inclusionAI/Ring-lite-rl-data huggingface dataset:
How can we solve the integral $$\int \limits_0^{2 \pi}e^{\cos t} \cos(nt - \sin t) \mbox{d}t$$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2 \pi}{n !}\)}
Let $ F_0=\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n+1}(x)=\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given the change of basis matrix \( S = \begin{pmatrix} p & q \\ 1 & 1 \end{pmatrix} \), the vector \( x = (8, 3) \) transforms to \( x' = (1, 2) \) and the vector \( y = (5, 2) \) transforms to \( y' = (1, 1) \). Determine the new coordinates of the vector \( z = (-1, 0) \) after the change of basis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, -1)}
Calculate the derivative of the conjugate of a complex number $c$ with respect to $c$ itself. Specifically, find $\frac{\partial (c^*)}{\partial c}$, where $c \in \mathbb{C}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that the moment generating function of a random variable $X$ is $M_X(u) = e^{5u + 2u^2}$, determine the probability that the stock price $S = e^X$ is between $70$ and $90$, given that $S > 50$. Assume $X$ follows a normal distribution with mean $\mu = 5$ and variance $\sigma^2 = 4$. Provide your answer as a probability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0675}
Five computers are connected in a network. For each pair of computers, a coin is flipped. If it is heads, a direct link is established between the two computers; if it is tails, no link is established. A message received by a computer is automatically forwarded to all computers directly linked to it. Calculate the probability that every computer can receive messages from every other computer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{91}{128}}
Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. Find the sum of the three digits that were then in the positions $2019, 2020, 2021$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Evaluate the line integral \( \int_{C} (x^{4}+5y^{2})\,dx + (xy-y^{5})\,dy \), where \( C \) is the polygonal path joining the vertices \( (-1, 0), (4, 0), (6, 4), (4, 8), (-1, 8), \) and \( (-1, 0) \), oriented counterclockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1728}
Each of the points $A,B,C,D,E,$ and $F$ in the figure below represents a different digit from $1$ to $6.$ Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is $47.$ What is the digit represented by $B?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
How many games must be played between two opponents to have 90% confidence that the overall win percentage approaches a single number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{271}
Given the expression \( y = \sin^{-1}(\sin 8) - \tan^{-1}(\tan 10) + \cos^{-1}(\cos 12) - \sec^{-1}(\sec 9) + \cot^{-1}(\cot 6) - \csc^{-1}(\csc 7) \), find the value of \( a - b \) if \( y \) simplifies to \( a\pi + b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
Circle  $\omega$  has radius  $5$  and is centered at  $O$ .  Point  $A$  lies outside  $\omega$  such that  $OA=13$ .  The two tangents to  $\omega$  passing through  $A$  are drawn, and points  $B$  and  $C$  are chosen on them (one on each tangent), such that line  $BC$  is tangent to  $\omega$  and  $\omega$  lies outside triangle  $ABC$ .  Compute  $AB+AC$  given that  $BC=7$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Determine the number of functions \( g: \{1, 2, \cdots, 10\} \rightarrow \{1, 2, \cdots, 10\} \) such that if \((x, y, z)\) satisfies \(xy = z\), then \((g(x), g(y), g(z))\) also satisfies this equation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{450}
Nine numbers  $a, b, c, \dots$  are arranged around a circle. All numbers of the form  $a+b^c, \dots$  are prime. What is the largest possible number of different numbers among  $a, b, c, \dots$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Arrange $n+1$ squares with a side length of 1 as shown in figure 4. Points $A, A_1, A_2, \cdots, A_{n+1}, M_0, M_1, M_2, \cdots, M_n$ are vertices of the squares. Connect $A M_i \ (i=1, 2, \cdots, n)$ and intersect with side $A_i M_{i-1}$ at point $N_i$. Let the area of quadrilateral $M_i N_i A_i A_{i+1}$ be $S_i \ (i=1,2, \cdots, n)$. What is $S_{2017}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4035}{4036}}
A square is drawn in the Cartesian coordinate plane with vertices at \((2, 2)\), \((-2, 2)\), \((-2, -2)\), \((2, -2)\). A particle starts at \((0,0)\). Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is \(1/8\) that the particle will move from \((x, y)\) to each of \((x, y + 1)\), \((x + 1, y + 1)\), \((x + 1, y)\), \((x + 1, y - 1)\), \((x, y - 1)\), \((x - 1, y - 1)\), \((x - 1, y)\), or \((x - 1, y + 1)\). The particle will eventually hit the square for the first time, either at one of the 4 corners of the square or at one of the 12 lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is \(m/n\), where \(m\) and \(n\) are relatively prime positive integers. What is \(m + n\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Let $k$ be a positive integer with the following property: For every subset $A$ of $\{1,2,\ldots, 25\}$ with $|A|=k$, we can find distinct elements $x$ and $y$ of $A$ such that $\tfrac23\leq\tfrac xy\leq\tfrac 32$.  Find the smallest possible value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find all twice differentiable functions \( f\left( x\right) \) such that \( {f}^{\prime \prime }\left( x\right)  = 0,f\left( 0\right)  = {19} \) ,and \( f\left( 1\right)  = {99} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( f(x) = 80x + 19 \)}
The denominator of the fraction $15 \cdot 18$ in simplest form is 30. Find the sum of all such positive rational numbers less than 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
On a circle, 40 red points and one blue point are marked. All possible polygons with vertices at the marked points are considered. Which polygons are more numerous, those with a blue vertex, or those without it, and by how many?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{780}
The product of four consecutive integers is a perfect square. What are these integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1, 0, 1, 2}
How many functions \( f : \mathbb{N} \rightarrow  \mathbb{N} \) are there such that \( f\left( 0\right)  = {2011},f\left( 1\right)  = {111} \) ,and \( f\left( {\max \{ x + y + 2,{xy}\} }\right)  = \min \{ f\left( {x + y}\right) ,f\left( {{xy} + 2}\right) \} \)for all non-negative integers \( x,y \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1

