inclusionAI/Ring-lite-rl-data huggingface dataset:
Smith has a small booth where he sells lottery tickets. Customers arrive according to a Poisson process of rate $\lambda$= 1 per minute. Smith will close the shop when the gap between the arrival of two consecutive customers exceeds $b$ minutes. Let $Y$ denote the time until Smith closes the shop. Find $E(Y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^b}
Select some numbers from \(1, 2, 3, \cdots, 9, 10\) such that each number from \(1, 2, 3, \cdots, 19, 20\) is equal to the sum of one or two of the selected numbers (they can be the same). Find the minimum number of selected numbers needed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In Smithtown, the ratio of right-handed people to left-handed people is 3 to 1 and the ratio of men to women is 3 to 2. If the number of right-handed men is maximized, what percentage of all the people in Smithtown are left-handed women?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\%}
Form a five-digit number without repeating digits using the numbers 0, 1, 2, 3, 4. How many of these numbers are even and have the digits 1 and 2 adjacent to each other? (Answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
The sum of the first $n$ terms of an arithmetic sequence $\{a\_n\}$ is $S\_n$, where the first term $a\_1 > 0$ and the common difference $d < 0$. For any $n \in \mathbb{N}^*$, there exists $k \in \mathbb{N}^*$ such that $a\_k = S\_n$. Find the minimum value of $k - 2n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
Given that a ( a - 4 ) = some number and b ( b - 4 ) = the same number , where a ≠ b , and a + b = 4. What is that number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
5. Žan decided to assign a single-digit number to each two-digit number, and he would do so only by multiplying the digits. For the numbers 91 and 66, he wrote:

$$
\begin{aligned}
& 91 \xrightarrow{9 \cdot 1} 9 \\
& 66 \xrightarrow{6 \cdot 6} 36 \xrightarrow{3 \cdot 6} 18 \xrightarrow{1 \cdot 8} 8
\end{aligned}
$$

How many two-digit numbers did he assign the number 0 to?

