inclusionAI/Ring-lite-rl-data huggingface dataset:
7. We call a four-digit number with this property a "central four-digit number": by rearranging the four digits of this number in any order, and arranging all the four-digit numbers (at least 2) formed in ascending order, the original four-digit number is exactly in the middle position. For example, 2021 is a "central four-digit number". Therefore, including 2021, there are $\qquad$ such "central four-digit numbers" in total.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Determine the value of $c$ that minimizes the maximum norm of the difference between $\cos(x)$ and a polynomial of degree $n=1$ over the interval $[-\pi/2,\pi/2]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A regular (equilateral) triangle $D E F$ has an inscribed circle with radius $r$. This circle touches three other circles of the same radius externally at the points where they touch the sides of the triangle. The centers of the external circles are $O_{1}, O_{2}, O_{3}$ respectively. Find the area of the hexagon formed by the intersection of triangles $D E F$ and $O_{1}, O_{2}, O_{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 r^{2} \sqrt{3}}
Elza draws $2013$ cities on the map and connects some of them with $N$ roads. Then Elza and Susy erase cities in turns until just two cities left (first city is to be erased by Elza). If these cities are connected with a road then Elza wins, otherwise Susy wins. Find the smallest $N$ for which Elza has a winning strategy.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006}
Given that p and q are two natural numbers that do not end in zero and their product is always a power of 10, find the last digit that p - q cannot be when p > q. Express your answer as a single digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The school principal decided that she wanted every class to have an equal number of boys and girls in each first-grade classroom. There are 7 classrooms. There are 68 boys and 53 girls. How many total students can be in each classroom while maintaining an equal distribution of boys and girls in each room?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In triangle \(ABC\), angle bisectors \(AA_{1}\), \(BB_{1}\), and \(CC_{1}\) are drawn. \(L\) is the intersection point of segments \(B_{1}C_{1}\) and \(AA_{1}\), \(K\) is the intersection point of segments \(B_{1}A_{1}\) and \(CC_{1}\). Find the ratio \(LM: MK\) if \(M\) is the intersection point of angle bisector \(BB_{1}\) with segment \(LK\), and \(AB: BC: AC = 2: 3: 4\). (16 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11/12}
Let \(D(n)\) denote the number of ways of writing the positive integer \(n\) as a product \(n = f_1 \cdot f_2 \cdots f_k,\) where \(k \ge 1\), the \(f_i\) are integers strictly greater than 1, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number 6 can be written as 6, \(2 \cdot 3\), and \(3 \cdot 2\), so \(D(6) = 3\). What is \(D(96)\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112}
Through the altitude of an equilateral triangle with side length \(a\), a plane is drawn perpendicular to the plane of the triangle, and within this plane a line \(l\) is taken parallel to the altitude of the triangle. Find the volume of the solid obtained by rotating the triangle around the line \(l\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi a^3 \sqrt{3}}{24}}
Let's inscribe a rectangle with the largest area in the ellipse $b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}(a>b)$ (the sides of the rectangle are parallel to the major axes of the ellipse). In this rectangle, inscribe an ellipse whose major axes are equal to the sides of the rectangle. In the thus obtained ellipse, inscribe a rectangle with the largest area, and so on. What is the sum of the areas of the infinitely many rectangles and ellipses thus obtained?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2ab\pi+4ab}
Let $f(w)$ be a bilinear transformation that maps the circle $|w|=1$ to the circle $|f(w)|=1$. If $f(0) = z_2$ and $f(1) = 2$, find the value of $|f(-1)|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compare the magnitude (indicate with ">" or "<"): $-(-1)$ and $-\left(+2\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{>}
What positive integer can $n$ represent if it is known that by erasing the last three digits of the number $n^{3}$, we obtain the number $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Of the students attending a school athletic event, $80\%$ of the boys were dressed in the school colors, $60\%$ of the girls were dressed in the school colors, and $45\%$ of the students were girls. Find the percentage of students attending the event who were wearing the school colors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
Let $p$ be a constant number such that $0 < p < 1$. Evaluate