}
Ali defines a pronunciation of any sequence of English letters to be a partition of those letters into substrings such that each
 substring contains at least one vowel. For example, \( \Lambda \left| {\mathrm{{THEN}} \mid  \Lambda ,\Lambda \mathrm{{TH}} \mid  \mathrm{E} \mid  \mathrm{{NA}},\text{ ATHENA,and AT }}\right| \mathrm{{HEN}} \mid  \Lambda \) are all
 pronunciations of the sequence ATHENA. How many distinct pronunciations does YALEMATHCOMP have? (Y is not a
 vowel.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45
}
Find the smallest integer \( n \) such that there exists a cyclic binary code of length \( n \) with the generator polynomial \( g(x) = 1 + x^4 + x^5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Solve for $X$ and $Y$ such that
\[\frac{Yx + 8}{x^2 - 11x + 30} = \frac{X}{x - 5} + \frac{7}{x - 6}\].
Find $X+Y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{22}{3}}
Evaluate the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n} \, dS \) for the surface \( S \) defined by the boundary of the triangle with vertices \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\). The vector field is \( \mathbf{F}(x,y,z) = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), and \( \mathbf{n} \) is the unit normal to \( S \) with a non-negative \( z \)-component. Use the parameterization \( \mathbf{r}(u,v) = (u+v)\mathbf{i} + (u-v)\mathbf{j} + (1-2u)\mathbf{k} \), where \( u \) and \( v \) are parameters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the sub-interval of \([0,1]\) on which the sequence of functions \(f_n(x) = x^{1/n}\) converges uniformly to the constant function 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[a, 1]}
Points \( A \), \( B \), \( C \), \( D \), and \( E \) are located in 3-dimensional space with \( AB = BC = CD = DE = 3 \) and \( \angle ABC = \angle CDE = 60^\circ \). Additionally, the plane of triangle \( ABC \) is perpendicular to line \( \overline{DE} \). Determine the area of triangle \( BDE \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.5}
In an old estate, the house is surrounded by tall trees arranged in a circle, including spruces, pines, and birches. There are 96 trees in total. These trees have a peculiar property: for any coniferous tree, among the two trees that are two trees away from it, one is coniferous and the other is deciduous; also, among the two trees that are three trees away from any coniferous tree, one is coniferous and the other is deciduous. How many birches have been planted around the house?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let $n(S)$ denote the number of subsets of a set $S$. Given that $n(X) + n(Y) + n(Z) = n(X \cup Y \cup Z)$ and $|X| = |Y| = 200$, find the minimum possible value of $|X \cap Y \cap Z|$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{197}
Find the area of one petal of the rose curve given by \( r = |\cos(2\theta)|^{1/2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
The CMU Kiltie Band is attempting to crash a helicopter using a grappling hook. The helicopter starts parallel to the ground at an angle of $0$ degrees. Each time the band members pull the hook, they tilt the helicopter forward by either $x$ or $x+1$ degrees, with equal probability, if the helicopter is currently at an angle $x$ degrees with the ground. Determine the expected number of pulls required to tilt the helicopter to $90$ degrees or beyond, causing it to crash.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{269}{32}}
Consider the Banach space $c_0$ with the supremum norm. Let $F$ be a subset of $c_0$ defined as:
$$F=\{x=(x_n)\in c_0:\sum_{n=1}^\infty 3^{-n}x_n=1\}.$$ Find the distance $d(F,r)$, where $r=(2,0,0,0,...)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Let \( \{ a_n \} \) be a sequence of non-negative real numbers such that the series \( \sum_{n=1}^{\infty} a_n \) is convergent. Determine the range of the real number \( q \) for which the series \( \sum \frac{a_n}{n^q} \) also converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, \infty)}
Find the smallest constant M, so that for any real numbers \( {a}_{1},{a}_{2},\ldots {a}_{2023} \in  \left\lbrack  {4,6}\right\rbrack \) and
 \( {b}_{1},{b}_{2},\ldots {b}_{2023} \in  \left\lbrack  {9,{12}}\right\rbrack \) following inequality holds:
 \( \sqrt{{a}_{1}^{2} + {a}_{2}^{2} + \cdots  + {a}_{2023}^{2}} \cdot  \sqrt{{b}_{1}^{2} + {b}_{2}^{2} + \cdots  + {b}_{2023}^{2}} \leq  M \cdot  \left( {{a}_{1}{b}_{1} + {a}_{2}{b}_{2} + \cdots  + {a}_{2023}{b}_{2023}}\right) \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$M = \frac{3}{2\sqrt{2}}$

