inclusionAI/Ring-lite-rl-data huggingface dataset:
You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4$}
A regular triangular prism $ABC A_1B_1C_1$ is inscribed in a sphere, with base $ABC$ and lateral edges $AA_1, BB_1, CC_1$. Segment $CD$ is a diameter of this sphere, and point $K$ is the midpoint of edge $AA_1$. Given that $CK = 2 \sqrt{6}$ and $DK = 4$, find the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Let $f$ be a function from the set $X = \{1,2, \dots, 10\}$ to itself. A partition $(S, T)$ of $X$ is called $f$-balanced if for all $s \in S$, $f(s) \in T$, and for all $t \in T$, $f(t) \in S$. (A partition $(S, T)$ is a pair of subsets $S$ and $T$ of $X$ such that $S \cap T = \emptyset$ and $S \cup T = X$. Note that $(S, T)$ and $(T, S)$ are considered the same partition.)

Let $g(f)$ be the number of $f$-balanced partitions, and let $m$ be the maximum value of $g(f)$ over all functions $f$ from $X$ to itself. If there are $k$ functions satisfying $g(f) = m$, determine $m+k$. 

Find the value of $m+k$. \( \text{(2016 CCA Math Bonanza Individual #12)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{961}
The teacher gave 210 pencils to 30 students in the first classroom and 20 students in the second classroom equally. How many pencils are left for each of the two classrooms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Using only the paths and the directions shown, count the number of different routes from M to N. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given that the lengths of all three sides of a triangle satisfy the equation $x^{2}-6x+8=0$, calculate all possible perimeters of this triangle. Express your answer as a list of whole numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6, 10, 12}
Find the sum of the infinite series for positive real numbers $a$, $b$, and $c$:

$$ \sum_{n=0}^\infty \frac{a^n}{\Gamma(b+nc)} $$

Express your answer in terms of the Gamma function, $\Gamma$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E_{c,b}(a)}
On a windless day, a polar bear found itself on a small ice floe that broke off from an iceberg, floating in still water. Rescuers from a helicopter hovering above the floe noted that the animal was walking in a circle with a diameter of 9.5 meters. They were surprised when later, in a photograph, they saw the bear's trail and the diameter of this trajectory was 10 meters. Estimate the mass of the ice floe, assuming the mass of the bear is 600 kg.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11400}
Calculate the volume of the solid $D$ defined by the inequalities: \(x^2+y^2+(z-1)^2 \le 1\) and \(z^2 \le x^2+y^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Let $M=\{0,1,2,\dots,2022\}$ and let $f:M\times M\to M$ such that for any $a,b\in M$,  
\[f(a,f(b,a))=b\]  
and $f(x,x)\neq x$ for each $x\in M$. How many possible functions $f$ are there $\pmod{1000}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider a regular hexagon with vertex set $M$. Determine the number of chains of subsets $\emptyset \subset A \subset B \subset C \subset D \subset M$ such that each subset is distinct and the chain begins with the empty set and ends with the full set $M$. What is the total number of such chains?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1800}
If $a^{2}+b^{2}=c^{2}$ $(a,b,c\in\mathbb{N}^{*})$, we call $a$, $b$, $c$ a Pythagorean triple. The following are several groups of Pythagorean triples: $3$, $4$, $5$; $5$, $12$, $13$; $7$, $24$, $25$; $9$, $40$, $41$. By analogy, the three numbers of the fifth group of Pythagorean triples are \_\_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11, 60, 61}
How can we calculate the surface area of the solid enclosed by the paraboloid $z=2+x^2+y^2$ and the cone $z=4-\sqrt{x^2+y^2}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{5\sqrt{5}-1}{6}\pi+\sqrt{2}\pi\)}
Four rays spread out from a point $O$ in a 3-dimensional space such that the angle between any two rays is $a$. Determine the value of $\cos a$. \(\boxed{\text{Your answer here}}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{3}}
Determine the number of 3-Sylow subgroups in a simple group with 168 elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Evaluate the limit \( \displaystyle \lim_{n\to \infty} \int_0^1 \frac{n^2x^{n-2}}{(x+1)^2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In the diagram, hexagon \( P Q R S T U \) has interior right angles at \( P, Q, S, T \), and \( U \) and an exterior right angle at \( R \). Also, \( P U = U T \), \( P Q = S T = 10 \), and \( Q S = 8 \). Calculate the perimeter of \( P Q R S T U \) and round your answer to the nearest whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
求$y = \sin ^ { 2 } x + 2 \sin x \cdot \cos x + 3 \cos ^ { 2 } x$的最值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2-\sqrt{2}$.}
Compare the sizes of the following numbers:

\[ a = 3^{25} \]
\[ b = 4^{20} \]
\[ c = 5^{15} \]