\[
\sum_{k=0}^{2004} \frac{p^k (1-p)^{2004-k}}{\int_0^1 x^k (1-x)^{2004-k} \, dx}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2005}
Let $A$ be a subset of $\{1, 2, 3, \ldots, 50\}$ with the property that for every $x, y \in A$ with $x \neq y$, it holds that \[\left| \frac{1}{x} - \frac{1}{y} \right| > \frac{1}{1000}.\] Determine the largest possible number of elements that the set $A$ can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Suppose \(a\) is a real number such that the equation 
\[ a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)} \]
has more than one solution in the interval \((0, \pi)\). The set of all such \(a\) can be written in the form \((p,q) \cup (q,r),\) where \(p, q,\) and \(r\) are real numbers with \(p < q < r\). What is the value of \(p+q+r\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
A traffic light cycles through red for 30 seconds, yellow for 10 seconds, and green for 40 seconds. You can proceed during the yellow light. Calculate the probability that your waiting time is no more than 10 seconds to proceed. Express your answer as a simplified fraction in the form $\boxed{\frac{a}{b}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
Suppose that $x$, $y$, and $z$ are complex numbers of equal magnitude that satisfy:
\[
x + y + z = -\frac{\sqrt{3}}{2} - i\sqrt{5}
\]
and
\[
xyz = \sqrt{3} + i\sqrt{5}.
\]
If $x = x_1 + ix_2$, $y = y_1 + iy_2$, and $z = z_1 + iz_2$ for real $x_1, x_2, y_1, y_2, z_1$, and $z_2$, then:
\[
(x_1x_2 + y_1y_2 + z_1z_2)^2
\]
can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1516}
Let $ABCD$ be a convex quadrilateral for which $DA = AB$ and $CA = CB$. Set $I_0 = C$ and $J_0 = D$, and for each nonnegative integer $n$, let $I_{n+1}$ and $J_{n+1}$ denote the incenters of $\triangle I_nAB$ and $\triangle J_nAB$, respectively.

Suppose that \[ \angle DAC = 15^{\circ}, \quad \angle BAC = 65^{\circ} \quad \text{and} \quad \angle J_{2013}J_{2014}I_{2014} = \left( 90 + \frac{2k+1}{2^n} \right)^{\circ} \] for some nonnegative integers $n$ and $k$. Compute $n+k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2021}
Calculate the mass of the material segment \( A B \), given \( A(-2,1,0) \) and \( B(-1,3,5) \), where the density at each point \( M \) on the segment is proportional to the distance from \( M \) to \( A \) with a proportionality coefficient \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15 k}
Given Jasmine has two types of bottles, one that can hold 45 milliliters and another that can hold 675 milliliters, and a vase that can hold 95 milliliters, determine the total number of small bottles she must buy to fill the large bottle as much as possible and the vase.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
A rubber ball falls from the roof of a house of height $10 \mathrm{~m}$. Each time it hits the ground, it rebounds to four-fifths of the height it fell from previously. How many times will the ball appear in front of a $1 \mathrm{~m}$ high window whose bottom edge is $5 \mathrm{~m}$ above the ground?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In the isosceles triangle \(ABC\) with the sides \(AB = BC\), the angle \(\angle ABC\) is \(80^\circ\). Inside the triangle, a point \(O\) is taken such that \(\angle OAC = 10^\circ\) and \(\angle OCA = 30^\circ\). Find the angle \(\angle AOB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70^\circ}
On the base \(AB\) of an isosceles trapezoid \(ABCD\), point \(P\) is chosen such that the base is divided in the ratio \(AP : BP = 4 : 1\). Given that \(\angle CPD = \angle PAD\), find the ratio \(PD / PC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
已知$a$为自然数，存在一个以$a$为首项系数的二次整数系数的多项式，它有两个小于1的不同的正根，那么，$a$的最小值是______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the smallest positive integer \( n \) such that every \( n \)-element subset of \( S = \{1, 2, \ldots, 150\} \) contains 4 numbers that are pairwise coprime (it is known that there are 35 prime numbers in \( S \)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{111}
Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=3$, and $|a_{n+1}-a_n|=2^n$ ($n\in\mathbb{N}^*$), and that $\{a_{2n-1}\}$ is an increasing sequence, $\{a_{2n}\}$ is a decreasing sequence, find the limit $$\lim_{n\to\infty} \frac{a_{2n-1}}{a_{2n}} = \_\_\_\_\_\_.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
Find all prime numbers $p, q, r$ such that $$p(p+1)+q(q+1) = r(r+1)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 2, 3)}
In $\triangle ABC$, $AB = 10$, $AC = 8$, and $BC = 6$. Circle $P$ is the circle with the smallest radius that passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection, distinct from $C$, of circle $P$ with sides $AC$ and $BC$, respectively. If the length of segment $QR$ is $x$, what is the value of $10x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Any positive integer \( n \) can be written in the form \( n = 2^b (2c+1) \). We call \( 2c+1 \) the odd part of \( n \). Given an odd integer \( n > 0 \), define the sequence \( a_0, a_1, a_2, \ldots \) as follows: \( a_0 = 2^n - 1 \), and \( a_{k+1} \) is the odd part of \( 3a_k + 1 \). Find \( a_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3^n - 1}{2}}
On an island, there are chameleons of five colors. When one chameleon bites another, the color of the bitten chameleon changes according to a certain rule, and the new color depends only on the color of the biting chameleon and the color of the bitten chameleon. It is known that 2023 red chameleons can agree on a sequence of bites, after which all of them will become blue. What is the smallest $k$ such that $k$ red chameleons can guarantee to agree on a sequence of bites so that they all become blue?

