inclusionAI/Ring-lite-rl-data huggingface dataset:
Solve the equation\( x \cdot  \left\lbrack  {x \cdot  \left\lbrack  {x \cdot  \left\lbrack  x\right\rbrack  }\right\rbrack  }\right\rbrack   = {88} \) in the set of real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{22}{7}$}
$设函数f(x)=3|x-2|+|x|.$
$求直线 y = a 与 f(x) 的图象围成的三角形的面积的最大值.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
$ABC$ is an isosceles triangle with base $\overline{AB}$. $D$ is a point on $\overline{AC}$ and $E$ is the point on the extension of $\overline{BD}$ past $D$ such that $\angle{BAE}$ is right. If $BD = 15, DE = 2,$ and $BC = 16$, then $CD$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
The sequence starts at 2,187,000 and each subsequent number is created by dividing the previous number by 3. What is the last integer in this sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
In triangle \( ABC \), a circle is described on side \( AC \) as its diameter, intersecting side \( AB \) at point \( M \) and side \( BC \) at point \( N \). Given that \( AC = 2 \), \( AB = 3 \), and \( AN = 1.8 \), find the cosine of angle \( BAC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.54 - 0.4 \sqrt{0.76}}
Given real numbers $x$, $y$, and $z$, such that \[\frac{\sin{x}+\sin{y}+\sin{z}}{\sin{(x+y+z)}}=\frac{\cos{x}+\cos{y}+\cos{z}}{\cos{(x+y+z)}}=2,\] find the value of \[\sum_{cyc}\sin x\sin y.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{3}{4}}
Determine the value of $x$ such that \( \lim\limits_{n\to\infty} \sqrt{1+\sqrt{x+\sqrt{x^2+\cdots+\sqrt{x^n}}}} = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the coefficient of \(x^{25}\) in the expansion of \((2x^3 + 3x^2 - 5)^{15}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7999230330000}
Find the function \( f(x) \) that satisfies the functional equation \( f(x+y) = f(x) + f(y) + y\sqrt{f(x)} \) for all real numbers \( x \) and \( y \), given that \( f'(0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the values of $k \in \mathbb{R}$ for which the fractional Sobolev space $W^{k,2}(\mathbb{R}^n)$ is continuously embedded into the space of Lebesgue-measurable functions $\mathcal{M}$ on $\mathbb{R}^n$. Specifically, for which $k$ does the identity map $C^\infty_c(\mathbb{R}^n) \hookrightarrow \mathcal{M}$ extend by continuity to a map $W^{k,2}(\mathbb{R}^n) \to \mathcal{M}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \geq 0}
Find the limit \( \lim_{x \to \infty} \left( e^{1/x}(x^3-x^2+x/2)-(x^3+x^6)^{0.5} \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{3}}
Fiona has a deck of cards labelled $1$ to $n$, laid out in a row on the table in order from $1$ to $n$ from left to right. Her goal is to arrange them in a single pile, through a series of steps of the following form:

If at some stage the cards are in $m$ piles, she chooses $1\leq k<m$ and arranges the cards into $k$ piles by picking up pile $k+1$ and putting it on pile $1$; picking up pile $k+2$ and putting it on pile $2$; and so on, working from left to right and cycling back through as necessary.
She repeats the process until the cards are in a single pile, and then stops. So for example, if $n=7$ and she chooses $k=3$ at the first step she would have the following three piles:

$\begin{array} {ccc}7 & & \\ 4 & 5 & 6 \\ 1 & 2 & 3\\ -- & -- & --\end{array}$

If she then chooses $k=1$ at the second stop, she finishes with the cards in a single pile with cards ordered $6352741$ from top to bottom.

How many different final piles can Fiona end up with?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-2}}
Determine the values of \( \alpha \) for which the series \( \sum_{n \geqslant 1} n^{\alpha} \cdot x^{n} \cdot (1-x) \) uniformly converges on the interval \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha < 0}
After moving his sticky toy spider one morning, Tony heads outside to play \u0022pirates\u0022 with his pal Nick, who lives a few doors down the street from the Kubiks.  Tony and Nick imagine themselves as pirates in a rough skirmish over a chest of gold.  Victorious over their foes, Tony and Nick claim the prize.  However, they must split some of the gold with their crew, which they imagine consists of eight other bloodthirsty pirates.  Each of the pirates receives at least one gold coin, but none receive the same number of coins, then Tony and Nick split the remainder equally.  If there are $2000$ gold coins in the chest, what is the greatest number of gold coins Tony could take as his share?  (Assume each gold coin is equally valuable.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
982
}
Find the real value \( \alpha \) such that the coordinate transformation \( \varphi(p,q) \rightarrow (P,Q) \), where \( Q = q^2 + \alpha\sqrt{q^2+p} \) and \( P = q + \sqrt{q^2+p} \), preserves the symplectic form \( dp \wedge dq \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a diagram, several circles are connected by segments. Choose a natural number \( n \) and arrange different natural numbers in the circles such that:

- If two numbers \( a \) and \( b \) are not connected by a segment, then \( |a - b| \) is coprime with \( n \).
- If two numbers \( a \) and \( b \) are connected by a segment, then \( |a - b| \) shares a common natural divisor greater than 1 with \( n \).

