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80,202 | Given $(x-1)^4(x+2)^8 = ax^{12} + a_1x^{11} + \ldots + a_{n}x + a_{12}$, find the value of $a_2 + a_4 + \ldots + a_{12}$. | 7 |
80,208 | During a physical examination, the heights (in meters) of 6 students were 1.71, 1.78, 1.75, 1.80, 1.69, 1.77, respectively. The median of this set of data is \_\_\_\_\_\_ (meters). | 1.76 |
80,250 | Let \( n \) be a natural number, \( a \) and \( b \) be positive real numbers, and satisfy \( a + b = 2 \). What is the minimum value of \(\frac{1}{1+a^n} + \frac{1}{1+b^n} \)? | 1 |
80,268 | Place the integers 1 through 9 on three separate cards, with three numbers on each card. The condition is that the difference between any two numbers on the same card cannot also be on the same card. Currently, the numbers 1 and 5 are on the first card, number 2 is on the second card, and number 3 is on the third card. Find the remaining number to be written on the first card. | 8 |
80,312 | 2. (15 points) For what values of the parameter $a$ is the sum of the squares of the roots of the equation $x^{2}+a x+2 a=0$ equal to $21?$ | -3 |
80,343 | If a convex polygon has exactly 4 obtuse angles, then the maximum number of sides, $n$, this polygon can have is ___. | 7 |
80,362 | The set $\{[x] + [2x] + [3x] \mid x \in \mathbb{R}\} \mid \{x \mid 1 \leq x \leq 100, x \in \mathbb{Z}\}$ has how many elements, where $[x]$ denotes the greatest integer less than or equal to $x$. | 67 |
80,372 | 5. Using the method described in the text for exchanging common keys, what is the common key that can be used by individuals with keys \( k_{1}=27 \) and \( k_{2}=31 \) when the modulus is \( p=101 \) and the base is \( a=5 \)? | 92 |
80,376 | Nini's report card shows the scores of 6 exams. The average score of the 6 exams is 74 points; the most frequent score among the 6 exams is 76 points; the median score of the 6 exams is 76 points; the lowest score is 50 points; the highest score is 94 points. There is only one score that appears twice, and no score appears more than twice. Assuming her scores are all integers, how many possible values are there for the second lowest score among the 6 exams? | 17 |
80,424 | There are \( R \) zeros at the end of \(\underbrace{99\ldots9}_{2009 \text{ of }} \times \underbrace{99\ldots9}_{2009 \text{ of } 9 \text{'s}} + 1 \underbrace{99\ldots9}_{2009 \text{ of } 9 \text{'s}}\). Find the value of \( R \). | 4018 |
80,427 | Given $a= \int_{ 0 }^{ \pi }(\sin x-1+2\cos ^{2} \frac {x}{2})dx$, find the constant term in the expansion of $(a \sqrt {x}- \frac {1}{ \sqrt {x}})^{6}\cdot(x^{2}+2)$. | -332 |
80,431 | There are six wooden sticks, each 50 cm long. They are to be connected end to end in sequence, with each connection section measuring 10 cm. After nailing them together, what is the total length of the wooden sticks? ( ) cm. | 250 |
80,456 | Master Li Si Cen makes fans. Each fan consists of 6 sectors, painted on both sides in red and blue (see figure). If one side of a sector is painted red, then the reverse side is painted blue and vice versa. Each pair of fans made by the master differ in coloring (if one coloring transforms into another by flipping the fan, they are considered the same). What is the maximum number of fans the master can make? | 36 |
80,489 | For a positive integer $n$, let, $\tau(n)$ be the number of positive integer divisors of $n$. How many integers $1 \leq n \leq 50$ are there such that $\tau(\tau(n))$ is odd? | 17 |
80,508 | The sum of the maximum and minimum values of the function $y=1- \frac {\sin x}{x^{4}+2x^{2}+1}$ ($x\in\mathbb{R}$) is \_\_\_\_\_\_. | 2 |
80,537 | 61. Bob Barker went back to school for a PhD in math, and decided to raise the intellectual level of The Price is Right by having contestants guess how many objects exist of a certain type, without going over. The number of points you will get is the percentage of the correct answer, divided by 10 , with no points for going over (i.e. a maximum of 10 points).
