that only one third of the total work is completed. Following this realization, he fires 4 less efficient workers and 3 more efficient workers. In how many more days will the work be finished?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112}
Let $n$ be a positive integer.  Let $(a, b, c)$ be a random ordered triple of nonnegative integers such that $a + b + c = n$, chosen uniformly at random from among all such triples.  Let $M_n$ be the expected value (average value) of the largest of $a$, $b$, and $c$.  As $n$ approaches infinity, what value does $\frac{M_n}{n}$ approach?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{18}}
Let \(P\) be a regular polygon with 2026 sides. A diagonal of \(P\) is said to be good if its endpoints divide the contour of \(P\) into two parts, each having an odd number of sides of \(P\). The sides of \(P\) are also said to be good. Suppose that \(P\) has been subdivided into triangles by 2023 diagonals, no two of which share a common point inside \(P\). What is the maximum number of such a subdivision that can contain isosceles triangles whose two sides are good?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1013}
Insert \( n \) real numbers between 1 and 100 such that these \( n+2 \) numbers form an increasing geometric sequence. Let the product of these \( n+2 \) numbers be denoted as \( T_n \). Furthermore, let \( a_n = \lg T_n \) for \( n \geq 1 \). Define \( b_n = \tan a_n \cdot \tan a_{n+1} \). Find the sum of the first \( n \) terms of the sequence \(\{b_n\}\) denoted as \( S_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{\tan (n+3) - \tan 3}{\tan 1} - n }
Olga Ivanovna, the homeroom teacher of class 5B, is organizing a "Mathematical Ballet". She wants to arrange the boys and girls so that exactly 2 boys are at a distance of 5 meters from each girl. What is the maximum number of girls that can participate in the ballet if it is known that 5 boys are participating?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
设四面体ABCD中，AB=41,AC=7,AD=18,BC=36,BD=27,CD=13，设d为AB 与CD两棱中点间的距离.求$d ^ { 2 }$的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{137}
In the convex quadrilateral \(ABCD\), the side lengths \(AB\) and \(BC\) are equal, \(DB\) is the angle bisector of \(\angle ADC\), and \(AD:DC = 4:3\). Find the cosine of \(\angle AKB\), if \(K\) is the intersection point of the diagonals \(AC\) and \(BD\), and \(BK:KD = 1:3\). (16 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Nadya wants to cut a sheet of paper into 48 identical rectangles for a game. What is the minimum number of cuts she needs to make, given that the pieces of paper can be rearranged but not folded, and Nadya is able to cut any number of layers of paper simultaneously? (Each cut is a straight line from edge to edge of the piece.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the number of spanning trees in a graph after contracting one edge in the middle of a triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the minimum value of \( n \) such that every subset of \( \{1, 2, \ldots, 999\} \) with \( n \) elements contains four distinct integers \( a, b, c, d \) satisfying \( a + 2b + 3c = d \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{835}
Consider two independent uniform random variables $X$ and $Y$ over $[a, b]$. Calculate $\mathbb{E}[X^2 - Y^2 \ | \ X < Y]$ using the correct approach.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{a^2 - b^2}{3}}
Determine the smallest real number \( \ell \) such that every hexagon \( H \) in the family \( \mathcal{F} \) can be covered by a strip of width \( \ell \). The family \( \mathcal{F} \) consists of hexagons with parallel opposite sides, and any three vertices of \( H \) can be covered with a strip of width 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
How can we find a closed-form expression for the following summation?