Determine the smallest natural number \( n \) for which such an arrangement is possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the number of elements in the quotient group \(\mathbb{Z}^2/\!\ker\varphi\), where \(\varphi: \mathbb{Z}^2 \to S_{15}\) is the homomorphism defined by \(\varphi(1,0)=(2,7,3)(3,11,5)(12,13)\) and \(\varphi(0,1)=(14,15)(1,4,6,8,9,10)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
A pendulum is suspended near a vertical wall and oscillates in a plane parallel to the wall. A thin nail is driven into the wall such that the middle of the pendulum's string hits it every time the pendulum passes through its equilibrium position from right to left. Given that the period of oscillation of this pendulum is \( T = 2.41 \) seconds and the acceleration due to gravity is \( g = 10 \, \text{m/s}^2 \), find the length of the pendulum's string.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.02}
Determine the minimum number of variables required to construct a sentence in a first-order logic language consisting solely of the equality predicate "=" that is valid in a structure if and only if the structure's domain contains exactly n elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n+1}
Determine whether the infinite product \( \prod_{k=2}^\infty \left(1 - \frac{1}{k^\alpha}\right) \) converges for a given \( \alpha \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > 1}
Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=1}^n\frac{k}{n}\sin\left(\frac{2\pi k}{n}\right)\sin\left(\frac{2\pi k-\pi}{n}\right)\sin\left(\frac{\pi}{n}\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Find the limit of the sequence \(\{a_n\}\) defined recursively by \(a_1 = 0\) and \(a_{n+1} = a_n + \sqrt{1 + a_n^2}\) as \(n\) approaches infinity. Express the limit in terms of the ratio \(\frac{a_n}{2^{n-1}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{\pi}}
Determine the number of infinite subsets of the set of natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
The highest power of 2 that is a factor of \(15.13^{4} - 11^{4}\) needs to be determined.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Determine the cardinality of the set $S$ of all convergent sequences of real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Compute the treewidth of the complete bipartite graph $K_{m,n}$ using the chordal graph characterization. Recall that the treewidth $tw(G)$ is given by $\min \{\omega(H)-1 : G\subseteq H \ \wedge \ H \ \text{is chordal}\}$. Determine the treewidth of $K_{m,n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\min(m, n)}
You want to create a three-digit number by removing one from the number cards 2, 5, 6, or 9 and using all remaining ones. Find the sum of the third largest and third smallest possible numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1221}
Paul had 50 books. He went to a garage sale where they were selling mystery, science fiction, and romance books. The mystery books were on sale: buy 2, get 1 free. For the science fiction books, there was a 20% discount, and romance books cost their original price. 

Here are the prices of the books:
- Mystery books: $4 each
- Science fiction books: $6 each (before discount)
- Romance books: $5 each