Let's see the first object for our contestants...a table of shape $(5,4,3,2,1)$ is an arrangement of the integers 1 through 15 with five numbers in the top row, four in the next, three in the next, two in the next, and one in the last, such that each row and each column is increasing (from left to right, and top to bottom, respectively). For instance:
```
6
13\quad14
15
```
is one table. How many tables are there? | 292864 |
80,555 | Computer viruses spread quickly. If a computer is infected, after two rounds of infection, there will be 121 computers infected. If on average one computer infects $x$ computers in each round of infection, then $x=\_\_\_\_\_\_$. | 10 |
80,564 | Five people each take a bucket to the tap to wait for water. If the time it takes for the tap to fill each person's bucket is 4 minutes, 8 minutes, 6 minutes, 10 minutes, and 5 minutes respectively, the minimum total waiting time for all the buckets to be filled is ______ minutes. | 84 |
80,611 | Problem 4. In an urn, there are 10 balls, among which 2 are red, 5 are blue, and 3 are white. Find the probability that a randomly drawn ball will be colored (event $A$). | 0.7 |
80,644 | Starting from 400,000, counting by increments of 50 up to 500,000 requires counting \_\_\_\_\_\_ times. | 2000 |
80,701 | In the same Cartesian coordinate system, the number of intersection points between the graph of the function $y=\sin \left(x+ \frac {\pi}{3}\right)$ $(x\in[0,2\pi))$ and the line $y= \frac {1}{2}$ is ______. | 2 |
80,855 | A school selects a sample from 90 students numbered sequentially from 01, 02, ..., 90 using systematic sampling. Given that the adjacent group numbers in the sample are 14 and 23, what is the student number from the fourth group in the sample? | 32 |
80,964 | In a magical land, each piglet either always lies or always tells the truth, and each piglet knows for sure whether others are liars. One day, Nif-Nif, Naf-Naf, and Nuf-Nuf met for tea, and two of them made statements. However, it is not known who said what. One of the three piglets said: "Nif-Nif and Naf-Naf both always lie." The other said: "Nif-Nif and Nuf-Nuf both always lie." Determine how many liars are there among the three piglets. | 2 |
80,974 | There are 10 cups on a table, 5 of them with the opening facing up and 5 with the opening facing down. Each move involves flipping 3 cups simultaneously. What is the minimum number of moves required to make all cup openings face the same direction? | 3 |
81,006 | If $x > 0$, $y > 0$, and $\frac{2}{x} + \frac{8}{y} = 1$, find the product $xy$ and the minimum value of $x+y$, and when are these minimum values attained? | 18 |
81,040 | Given that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and the sequence $\left\{{\frac{{{S_n}}}{n}}\right\}$ forms an arithmetic sequence with a first term of $\frac{1}{3}$ and a common difference of $\frac{1}{3}$. If $\left[x\right]$ represents the greatest integer not exceeding $x$, such as $\left[0,1\right]=0$, $\left[1,9\right]=1$, then the sum of the first $35$ terms of the sequence $\{[a_{n}]\}$ is ____. | 397 |
81,140 | There are 900 three-digit numbers (from 100 to 999) printed on cards, with each card displaying one three-digit number. Some of these numbers, when viewed upside down, still form a three-digit number, such as 198 which becomes 861 when flipped; however, others do not, like 531 which becomes unrecognizable when flipped. Therefore, some cards can be dual-purpose, allowing for a reduction in the total number of cards printed. At most, how many fewer cards can be printed? | 34 |
81,142 | A travel agency rents two types of buses to arrange a trip for 900 guests. The two types of vehicles, A and B, have capacities of 36 and 60 passengers respectively, and their rental costs are 1600 yuan per vehicle and 2400 yuan per vehicle respectively. The travel agency requires that the total number of rented vehicles does not exceed 21, and the number of type B vehicles should not exceed the number of type A vehicles by more than 7. How should they arrange the rental to minimize the cost, and what is the minimum rental cost? | 36800 |
81,153 | Sergey arranged several (more than two) pairwise distinct real numbers in a circle in such a way that each number is equal to the product of its neighbors. How many numbers could Sergey have arranged? | 6 |
81,279 | 72 vertices of a regular 3600-gon are painted red in such a way that the painted vertices form the vertices of a regular 72-gon. In how many ways can 40 vertices of this 3600-gon be chosen so that they form the vertices of a regular 40-gon and none of them are red? | 81 |
81,296 | Given two concentric circles with radii 1 and 3 and a common center $O$. A third circle is tangent to both of them. Find the angle between the tangents to the third circle drawn from the point $O$.