$$ \sum_{m\geq 0} \frac{m r^m \Gamma(m+c)}{\Gamma(m+1)} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c!\,{r \over \pars{1 - r}^{c + 1}}}
John was given $14.00 in order to buy a gift. This covered half of the cost. The store was also offering a 15% discount on the gift. Additionally, there is an 8% sales tax applied on the discounted price. Calculate the original price of the gift before the discount and tax were applied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$30.50}
Determine the smallest positive integer \(n\) such that the sum \(S = \frac{a_{1}}{1} + \frac{a_{2}}{2} + \ldots + \frac{a_{n}}{n}\), where \(a_1, a_2, \ldots, a_n\) is a permutation of \(1, 2, \ldots, n\), can take on all integer values from \(n\) to \(n+100\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Using the divergence theorem, find the flux of the vector field \( \mathbf{F} = \left(x(z - 1), y(z - 1), xy\right) \) over the region defined by the inequalities:
\[
(z-1)^2 \le 4-x^2-y^2, \quad (z-1)^2 \ge x^2+y^2, \quad z \ge 1
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Four consecutive digits are written one after another, and then the first two digits are swapped. The resulting four-digit number is a perfect square of a natural number. Find this number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4356}
Let $a,b,c$ be real numbers such that $$a\sin^2{x}+b\cos^2{x}=c,~~~\dfrac{a}{\sin^2{x}}+\dfrac{b}{\cos^2{x}}=c$$ find the value of $$\dfrac{c}{a-b}+\dfrac{a}{b-c}+\dfrac{b}{c-a}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the dimension of the orthogonal complement of the subspace $V_k = \mathrm{span}\{x^k, x^{k+1}, x^{k+2}, \dots\}$ in the space $L^2[-1,1]$, where $k$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k}
On the first day, $2^n$ students participated in a knockout table tennis tournament: initially, two students played, then the winner played against the third student, the winner of that match played against the fourth student, and this continued until the last student played. On the second day, the same students competed in a cup tournament: they were randomly paired and played in pairs, with losers being eliminated, and winners being randomly paired again until a single winner remained. It was found that the sets of pairs that played on the first and second days were identical (though possibly with different winners). Determine the largest possible value of $n$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of terms required from the series \( \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)n!} \) to approximate \( \int_{0}^{1}e^{-x^2} \, dx \) with an error less than \( 10^{-6} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In the Cartesian coordinate plane, let the vertices of triangle $ABC$ be $A(0,a)$, $B(b,0)$, $C(c,0)$, and point $P(0,p)$ be on line segment $AO$ (and not an endpoint). Given that $a$, $b$, $c$, and $p$ are non-zero real numbers, lines $BP$ and $CP$ intersect $AC$ and $AB$ at points $E$ and $F$, respectively. One student has correctly derived the equation of $OE$ as $(\frac{1}{b} - \frac{1}{c})x + (\frac{1}{p} - \frac{1}{a})y = 0$. Find the equation of $OF$: $(\_\_\_\_\_\_\_)x + (\frac{1}{p} - \frac{1}{a})y = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{c} - \frac{1}{b}}
Let $p(x)$ be a monic polynomial of degree 7 such that $p(0) = 0,$ $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ $p(6) = 6,$ and $p(7) = 7.$  Find $p(8).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40328}
I am interested in evaluating the integral $$\int_{-\infty}^{\infty} \frac{\exp( - a x^2 ) \sin( bx )}{x} \,\mathrm{d}x$$ for real $a,b>0$. I would like to find a closed-form expression for the integral, or an efficient numerical method for evaluating it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \mathrm{erf}\left(\frac{b}{2\sqrt{a}}\right)}
In triangle $\triangle ABC$, if $\tan A \tan B = 1$ and $AB = \sqrt{3}$, calculate the size of angle $A$ in radians when the area of $\triangle ABC$ reaches its maximum value. Express your answer in radians, rounded to 4 decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
【Example 2】Using 4 different colors to color the 4 sides of a small square wooden block, with each side painted a different color, how many different coloring patterns can there be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
What will be the cost of building a fence around an irregular shaped plot which has a square, a rectangle, and an equilateral triangle combined, if the price per foot of building the fence varies for each side? The square has an area of 289 sq ft, the rectangular side attached to the square is 45 ft long and has a height of 15 ft, and the equilateral triangle has a side length of 20 ft. The cost of fencing per foot is Rs. 55 for the square sides, Rs. 65 for the rectangular sides, and Rs. 70 for the triangular sides. Additionally, a gate will be installed on one side of the rectangle, which costs Rs. 750. Calculate the total fencing cost for the plot.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13565}
Inside triangle \( ABC \) with angles \(\angle A = 50^\circ\), \(\angle B = 60^\circ\), \(\angle C = 70^\circ\), a point \( M \) is chosen such that \(\angle AMB = 110^\circ\) and \(\angle BMC = 130^\circ\). Find \(\angle MBC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20^\circ}
Rectangle $EFGH$ has area $4024$. An ellipse with area $4024\pi$ passes through $E$ and $G$ and has foci at $F$ and $H$. What is the perimeter of the rectangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\sqrt{503}}
Determine the digits $a, b, c, d, e$ such that the two five-digit numbers formed with them satisfy the equation $\overline{a b c d e} \cdot 9 = \overline{e d c b a}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10989}
Given \( x \geq y > 0 \). If there exist real numbers \( a \) and \( b \) such that \( 0 \leq a \leq x \), \( 0 \leq b \leq y \), and 
\[
(x - a)^{2} + (y - b)^{2} = x^{2} + b^{2} = y^{2} + a^{2},
\]
what is the maximum value of \( \frac{x}{y} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 \sqrt{3}}{3}}
There are 150 'zeroes' and 151 'ones' written on a blackboard. A step involves choosing 3 digits and replacing them with a single digit. If all the 3 digits are identical, they are replaced with the same digit. Otherwise, the digit that appears twice replaces the 3 chosen digits. What is the minimum number of steps after which there will be no zero written on the blackboard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}