If Paul spent $90 at the garage sale and bought the same number of books in each category, how many books did he buy in total, considering the discounts and promotions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Determine the number of non-real roots of the equation \(z^2 = \sin(z)\) within the region \(\{z \in \mathbb{C} \setminus \mathbb{R} \mid |z| < 2\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $X$ be a set of 9 positive integers. Define $E \subseteq X$, and let $S(E)$ represent the sum of the elements in the subset $E$. Determine the maximum value of an element $n$ in $X$ such that there exist at least two distinct subsets $A$ and $B$ of $X$ for which $S(A) = S(B)$. Find the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255}
The curve \( y = \sin(4x) \cdot 10e^{-x/2} \) for \( x \ge 0 \) has stationary points labeled as \( T_1, T_2, T_3, \ldots \). The x-coordinates of \( T_1 \) and \( T_2 \) are \( 0.362 \) and \( 1.147 \) respectively. Given that the x-coordinate of \( T_n \) is greater than 25, find the least possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Find the minimum value of the integral \( \int_0^{\infty} |a+bx+cx^2+x^3|^2 e^{-x} \, dx \) with respect to \(a, b, c \in \mathbb{C}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Evaluate the triple integral \( \iiint\limits_{V} xe^{-z} \, dV \) where \( V \) is the portion of the cone \( z=\sqrt{x^2+y^2} \) for \( x \geq 0 \) using cylindrical polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the minimum value of the integral \( \int_{-1}^{1} |x^3-a-bx-cx^2|^2 \, dx \) over all real numbers \( a, b, c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{175}}
How many tori are in the connected sum obtained by encircling the wireframe of an octahedron with a small tube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
What is the maximal number of elements in a sigma-algebra generated by a collection of N subsets of a set X?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^N}}
A weight is to be suspended 10 feet below a horizontal line AB by a wire in the shape of a Y. If the points A and B are 8 feet apart, determine the shortest total length of wire that can be used.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10 + 4\sqrt{3}}
Determine the set of points where the partial derivative $f_x$ is continuous for the function \( f(x,y) = \frac{x^2-y^2}{\sqrt[3]{x^2+y^2}} \) for \((x,y) \neq (0,0)\) and \( f(x,y) = 0 \) for \((x,y)=(0,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}^2}
What is the minimum time complexity for checking whether a set of integers from 1 to n can be partitioned into two subsets with equal sums?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(1)}
Given 10 distinct points on a plane, consider the midpoints of all segments connecting all pairs of points. What is the minimum number of such midpoints that could result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Let $ABC$ be a triangle such that angle $C$ is obtuse. Let $D \in [AB]$ and $[DC] \perp [BC]$. If $m(\widehat{ABC}) = \alpha$, $m(\widehat{BCA}) = 3\alpha$, and $|AC| - |AD| = 10$, find the length of $|BD|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
For a positive integer \( n \), define \( S_{n} \) as the minimum value of the sum
\[
\sum_{k=1}^{n} \sqrt{(2k-1)^{2}+a_{k}^{2}}
\]
where \( a_{1}, a_{2}, \cdots, a_{n} \) are positive real numbers whose sum is 17. There exists a unique positive integer \( n \) for which \( S_{n} \) is also an integer. Find this \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A $\textit{palindrome}$ is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a time with a single-digit hour value. Therefore, 01:10 may not be used.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{57}
Two trains are moving at 80 kmph and some speed in opposite directions. Their lengths are 150 m and 100 m respectively. The time they will take to pass each other completely is 5.999520038396928 seconds. What is the speed of the second train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
12.418 Through the vertex of the base of a regular triangular pyramid, a plane is drawn perpendicular to the opposite lateral face and parallel to the opposite side of the base. This plane forms an angle $\alpha$ with the plane of the pyramid's base. Find the plane angle at the vertex of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\arctan(\sqrt{3}\cdot\sin\alpha)}
The function $f(x)=(m^{2}-m-1)x^{m^{2}+m-3}$ is a power function, and when $x\in (0,+\infty)$, $f(x)$ is a decreasing function. Find the real number $m=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
In convex quadrilateral \(ABCD\) with \(AB=11\) and \(CD=13\), there is a point \(P\) for which \(\triangle ADP\) and \(\triangle BCP\) are congruent equilateral triangles. Compute the side length of these triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial \[q(x) = c_3x^3+c_2x^2+c_1x+c_0\]such that $D(m) = q(m)$ for all odd integers $m\ge 5$. What is $c_1?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$11$}
What is the area of the smallest circular region that can cover the triangle ABC with sides a = 2, b = 2√2, and c = √2 + √6?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2+\sqrt{3})\pi}
the h.c.f. of two numbers is 50 and the other three factors of their l.c.m. are 13, 23, and 31. The larger of the two numbers is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{463450}
$A = 1 \cdot 4 \cdot 7 \cdots 2014$. Find the last non-zero digit of $A$ if it is known that $A \equiv 1 \pmod{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
24. [12] Find the number of subsets $S$ of $\{1,2, \ldots 6\}$ satisfying the following conditions:
- $S$ is non-empty.
- No subset of $S$ has the property that the sum of its elements is 10 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Adam’s wardrobe is too crowded so he decides to donate some of his clothes to a charity shop. He takes out 4 pairs of pants, 4 jumpers, 4 pajama sets (top and bottom), and some t-shirts, then asks his friends if they have anything they want to donate. 3 of his friends donate the same amount of clothing as Adam each. Then he takes another look over his clothing and decides that he actually wants to keep half of his clothes. In total, 126 articles of clothing are being donated. How many t-shirts did Adam initially take out?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In a speech contest held in a class, there are a total of $5$ contestants, including $3$ females (among which is female A) and $2$ males. If the two males cannot appear consecutively, and female A cannot be the first to appear, what is the number of ways for the appearance order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Two trains of different lengths run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. One train is 450 m long, and they take 26.99784017278618 seconds to cross each other. How long is the other train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Let $f(x)=x^3+7x^2+9x+10$. Find the value of $p$ such that for every integer $a,b$, the statement
\[
f(a) \equiv f(b) \ (\text{mod } p) \Rightarrow a \equiv b \ (\text{mod } p)
\]
is satisfied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
What is the measure, in degrees, of the smallest positive angle \( x \) for which \( 4^{\sin ^{2} x} \cdot 2^{\cos ^{2} x} = 2 \sqrt[4]{8} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
A positive integer  $N$  is given. Panda builds a tree on  $N$  vertices, and writes a real number on each vertex, so that  $1$  plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be  $M$  and the minimal number written  $m$ . Mink then gives Panda  $M-m$  kilograms of bamboo. What is the maximum amount of bamboo Panda can get? 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ (N-1)^2 }
S is a subset of {1, 2, 3, ... , 16} which does not contain three integers which are relatively prime in pairs. How many elements can S have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
On a race track a maximum of some horses can race together at a time. There are a total of 25 horses. There is no way of timing the races. The minimum number of races needed to get the top 3 fastest horses is 7. What is the maximum number of horses that can race together at a time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given |-x| = 2023, find the value of x. Express your answer using the ± symbol to represent both solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±2023}
Determine the minimum value of  $$ x^{2014} + 2x^{2013} + 3x^{2012} + 4x^{2011} +\ldots + 2014x + 2015 $$  where  $x$  is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1008}
What is the minimum value of the expression 