## 51st Mathematical Competition for High School Students in Slovenia

Optional Competition, April 4, 2007

## Problems for 3rd Year Students
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given the quadratic function $f(x) = x^2 - ax + 4$, if $f(x+1)$ is an even function, find the value of the real number $a$. Express your answer as a single real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Two adjacent faces of a tetrahedron, which are isosceles right triangles with a hypotenuse of 2, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $f(n) = \frac{n}{3}$ if $n$ is divisible by $3$, and $f(n) = 4n - 10$ otherwise. Find the sum of all positive integers $c$ such that $f^5(c) = 2$. (Here, $f^5(x)$ means $f(f(f(f(f(x)))))$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{748}
Given the irrational number e = 2.71828182845..., define the function f(n) = k, where the domain is the set of positive integers, and k is the n-th digit after the decimal point of e. Given f(0) = 2, determine the range of f(f(f(n))). Express your answer as a set of integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1,2,7,8\}}
Let $N$ be the greatest positive integer that can be expressed using all seven Roman numerals $I$, $V$, $X$, $L$, $C$, $D$, and $M$ exactly once each, and let $n$ be the least positive integer that can be expressed using these numerals exactly once each. Find $N - n$. Note that the arrangement $CM$ is never used in a number along with the numeral $D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{222}
If the power function $y=(m^2-4m+1)x^{m^2-2m-3}$ is an increasing function on $(0,+\infty)$, then the value of the real number $m$ is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let the set \( \mathrm{S} = \{1, 2, 3, \ldots, 10\} \). The subset \( \mathrm{A} \) of \( \mathrm{S} \) satisfies \( \mathrm{A} \cap \{1, 2, 3\} \neq \emptyset \) and \( \mathrm{A} \cup \{4, 5, 6\} \neq \mathrm{S} \). Find the number of such subsets \( \mathrm{A} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{888}
On a plot of land, there are 10,000 trees planted in a grid with 100 rows and 100 columns, forming a neat square grid. What is the maximum number of trees that can be cut down such that from the position of any remaining tree, no other tree is obscured? You can assume the trees are thin enough that this rule is strictly followed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2500}
As shown in the figure, \( C \) and \( D \) are two points on segment \( AB \), and \( M \) and \( N \) are the midpoints of \( AC \) and \( DB \) respectively. If \( AB = 10 \) and \( CD = 2 \), then calculate the length of \( MN \). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. If $2b\cos B = a\cos C + c\cos A$, then $B=$ __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
162 pluses and 144 minuses are placed in a $30 \times 30$ table such that each row and each column contains at most 17 signs. (No cell contains more than one sign.) For every plus, count the number of minuses in its row, and for every minus, count the number of pluses in its column. Find the maximum of the sum of these numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2592}
Let \( S = \left\{ A = \left(a_{1}, a_{2}, \cdots, a_{8}\right) \mid a_{i} = 0 \text{ or } 1, i = 1,2, \cdots, 8 \right\} \). For any two elements \( A = \left(a_{1}, a_{2}, \cdots, a_{8}\right) \) and \( B = \left(b_{1}, b_{2}, \cdots, b_{8}\right) \) in \( S \), define \( d(A, B) = \sum_{i=1}^{8} \left| a_{i} - b_{i} \right| \), which is called the distance between \( A \) and \( B \). What is the maximum number of elements that can be chosen from \( S \) such that the distance between any two of them is at least \( 5 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A trapezoid has side lengths $24$, $25$, $26$, and $27$ in some order. Find its area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{612}
A recipe that makes $5$ servings of hot chocolate requires $2$ squares of chocolate, $\frac{1}{4}$ cup sugar, $1$ cup water and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, lots of water, and $7$ cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8 \frac{3}{4}}
In a tetrahedron \(ABCD\), \(AB = AC = AD = BC = 1\), \(BD = \sqrt{3}\), and \(CD = \sqrt{2}\). Find the angle between \(AD\) and \(BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
John scores 93 on this year's AHSME. Had the old scoring system still been in effect, he would score only 84 for the same answers. 
How many questions does he leave unanswered?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given \( f(x) = |1 - 2x| \) for \( x \in [0, 1] \), find the number of solutions to the equation \( f(f(f(x))) = \frac{1}{2} x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
If each edge of a cube is increased by different percentages - 10% for edge A, 15% for edge B, and 20% for edge C, calculate the percentage increase in the surface area of the transformed rectangular prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32.17\%}
A quadrilateral is given, where two of its sides have lengths of $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Determine the perimeter of this quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Compute the integral $$\int_0^1 \frac{\arctan x}{(1+x)^2}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$\frac{1}{4}\ln(2)$$}
Find the limit of $$\lim_{x\to0}\frac{(1+x)^{\sin x}-1}{x}$$ using L'Hopital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed, leaving the remaining six cards in either ascending or descending order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
Vasya thought of a four-digit number and wrote down the product of each pair of its adjacent digits on the board. After that, he erased one product, and the numbers 20 and 21 remained on the board. What is the smallest number Vasya could have in mind?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3745}
In a group of 150 people who own pets, the following information is given: 
- 30 people own only dogs
- 25 people own only cats
- 10 people own only birds
- 7 people own only snakes
- 15 people own both cats and dogs
- 12 people own both birds and dogs
- 8 people own both birds and cats
- 3 people own both snakes and dogs
- 4 people own both snakes and cats
- 2 people own both snakes and birds
- 5 people own cats, dogs, and snakes
- 4 people own cats, dogs, and birds
- 6 people own cats, birds, and snakes
- 9 people own dogs, birds, and snakes
- 10 people own all four pets