Determine the order of \(a\), \(b\), and \(c\) using the symbols \(<\), \(=\), or \(>\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c < a < b}
Denis painted some faces of his cubes gray. Vova selected 10 cubes so that all of them had different paint patterns, and then he stacked them as shown in the picture. How many white faces does the top cube have in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the Theta class for the recursion \( T(n) = T(3n/4) + T(n/6) + 5n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n)}
Calculate the sum of the series: \( \cos \frac{2\pi}{2013} + \cos \frac{4\pi}{2013} + \cdots + \cos \frac{2010\pi}{2013} + \cos \frac{2012\pi}{2013} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Find the solution set for \( x > 0 \) that satisfies the inequality \( af(x) + bx^2 f\left( \frac{1}{x} \right) \geq \sin(x) \), given that \( ab \neq 0 \) and \( a^2 \neq b^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \infty)}
Given that $E[X^n] = 3n$ for $n \ge 1$, find the expected value $E[e^X]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 3e}
In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares. Calculate the measure of angle GDA in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Evaluate the integral \( \int_0^{\infty} x^{4n+3} e^{-x} \sin x \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Establish necessary and sufficient conditions on the constant  $k$  for the existence of a continuous real valued function  $f(x)$  satisfying   $$ f(f(x))=kx^9 $$   for all real  $x$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ k \geq 0 }
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \(12^2\) and \(13^2\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}
Find the smallest positive integer \( k \) such that \( 3^k \equiv 1 \pmod{3215} \), given that \( \phi(3215) = 2568 \) and \( 3^{2568} \equiv 1 \pmod{3215} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{428}
In the diagram, each of the integers 1 through 9 is to be placed in one circle so that the integers in every straight row of three joined circles add to 18. The 6 and 1 have been filled in. Determine the value of the number represented by x. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
If we replace the two middle digits $\overline{00}$ in 2009 with the two-digit numbers $\overline{ab}$ and $c\bar{d}$ respectively, we obtain perfect squares: $\overline{2ab9}=n^{2}$ and $\overline{2cd9}=m^{2}$, where $m > n$ and $m, n \in \mathbb{N}$. Determine the array $(m+n, \overline{ab}+\overline{cd})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(100,100)}
Given a acute triangle $PA_1B_1$ is inscribed in the circle $\Gamma$ with radius $1$. for all integers $n \ge 1$ are defined:
$C_n$ the foot of the perpendicular from $P$ to $A_nB_n$
$O_n$ is the center of $\odot (PA_nB_n)$
$A_{n+1}$ is the foot of the perpendicular from $C_n$ to $PA_n$
$B_{n+1} \equiv PB_n \cap O_nA_{n+1}$

If $PC_1 =\sqrt{2}$, find the length of $PO_{2015}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$PO_{2015} = \frac{1}{2^{1007}}$

}
Find all finite sets \(A\) such that for every \(x \in A\), the expression \(x^3 - 3|x| + 4\) is also in \(A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Find a constant \( M \) such that for all functions \( f \) in \( L^2(-1,1) \), the inequality \( \Vert f \Vert_{L^2} \leq M \Vert f \Vert_{H} \) holds, where \( \Vert f \Vert_H^{2} = \int_{-1}^1 \vert f(x) \vert^2 \frac{1}{1+x^2}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
[ $\left.\frac{\text { angles between bisectors }}{\text { Dihedral angle }}\right]$