\[
\sqrt{x^{2}+y^{2}-2x-2y+2} + \sqrt{x^{2}+y^{2}-2x+4y+2\sqrt{3}y+8+4\sqrt{3}} + \sqrt{x^{2}+y^{2}+8x+4\sqrt{3}x-4y+32+16\sqrt{3}}
\]

for \( (x, y) \in \mathbb{R} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{6} + 4\sqrt{2}}
Call a three-digit number  $\overline{ABC}$   $\textit{spicy}$  if it satisfies  $\overline{ABC}=A^3+B^3+C^3$ . Compute the unique  $n$  for which both  $n$  and  $n+1$  are  $\textit{spicy}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{370}
In how many ways can a square with a side length of 5 units be placed on an $8 \times 8$ chessboard so that each of its vertices falls on the center of a square? (Solutions that can be transformed into each other by reflection or rotation are not considered different.)





Translating the text as requested, while preserving the line breaks and formatting.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Positive real numbers \( x \) and \( y \) satisfy the following conditions: there exist \( a \in [0, x] \) and \( b \in [0, y] \) such that \( a^2 + y^2 = 2 \), \( b^2 + x^2 = 1 \), and \( ax + by = 1 \). What is the maximum value of \( x + y \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
Given $7\sqrt{3}sinθ=1+7cosθ$, find $sin(2θ+\frac{π}{6})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{97}{98}}
17 Given positive numbers $a$, $b$, $c$ satisfy: $2a + 4b + 7c \leqslant 2abc$, find the minimum value of $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{2}}
If the four-digit number $\overline{2 \mathrm{ABC}}$ is divisible by 13, what is the maximum value of $A + B + C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
There are three identical red balls, three identical yellow balls, and three identical green balls. In how many different ways can they be split into three groups of three balls each?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A 290 meter long train running at a certain speed crosses another train running in the opposite direction at the speed of 80 kmph in 9 seconds. The length of the other train is 210.04 meters. What is the speed of the first train in kmph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120.016}
A natural number is written on the board. If its last digit (in the units place) is erased, the remaining non-zero number is divisible by 20. If the first digit is erased, the remaining number is divisible by 21. What is the smallest number that could be on the board if its second digit is not 0?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1609}
The area of the part of the surface $x^2+y^2+z^2=4$ which lies inside the surface $x^2+y^2=2x$ is equal to $n(\pi-2)$ for an integer n. What is the value of n?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the degree of the splitting field of the polynomial $g(x) = x^4 + 2$ over $\mathbb{Q}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
正 2015 边形 $A_{1} A_{2} \cdots A_{2015}$ 内接于单位圆 $O$, 任取它的两个不同的顶点 $A_{i}, A_{j}$,则 $\left|\overrightarrow{O A_{i}}+\overrightarrow{O A_{j}}\right| \geqslant 1$ 的概率是 $\qquad$.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{1678}