How many total snakes are owned by the people in this group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
设整数数列 $a_{1}, a_{2}, \cdots, a_{10}$ 满足 $a_{10}=3 a_{1}, a_{2}+a_{8}=2 a_{5}$, 且 $a_{i+1} \in\left\{1+a_{i}, 2+a_{i}\right\}, i=$ $1,2, \cdots, 9$, 则这样的数列的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
A bag contains 4 red, 3 blue, and 6 yellow marbles. One marble is drawn and removed from the bag but is only considered in the new count if it is yellow. What is the probability, expressed as a fraction, of then drawing one marble which is either red or blue from the updated contents of the bag?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{13}}
已知整数 $a, b, c, d$ 满足 $a+b+c+d=6$ ，则 $a b+a c+a d+b c+b d+c d$ 的正整数取值个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given an ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a>b>0)$ with two vertices A(a, 0) and B(0, b). Draw perpendicular lines to AB through A and B, intersecting the ellipse at distinct points C and D. If 2|BD| = 3|AC|, what is the eccentricity of the ellipse?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
对于有理数加群 (\bQ,+)，记所有分母不超过 10 有理数对应的象构成的集合是 A. 设包含 A 的最小群是 G，则 G 的阶为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2520}
A refrigerator's price was increased by 10% twice consecutively, and then by 5% once. Calculate the percentage by which the price should be reduced to restore the original price. Express your answer as an integer percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In an isosceles triangle \(ABC\), the angle at the vertex \(B\) is \(20^\circ\). Points \(D\) and \(K\) are taken on the sides \(AB\) and \(BC\) respectively such that \(\angle KAC = 50^\circ\) and \(\angle DCA = 60^\circ\). Calculate \(\angle CDK\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Several energy-saving devices with a total weight of 120 kg were delivered to the factory. It is known that the total weight of the three lightest devices is 31 kg, and the total weight of the three heaviest devices is 41 kg. How many energy-saving devices were delivered to the factory if the weights of any two devices are different?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
All arrangements of letters $VNNWHTAAIE$ are listed in lexicographic (dictionary) order. If $AAEHINNTVW$ is the first entry, what entry number is $VANNAWHITE$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{738826}
How many multiples of 15 are between 25 and 225?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
How many perfect squares are between 100 and 400?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A cistern is two-thirds full of water. Pipe A can fill the remaining part in some minutes and Pipe B in 8 minutes. Once the cistern is emptied, they take 14.4 minutes to fill it together completely. How many minutes does Pipe A take to fill the remaining part?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Find the minimum value of the distance $|PQ|$ where point P lies on the curve $y=\frac{1}{2}e^x$, and point Q lies on the curve $y=\ln(2x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}(1 - \ln 2)}
The ends of a segment are marked with two ones. In the middle between them, their sum, the number 2, is written. Then in the middle between every two adjacent numbers, again their sum is written, and so on for 1973 times. How many times will the number 1973 be written?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1972}
Given 5 points \( A, B, C, D, E \) on a plane, with no three points being collinear. How many different ways can one connect these points with 4 segments such that each point is an endpoint of at least one segment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
设 $A=\{1,2,3,4,5,6\}, B=\{7,8,9, \cdots, n\}$, 在 $A$ 中任取 3 个数, 在 $B$ 中取 2个数,组成含有 5 个元素的集合 $A_i(i=1,2,3, \cdots, 20)$, 使得 $\left|A_i \cap A_j\right| \leqslant 2,1 \leqslant i<j \leqslant 20$.求 $n$ 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
On the hypotenuse \( BC \) of a right triangle \( ABC \), points \( D \) and \( E \) are marked such that \( AD \perp BC \) and \( AD = DE \). On the side \( AC \), a point \( F \) is marked such that \( EF \perp BC \). Find the angle \( ABF \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Given in rectangle ABCD, AB=8, BC=6, folding the rectangle along AC to form a dihedral angle B-AC-D, the surface area of the circumscribed sphere of the tetrahedron ABCD is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100\pi}
On the extension of the side \(AC\) of triangle \(ABC\) beyond point \(A\), there is a point \(T\) such that \(\angle BAC = 2 \angle BTC\). Find the area of triangle \(ABC\), given that \(AB = AC\), \(BT = 70\), and \(AT = 37\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
Evaluate the improper integral $$\int_0^{\frac{\pi}{2}} \cos(2nx) \log(\sin(x)) \, dx$$ for $n > 1$, considering its connection to Fourier series and utilizing the identity for $\log(\cos(x))$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{\pi}{4n}}
Determine the number of distinct ring homomorphisms $\psi$ $:$ $\Bbb Z$$_1$$_2$ $\rightarrow$ $\Bbb Z$$_3$. Let $\Bbb Z$$_1$$_2$ $=$ $<x>$ and $\Bbb Z$$_3$ $=$ $<y>$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The domain of the function $y=\sin x$ is $[a,b]$, and its range is $\left[-1, \frac{1}{2}\right]$. Calculate the maximum possible value of $b-a$. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\pi}{3}}
Consider a polynomial  $P(x) \in \mathbb{R}[x]$ , with degree  $2023$ , such that  $P(\sin^2(x))+P(\cos^2(x)) =1$  for all  $x \in \mathbb{R}$ . If the sum of all roots of  $P$  is equal to  $\dfrac{p}{q}$  with  $p, q$  coprime, then what is the product  $pq$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4046}
Rotating an isosceles trapezoid around its axis of symmetry forms a solid of revolution which is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{frustum}}
Given the parabola $y^{2}=2px (p > 0)$, its focus $F$ forms a line $l$ with an inclined angle of $60^{\circ}$ intersecting the parabola at points $A$ and $B$, with $|AF| > |BF|$. Calculate the value of $\frac{|AF|}{|BF|}$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \( d \) be a positive divisor of 2015. Then the maximum value of the unit digit of \( d^{\frac{2005}{d}} \) is \(\qquad\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
If $n$ is an odd number, find the remainder when ${7^{n}}+C_{n}^{1}{{7}^{n-1}}+C_{n}^{2}{{7}^{n-2}}+ \cdots +C_{n}^{n-1}7$ is divided by $9$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The area of the enclosed figure formed by the line $x+y-2=0$, the curve $y=x^3$, and the x-axis is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
Alice and Bob play a game together as a team on a  $100 \times 100$  board with all unit squares initially white. Alice sets up the game by coloring exactly  $k$  of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least  $10$  red squares and color all of the remaining squares in it red. What is the
smallest  $k$  such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?