For example, the rules could be such: if a red chameleon bites a green one, the bitten chameleon changes color to blue; if a green chameleon bites a red one, the bitten chameleon remains red, that is, "changes color to red"; if a red chameleon bites a red one, the bitten chameleon changes color to yellow, and so on. (Specific rules for color changes may be different.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that, the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers.

What is the smallest value of $n$ for which the second player can ensure that he completes both his moves?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $S_4 \geq 10$, $S_5 \leq 15$, and $S_7 \geq 21$, then find the value range of $a_7$. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[3, 7]}
Let $g(x)$ be a function satisfying $g(2x)=g(x+y)g(x-y)$ for all $x,y \in \mathbb{R}$. If $g(5)=2$, $g(0)=1$, and $g'(0)=1$, what is $g'(5)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Preston has a sandwich shop where he charges $5 for each sandwich, $3 for a side dish, and $1.50 for a drink. He also charges a $20 delivery fee. Abra Company orders 18 sandwiches, 10 side dishes, and 15 drinks from Preston, then adds a 10% tip to the total cost. Additionally, Abra Company has a 15% discount voucher that applies only to the food items. Calculate the total amount in dollars that Preston received after applying the discount and the tip.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$158.95}
How many distinct four-digit even numbers can be formed using the digits 0, 1, 2, 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let  $n$  be a positive integer. An  $n\times n$  board consisting of  $n^2$  cells, each being a unit square colored either black or white, is called *convex* if for every black colored cell, both the cell directly to the left of it and the cell directly above it are also colored black. We define the *beauty* of a board as the number of pairs of its cells  $(u,v)$  such that  $u$  is black,  $v$  is white, and  $u$  and  $v$  are in the same row or column. Determine the maximum possible beauty of a convex  $n\times n$  board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n^3 - n}{3}}
Polly and Peter take part in an intense chess tournament. They played four games against each other. In each game, an average of 38 moves were made. During the opening phase of each game (approximated as the first 12 moves), Polly took an average of 35 seconds per move and Peter took an average of 45 seconds per move. In the middle game (next 18 moves), Polly sped up to an average of 30 seconds per move, while Peter remained steady at 45 seconds per move. In the few endgame scenarios they reached (remaining 8 moves), Polly's average time went up to 40 seconds per move, while Peter slowed down further to 60 seconds per move. What is the total time in hours that they spent playing the four games?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.46}
Find the symmetry center of the function $y= \frac {1}{3}\tan (-7x+ \frac {\pi}{3})$. Express your answer as an ordered pair (x, y).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac {\pi}{21},0\right)}
Students at Highridge High earn 2 points for each correct answer during a quiz bowl. The quiz bowl follows a multiple-choice format with 4 possible answers per question. If a student incorrectly answers a question, they will lose 1 point. If a student correctly answers all the questions in a round within 1 minute, the student is awarded an additional 4 point bonus. If a student answers a question correctly within 10 seconds, they receive an extra 1 point. They played a total of five rounds, each consisting of five questions. If James only missed one question and did not attempt to answer it, and answered four questions within 10 seconds each, how many points did he get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
A semicircle with diameter  $d$  is contained in a square whose sides have length  $8$ . Given the maximum value of  $d$  is  $m- \sqrt{n}$ , find  $m+n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{544}
$ABCDE$ is a convex pentagon. Given the following measurements: \( m(\widehat{B}) = m(\widehat{D}) = 90^\circ \), \( m(\widehat{C}) = 120^\circ \), \( |AB| = 2 \), \( |BC| = |CD| = \sqrt{3} \), and \( |ED| = 1 \). Find the length of \( |AE| \). The original answer is in the form \(\sqrt{k}\), please give the value of \(k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A factory packs its products in cubic boxes. In one store, they put  $512$  of these cubic boxes together to make a large  $8\times 8 \times 8$  cube. When the temperature goes higher than a limit in the store, it is necessary to separate the  $512$  set of boxes using horizontal and vertical plates so that each box has at least one face which is not touching other boxes. What is the least number of plates needed for this purpose?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Given that $\textstyle\binom{2k}k$ results in a number that ends in two zeros, find the smallest positive integer  $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
The [i]Purple Comet! Math Meet[/i] runs from April 27 through May 3, so the sum of the calendar dates for these seven days is $27 + 28 + 29 + 30 + 1 + 2 + 3 = 120.$ What is the largest sum of the calendar dates for seven consecutive Fridays occurring at any time in any year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{142}
设 $x 、 y 、 z$ 为正实数. 求
$$
\left(x+\frac{1}{y}+\sqrt{2}\right)\left(y+\frac{1}{z}+\sqrt{2}\right)\left(z+\frac{1}{x}+\sqrt{2}\right)
$$
的最小值。请给出整数部分和非整数部分系数的和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Find a polynomial $ p\left(x\right)$ with real coefficients such that