The base of the pyramid $S A B C$ is a right triangle $A B C$ ( $C$ - the vertex of the right angle). All lateral faces of the pyramid are inclined to its base at the same angle, equal to $\arcsin \frac{5}{13}$. Find the area of the lateral surface of the pyramid, if $S O-$ is the height of the pyramid, $A O=1, B O=3 \sqrt{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{91}{25}}
Determine the number of real solutions to the equation \( f(f(f(f(x)))) = \frac{1}{\sqrt{2}} \), where \( f(x)=\frac{1}{(1-x^{2011})^{\frac{1}{2011}}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a labyrinth game, the largest room measures 10x10x8 and the smallest room measures 4x4x4. Assuming there is no empty volume between rooms and adjacency is defined by overlapping faces of at least 1 square, determine the maximum number of adjacent rooms possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Find the maximum value of \( p + q \) such that the inequality \( 2px^2 + qx - p + 1 \geq 0 \) holds for all \( |x| \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In triangle \(ABC\), the bisectors \(AM\) and \(BN\) intersect at the point \(O\). Vertex \(C\) lies on the circle passing through the points \(O, M, N\). Find \(OM\) and \(ON\) if \(MN = \sqrt{3}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the fractal dimension of a set using the following data for the number of boxes needed to cover the set at different shrink values: $N(1) = 7$, $N(0.3) = 201$, $N(0.22) = 478$, $N(0.12) = 2595$, and $N(0.06) = 17950$. Use these values to determine the fractal dimension.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.79}
Determine the curves along which the function $u(x,y)$ remains constant for the partial differential equation $(x\partial_y - y\partial_x)u = 0$. Use the fact that the directional derivative along these curves is zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + y^2 = C}
A lattice point is a point with integer coordinates. The line y = 3x - 5 passes through square PQRS. If the coordinates of R are (2009, 2009), then find the number of lattice points on the line which are inside the square. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{670}
The total capacity of three tanks is 1620 liters. Two of them are filled with kerosene, and the third one is empty. To fill the empty tank, either all the contents of the first tank plus $1/5$ of the contents of the second tank, or all the contents of the second tank plus $1/3$ of the contents of the first tank is needed. Find the capacity of each tank.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{540, 450, 630}
Using the Wright-Fisher model and the mean persistence time formula, calculate the probability of extinction of an allele in 120 generations, given that the allele starts at a frequency of 0.2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8}
Given 4 equilateral triangles on a plane, each with distinct vertices, forming a set S of 12 vertices, and no three points in S are collinear, determine the maximum number of squares that can be formed using the vertices from S.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In the diagram, points \( P, Q, R, \) and \( S \) are at intersections of gridlines in a \( 6 \times 6 \) grid. Find the perimeter of parallelogram \( PQRS \). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find the minimum value of $p$ for which the series \( \sum_{n=1}^{\infty}(1n^{27}+8)^{27}(1+27n^{29}+8n^{30})^p \) is divergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{73}{3}}
Three numbers are chosen at random between 0 and 2. What is the probability that the difference between the largest number and the smallest number is less than \( \frac{1}{4} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{256}}
Consider and odd prime  $p$ . For each  $i$  at  $\{1, 2,..., p-1\}$ , let  $r_i$  be the rest of  $i^p$  when it is divided by  $p^2$ .
Find the sum: $r_1 + r_2 + ... + r_{p-1}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{p^2(p-1)}{2}}
Find \( \lim_{x \to 0} g'(x) \) where the function \( g(x) \) is defined as follows:

\[
g(x) = \begin{cases} x^3 \left\lfloor \dfrac{1}{x^2} \right\rfloor & x \neq 0 \\
0 & x = 0 \end{cases}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the minimum number of quadratic equations required to define the 10-dimensional Spinor variety in a scheme-theoretic sense.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given that \(a, b, c\) are unit vectors satisfying \(|a-b|^2+|b-c|^2+|c-a|^2=9\), find the value of \(|2a+5b+5c|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the limit \( \lim_{x\to0}\int_{1}^{\infty} \frac{\sin^2(xy/2)}{x+y^2} \, dy \) using the change of variable technique.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the values of \(\alpha > 1\) for which the series \(\sum_{n=1}^{\infty} |\sin n|^{cn^\alpha}\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > 2}
The lengths of the sides of a triangle are consecutive integers. Find these numbers, given that one of the medians is perpendicular to one of the angle bisectors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2, 3, 4}
Integers with absolute value less than $4$ include ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, \pm 1, \pm 2, \pm 3}
7. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{0}=0$,
$$
a_{n+1}=\frac{8}{5} a_{n}+\frac{6}{5} \sqrt{4^{n}-a_{n}^{2}}(n \geqslant 0, n \in \mathbf{N}) \text {. }
$$