Proposed by *Nikola Velov, Macedonia*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
A gardener is planning to plant a row of 20 trees. There are two types of trees to choose from: maple trees and sycamore trees. The number of trees between any two maple trees (excluding the two maple trees) cannot be equal to 3. What is the maximum number of maple trees that can be planted among these 20 trees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Two neon signs are turned on at the same time. Both signs blink as they are turned on. One sign blinks every 9 seconds. The other sign blinks every few seconds. They will blink together again in 45 seconds. How often does the second sign blink?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
The real numbers \( x, y, z \) satisfy the equations \( x + y + z = 2 \) and \( xy + yz + zx = 1 \). Find the maximum possible value of \( x - y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 \sqrt{3}}{3}}
Find the smallest constant $K$ such that for all real numbers $x_1, x_2,\ldots,x_6$ the inequality $$(x_1 + x_2 + ...+ x_6)^2 \leq K \cdot (x_1 (x_2 + x_3) + x_2 (x_3 + x_4) + \ldots+ x_6 (x_1 + x_2))$$ holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
8. Let set $S$ consist of all positive integers not exceeding $2^{40}$, and whose binary representation contains exactly two 1s. If a number is randomly selected from $S$, the probability that this number is divisible by 9 is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{133}{780}}
a and b can do a work in 8 days, b and c can do the same work in some days. a, b, and c together can finish it in 6 days. a and c together can finish the work in 8 days. In how many days can b and c do the work?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Let $f(n)$ and $g(n)$ be polynomials of degree 2014 such that $f(n)+(-1)^{n} g(n)=2^{n}$ for $n=1,2, \ldots, 4030$. Find the coefficient of $x^{2014}$ in $g(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3^{2014}}{2^{2014} \cdot 2014!}}
Find the range of values of $I=\int _{-1}^{1} f(x)g(x) dx$ where $f$ and $g$ are functions defined on $[-1,1]$ that satisfy the following conditions:

* $f(x)f(-x)=1$ for all $x$
* $g(x)$ is an even function
* $\int _{-1}^{1}g(x)= 1$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty,\infty)}
Using the digits 1, 2, 3, 4, 5, 6, form a six-digit number without repeating any digit, such that any two adjacent digits have different parity, and 1 and 2 are adjacent. Calculate the number of such six-digit numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
对正整数 $n$ 及实数 $x(0 \leqslant x<n)$, 定义  
$$  
f(n, x)=(1-\{x\}) \cdot C_{n}^{[x]}+\{x\} \cdot C_{n}^{[x]+1},  
$$  
其中 $[x]$ 表示不超过实数 $x$ 的最大整数， $\{x\}=x-[x]$ .若整数 $m, n \geqslant 2$ 满足  
$$  
f\left(m, \frac{1}{n}\right)+f\left(m, \frac{2}{n}\right)+\cdots+f\left(m, \frac{m n-1}{n}\right)=123,  
$$  
求 $f\left(n, \frac{1}{m}\right)+f\left(n, \frac{2}{m}\right)+\cdots+f\left(n, \frac{m n-1}{m}\right)$ 的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
若 $\triangle A B C$ 的三边长分别是 $2,3,4$, 其外心为 $O$, 求 $\overrightarrow{O A} \cdot \overrightarrow{A B}+\overrightarrow{O B} \cdot \overrightarrow{B C}+$ $\overrightarrow{O C} \cdot \overrightarrow{C A}$ 的值。答案的形式为\frac{k}{m}，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-27}
已知正三棱柱$ABC-{{A}_{1}}{{B1}}{{C}_{1}}$的所有棱长为2，点$M,N$分别在侧面$AB{{B1}}{{A}_{1}}$和$AC{{C1}}{{A}_{1}}$内，$B{{C1}}$与${{B}_{1}}C$交于点$P$，则$\Delta MNP$周长的最小值为_______．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the Cesàro sum of the series $\sum\limits_{n = 1}^\infty {\cos (2n)}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
设复数 $z_{1}, z_{2}$ 满足 $\operatorname{Re}\left(z_{1}\right)>0, \operatorname{Re}\left(z_{2}\right)>0$, 且 $\operatorname{Re}\left(z_{1}^{2}\right)=\operatorname{Re}\left(z_{2}^{2}\right)=2$, 其中 $\operatorname{Re}(z)$ 表示复数 $z$ 的实部.
求 $\left|z_{1}+2\right|+\left|\overline{z_{2}}+2\right|-\left|\overline{z_{1}}-z_{2}\right|$ 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4 \sqrt{2}$}