$ \left(x\plus{}10\right)p\left(2x\right)\equal{}\left(8x\minus{}32\right)p\left(x\plus{}6\right)$

for all real $ x$ and $ p\left(1\right)\equal{}210$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(x + 4)(x - 4)(x - 8)}
$ f\left( x \right) \equal{} \frac {x^5}{5x^4 \minus{} 10x^3 \plus{} 10x^2 \minus{} 5x \plus{} 1}$ .  $ \sum_{i \equal{} 1}^{2009} f\left( \frac {i}{2009} \right) \equal{} ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1005}
For which values of $p$ does the following integral converge:
$$\int_{0}^{\infty}\frac{x^{-1}-e^{-x}}{x^{p}}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < p < 1}
A sum of money deposited at compound interest amounts to Rs. 2442 in 2 years and to Rs. 2926 in 3 years. Find the rate percent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19.82\%}
Suppose that the angles of triangle $ABC$ satisfy
\[\cos 3A + \cos 3B + \cos 3C = 1.\]
Two sides of the triangle have lengths 8 and 15. Find the maximum length of the third side assuming one of the angles is $150^\circ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{289 + 120\sqrt{3}}}
The graphs of the equations \[ y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k, \] are drawn in the coordinate plane for  $k=-10,-9,-8,\ldots,9,10.$   These 63 lines cut part of the plane into equilateral triangles of side  $2/\sqrt{3}.$   How many such triangles are formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{660}
For a given number $n$, let us denote by $p_n$ the probability that when randomly selecting a pair of integers $k, m$ satisfying the conditions $0 \leq k \leq m \leq 2^n$ (the selection of each pair is equally probable), the number $\binom{m}{k}$ will be even. Calculate $\lim_{n \to \infty} p_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $x_1\leq x_2\leq \cdots\leq x_{100}$ be real numbers such that $|x_1| + |x_2| + \cdots + |x_{100}| = 1$ and $x_1 + x_2 + \cdots + x_{100} = 0$. Among all such $100$-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{841}
Inside an equilateral triangle with base 2, a circle is inscribed such that it is tangent to all three sides. If the area of the equilateral triangle is subtracted from its base, and the perimeter is then added to this total, the result is 4. Additionally, the radius of the inscribed circle is equal to one-half the height of the triangle. What is the height of the equilateral triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Let $x$ be a positive real number. Define
\[
  A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
  B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad \text{and} \quad
  C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\] 
Given that $A^3+B^3+C^3 + 8ABC = 2014$, compute $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{183}
Let $A$ be a $2\times2$ matrix with real entries such that $\det(A) = d \neq 0$. If $\det(A + d\cdot\text{Adj}(A)) = 0$, determine $\det(A - d\cdot\text{Adj}(A))$, where $\text{Adj}(A)$ denotes the adjugate of $A$. Use only basic properties of matrices and determinants; do not rely on eigenvalues or the Cayley-Hamilton theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Sally picked 7 lemons, Mary picked 9 lemons, and Tom picked 12 lemons from the lemon tree. However, 2 lemons fell on the ground and were eaten by animals, and the total number of lemons originally on the tree is unknown. How many lemons were remaining on the lemon tree after Sally, Mary, and Tom picked the lemons and the animals ate the fallen ones?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Five soccer teams play a match where each team plays every other team exactly once. Each match awards 3 points to the winner, 0 points to the loser, and 1 point to each team in the event of a draw. After all matches have been played, the total points of the five teams are found to be five consecutive natural numbers. Let the teams ranked 1st, 2nd, 3rd, 4th, and 5th have drawn $A$, $B$, $C$, $D$, and $E$ matches respectively. Determine the five-digit number $\overline{\mathrm{ABCDE}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13213}
Simplify the following fraction:

$$
\frac{6 a^{2}-2 b^{2}+6 c^{2}+a b-13 a c-4 b c-18 a-5 b+17 c+12}{4 a^{2}-b^{2}+9 c^{2}-12 a c-16 a+24 c+16}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3a - 2c - 3 + 2b}{2a - 3c - 4 + b}}
Given $2^x = 50^y = 100$, find the value of $x^{-1} + y^{-1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let diamond \( A_{1} A_{2} A_{3} A_{4} \) have side length \( 1 \) and \(\angle A_{1} A_{2} A_{3} = \frac{\pi}{6} \). Point \( P \) lies in the plane of diamond \( A_{1} A_{2} A_{3} A_{4} \). Determine the minimum value of \( \sum_{1 \leqslant i < j \leqslant 4} \overrightarrow{P A_{i}} \cdot \overrightarrow{P A_{j}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}