Then $a_{10}=$ . $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{24576}{25}}
What is the minimal possible value of the Shannon entropy \(H(X_1 X_2 X_3 X_4)\) for 4 pairwise independent unbiased Bernoulli random variables \(X_1, X_2, X_3, X_4\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A hexagon is inscribed in a circle . Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$ , has length $31$ . Find the sum of the lengths of the three diagonals that can be drawn from $A$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{384}
Evaluate the limit: \[ \lim_{n \to +\infty} \sum_{k=0}^n \left(\sqrt[n]{k} - \sqrt[n]{n}\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
In the $n$-dimensional non-negative orthant $\mathbb{R}_+^n$ with $n \geq 1$, consider an $(n-1)$-simplex $S$ defined by \[ S = \left\{x \in \mathbb{R}_+^n : \sum_{i=1}^n x_i = 1\right\} \] and its relative interior $\mathring{S}$. Let $E$ be a linear subspace of dimension $n-2$ in $\mathbb{R}^n$ such that $E \cap \mathring{S} \neq \emptyset$. The intersection $E \cap \mathring{S}$ is a convex polyhedron with a relative interior of dimension $n-3$ and has at least $n-2$ extremal points. Determine the maximum possible number of extremal points for $E \cap S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
What is the cardinality of the set of continuous functions from the closed interval [0, 1] to itself?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
What is the cardinality of the set of all intervals of real numbers \( \{(a,b) \mid a,b \in \mathbb{R}\} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Given a number $m$, such as $9874$, determine the number of natural numbers less than $m$ that do not contain the digit $3$. Provide your answer for $m = 9874$. Note that $1 \leq m < 10^{16}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6455}
Evaluate the integral: \[ \int\limits_{0}^\infty \frac{\cos(\pi x)}{x}  \delta \left[ (x^2-1)(x-2) \right]  \mathrm{d}x \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{3}}
Calculate the derivative of \( \nabla_{\beta} \| g(X \beta)\|_{2}^{2} \), where \( g \) is the Step function, \( X \in \mathbb{R}^{n \times d} \), and \( \beta \in \mathbb{R}^d \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider a surjective and non-decreasing function \( f: \{1, 2, 3, \ldots, 13\} \rightarrow \{1, 2, 3, \ldots, 9\} \). What is the probability that \( f(7) = 4 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{20}{99}}
Evaluate the sum \( \tan^2(1^\circ) + \tan^2(3^\circ) + \tan^2(5^\circ) + \tan^2(7^\circ) + \ldots + \tan^2(89^\circ) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4005}
In $\triangle ABC$, $\angle A=30^{\circ}$, $BC=13$. Given $2$ circles $\gamma_1, \gamma_2$ ith radius $r_1,r_2$ contain $A$ and touch $BC$ at $B$ and $C$ respectively. Find $r_1r_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
169
}
Determine the number of integers \(x\) in the range \(1 \leq x \leq n\) that satisfy the inequality:
\[\left\lfloor \frac{(x+1)r}{n}\right\rfloor > \left\lfloor \frac{xr}{n}\right\rfloor\]
where \(r\) and \(n\) are integers such that \(r < n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r}
Let $(X_1, X_2, \ldots)$ be an independent sequence of random variables, where each $X_n$ is uniformly distributed on the interval $[0, n]$. Determine the probability:

$$ P\left(\{w: \lim_{n \to \infty} X_n(w) = \infty\}\right) $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to  $2$  and the sum of its terms is  $200$ . How many progressions does Omar's list have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate the line integral \( \int\limits_C (-y^3dx + x^2dy-z^3dz) \) using Stokes' Theorem, where \( C \) is the intersection of the cylinder \( x^2 + y^2 = 1 \) and the plane \( x + y + z = 1 \). Determine the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{4}}
Evaluate the limit: \[ \lim_{{n \to \infty}} \left( \frac{10^n+5^n}{2^n+5^n} - \frac{8^n-4^n+\sin(4^n)}{(-1)^n+4^n} \right) \] without using L'Hôpital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
给定整数 $m 、 n(2 \leqslant m<n,(m, n)=1)$.求最小的整数 $k$ ，满足对集合 $\{1,2, \cdots, n\}$ 的任意 $m$ 元子集 $I$, 若 $\sum_{i \in I} i>k$, 则存在 $n$ 个实数 $a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n}$, 使得
$$
\frac{1}{m} \sum_{i \in I} a_{i}>\frac{1}{n} \sum_{i=1}^{n} a_{i}
$$
若$m=3,n=5$，求$k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Calculate the limit: $$\lim_{n\rightarrow\infty}\left(\frac{\prod\limits_{r=1}^n\left(1 +\frac{r}{n^2}\right)}{e^{1/2}}\right)^n.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{1}{3}}}
Let \( a, b, c, d, e \) be positive integers such that their sum is 2018. Define \( M = \max (a+b, b+c, c+d, d+e) \). Determine the smallest possible value of \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{673}
Determine the number of real solutions a to the equation:
$$\left[\frac{1}{2}a\right]+\left[\frac{1}{3}a\right]+\left[\frac{1}{5}a\right]=a.$$
$\mathrm{Here,if}x\text{is a real number,then}[x]\text{denotes the greatest integer that is less than or equal to}x.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
30
}
Through which point of triangle ABC does the trajectory of P necessarily pass, given that $O$ is a fixed point on the plane, $A$, $B$, and $C$ are three non-collinear points on the plane, and a moving point $P$ satisfies $\overrightarrow{OP} = \overrightarrow{OA} + \lambda\left(\frac{\overrightarrow{AB}}{|\overrightarrow{AB}|} + \frac{\overrightarrow{AC}}{|\overrightarrow{AC}|}\right), \lambda \in [0, +\infty)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Incenter}
Determine the smallest integer \( n \) such that the error in the nth degree Taylor polynomial approximation of \( \ln(1-x^2) \) about \( x=0 \) is less than 0.1 for \( 0 \leq x \leq 0.5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}