}
Determine the number of functions \( f: A \to A \), where \( A = \{1, 2, 3, 4, 5, 6, 7\} \), such that \( f(f(f(x))) = x \) for all \( x \in A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{351}
In the center of a circular field, there is a house of geologists. Eight straight roads emanate from the house, dividing the field into 8 equal sectors. Two geologists set off on a journey from their house at a speed of 4 km/h choosing any road randomly. Determine the probability that the distance between them after an hour will be more than 6 km.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.375}
Consider the sequence \(a_{1}, a_{2}, a_{3}, \cdots, a_{21}\) such that \(\left|a_{n+1}-a_{n}\right| = 1\) for \(n = 1, 2, 3, \cdots, 20\). Additionally, \(a_{1}, a_{7}, a_{21}\) form a geometric sequence. Given that \(a_{1}=1\) and \(a_{21}=9\), determine the number of distinct sequences that satisfy these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15099}
Evaluate the limit of the fractional derivative of the softplus activation function at $x=0$ and $a=0.9$. What is the value of this limit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a rectangular parallelepiped \( ABCD A_{1} B_{1} C_{1} D_{1} \) with edges \( AB=4 \), \( AD=2 \), \( AA_{1}=3\sqrt{2} \), a plane passes through the diagonal \( BD_{1} \) and intersects the edge \( AA_{1} \) such that the cross-section of the parallelepiped with this plane has the smallest perimeter. Find the area of this cross-section.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8 \sqrt{2}}
Let $X_{1}, X_{2}, \ldots, X_{n}, \ldots$ be independent and identically distributed random variables with distribution $P(X_{n}=0)=P(X_{n}=1)=P(X_{n}=2)=P(X_{n}=3)=\frac{1}{4}$. Assume that the sequence $\{Y_n\}$ is defined as: $Y_{0}=0$ and for all $n\in\mathbb{N}$ we have $Y_n=\begin{cases} 3 &\text{if } X_n=3,\\\min{\{Y_{n-1},X_n\}} &\text{if } X_n<3. \end{cases} $ Compute $\displaystyle \lim_{n\to +\infty}E[Y_{n}Y_{n-1}]$. \(\text{Express your answer as a single number.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{19}{12}}
Determine all values of \( \alpha \) for which some non-trivial solutions of the differential equation \( y'' + (2\alpha - 3)y' + \alpha(\alpha - 3)y = 0 \) remain bounded and some become unbounded as \( t \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 3)}
In a free group with two generators $a$ and $b$, you start with a word of length 1 at time $0$. Every second, you multiply the current word by one of the elements $a$, $b$, $a^{-1}$, or $b^{-1}$, each chosen with probability $\frac{1}{4}$. What is the probability that the reduced word $a$ ever appears in this process?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Determine the number of integer-coordinate points inside the \(\ell_1\) ball with radius \(m\), excluding the boundary, where \(m\) is a natural number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2m^2 - 2m + 1}
Find the function \( \phi(x) \) that minimizes the integral \[ I = \int_a^b \left\{ \frac{1}{4} m \left( \frac{d\phi(x)}{dx} \right)^2 + \frac{1}{4} n^2 \cdot \omega \cdot \sin^2 \phi(x) \right\} dx, \] where \( m, n, \omega, a, \) and \( b \) are constants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\phi(x) = k\pi}
We color the numbers $1, 2, 3, \ldots, 20$ with two colors: white and black, ensuring that both colors are used. Determine the number of ways to perform this coloring such that the greatest common divisor (GCD) of the product of the white numbers and the product of the black numbers is equal to $1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
Let \(\alpha \in \mathbb{R}\). If \((3,0,0,\beta)\) is an optimal solution of the linear programming problem, minimize \(x+y+z-\alpha t\) subject to the constraints \(2x-y+z=6\), \(-x+y+t=3\), and \(x,y,z,t \ge 0\). What is the maximum value of \(\beta - \alpha\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
What is the greatest number of identical bouquets that can be made out of 21 white tulips, 91 red tulips, 37 blue tulips, and 67 yellow tulips if no flowers are to be left out and each bouquet has a ratio of 3 white tulips: 7 red tulips: 5 blue tulips: 9 yellow tulips? Additionally, each bouquet must have at least 24 flowers and no more than 50 flowers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Solve for $x$ in the equation:

$$26280 = 2 \sqrt{\frac{149598000000 - x}{1.4}} + \frac{x}{10217588910387300000}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{149356276560}
From point $M$ on the circle, three chords are drawn: $M N=1, M P=6, M Q=2$. The angles $N M P$ and $P M Q$ are equal. Find the radius of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{\frac{34}{15}}}
Cole wants to fence his backyard on three sides. His backyard is 9 feet along the sides and 18 feet along the back. The neighbor behind him agreed to pay for a portion of their shared side of his fence, and the neighbor on his left agreed to pay for a third of their shared side. Fencing costs $3 per foot. Cole has to pay $72 for the fence. What is the ratio of the amount the neighbor behind him pays for their shared side to the total cost of that side?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
Calculate \( \frac{d}{dx}f(x,f(x,x^2)) \) at \( x=1 \), given that \( f(x,y) \) is differentiable at \((1,1)\), and \( f(1,1)=\frac{\partial f}{\partial x}(1,1)=\frac{\partial f}{\partial y}(1,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given two random variables $x$ and $y$, consider the inequality:

$$-g(x) < f(x, y) < g(x)$$

where $g(x) > 0$. The probability of each part of the inequality is the same, denoted as $\alpha$:

$$P[-g(x) < f(x, y)] = P[f(x, y) < g(x)] = \alpha$$

Determine the probability that the inequality holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\alpha - 1}
Vera has a set of weights, each of which has a distinct mass and weighs an integer number of grams. It is known that the lightest weight in the set weighs 71 times less than the sum of the weights of all other weights in the set. It is also known that the two lightest weights together weigh 34 times less than the sum of the weights of all other weights in the set. What is the minimum number of grams the lightest weight can weigh?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Find a set $B \subseteq \mathbb{R}$ such that for every closed set $F$ containing $B$, the intersection $|B \cap F|$ is infinite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{N}}
Lil writes one of the letters P, Q, R, S in each cell of a 2x4 table. She does this in such a way that, in each row and in each 2x2 square, all four letters appear. How many distinct ways can she fill this table? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Denote by $ S$ the set of all positive integers. Find all functions $ f: S \rightarrow S$ such that
\[ f (f^2(m) + 2f^2(n)) = m^2 + 2 n^2\]
for all $ m,n \in S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(n)=n}
Evaluate the limit \(\lim_{x\to y}\frac{f_1'(y)-f_3'(y)}{f_2'(y)}\) where \(f_1(y)=x^y\), \(f_2(y)=y^x\), and \(f_3(y)=x^x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln y}
Nancy now drinks the equivalent of 75% of her body weight (W) in water, in addition to consuming 2 pounds of water from her food intake. Furthermore, some of that water evaporates during the day, and the evaporated water (E) makes up 5% of her total daily water intake. If Nancy's total remaining daily water intake (R), excluding the evaporated water, is 64 pounds, calculate her weight (W) using this system of equations:

W = R - 2 (amount of water excluding food)
0.05(W+2) = E (amount of water evaporated)
R - E = 64 (remaining daily water intake)

Use these equations to find Nancy's weight (W).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65.37}
A firm is about to hire one of three candidates who take a test with scores that are continuous random variables distributed as \( \text{Unif}[0,100] \). Candidates apply sequentially, and the firm must decide to either select a candidate immediately or reject them permanently. If the first two candidates are rejected, the third candidate must be selected. What is the score threshold below which the first candidate will be rejected to maximize the expected score of the admitted candidate?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62.5}
A boulevard has 25 houses on each side, for a total of 50 houses. The addresses on the east side of the boulevard follow an arithmetic sequence, as do the addresses on the west side. On the east side, the addresses start at 5 and increase by 7 (i.e., 5, 12, 19, etc.), while on the west side, they start at 2 and increase by 5 (i.e., 2, 7, 12, etc.). A sign painter charges $\$1$ per digit to paint house numbers. If he paints the house number on each of the 50 houses, how much will he earn?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{113}
Let us call a number \( \mathrm{X} \) "50-podpyirayushchim" if for any 50 real numbers \( a_{1}, \ldots, a_{50} \) whose sum is an integer, there exists at least one \( a_i \) such that \( \left|a_{i}-\frac{1}{2}\right| \geq X \).

Find the greatest 50-podpyirayushchee \( X \), rounded to the nearest hundredth according to standard mathematical rules.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.01}