# | 60 |
81,302 | In front of an elevator are people weighing 130, 60, 61, 65, 68, 70, 79, 81, 83, 87, 90, 91, and 95 kg. The elevator has a capacity of 175 kg. What is the minimum number of trips the elevator must make so that all the people can be transported? | 7 |
81,312 | There are 5 chairs in a row for 3 people to sit on. It is required that persons A and B must sit next to each other, and the three people cannot all sit next to each other. How many different seating arrangements are there? | 12 |
81,328 | Given the functions $f(x)=ax^{2}-2ax+a+1 (a > 0)$ and $g(x)=bx^{3}-2bx^{2}+bx- \frac {4}{27} (b > 1)$, determine the number of zeros of the function $y=g(f(x))$. | 2 |
81,339 | Solve the inequality
$$
\sqrt{10x - 21} - \sqrt{5x^2 - 21x + 21} \geq 5x^2 - 31x + 42
$$
Provide the sum of all integer solutions for \(x\) that satisfy the inequality. | 7 |
81,410 | Determine the value of $c$ for which the distance between two parallel lines $3x-2y-1=0$ and $3x-2y+c=0$ is $\frac {2 \sqrt {13}}{13}$. | -3 |
81,435 | $11.1951^{1952}-1949^{1951}$ The last two digits of the difference are ( ).
The translation is provided as requested, preserving the original format and line breaks. | 52 |
81,437 | Given 10 positive integers, the sums of any 9 of them take exactly 9 different values: 86, 87, 88, 89, 90, 91, 93, 94, 95. After arranging these 10 positive integers in descending order, find the sum of the 3rd and the 7th numbers. | 22 |
81,458 | When $\begin{pmatrix} a \\ b \end{pmatrix}$ is projected onto $\begin{pmatrix} \sqrt{3} \\ 1 \end{pmatrix},$ the resulting vector has magnitude $\sqrt{3}.$ Also, $a = 2 + b \sqrt{3}.$ Enter all possible values of $a,$ separated by commas. | 2 |
81,467 | There are 20 people - 10 boys and 10 girls. How many ways are there to form a company where the number of boys and girls is equal?
# | 184756 |
81,516 | The line $y=ax+1$ intersects the curve $x^2+y^2+bx-y=1$ at two points, and these two points are symmetric about the line $x+y=0$. Find the value of $a+b$. | 2 |
81,530 | The seventh question: For an integer $n \geq 2$, non-negative real numbers $a_{1}, a_{2}, \ldots, a_{n}$ satisfy $\sum_{i=1}^{n} a_{i}=4$. Try to find the maximum possible value of $2 a_{1}+a_{1} a_{2}+a_{1} a_{2} a_{3}+ \ldots +a_{1} a_{2} \ldots a_{n}$. | 9 |
81,710 | For integers \(a, b, c, d\), let \(f(a, b, c, d)\) denote the number of ordered pairs of integers \((x, y) \in \{1,2,3,4,5\}^{2}\) such that \(ax + by\) and \(cx + dy\) are both divisible by 5. Find the sum of all possible values of \(f(a, b, c, d)\). | 31 |
81,753 | A square is inscribed in a circle, and another square is inscribed in the segment cut off from the circle by one of the sides of this square. Find the ratio of the side lengths of these squares. | 5 |
81,808 | The height of an isosceles trapezoid, dropped from the vertex of the smaller base to the larger base, divides the larger base into segments that are in the ratio of 2:3. How does the larger base relate to the smaller base? | 5 |
81,834 | Given positive integers \( a, b, \) and \( c \) such that
\[ 2019 \geqslant 10a \geqslant 100b \geqslant 1000c, \]
determine the number of possible tuples \((a, b, c)\). | 574 |
81,842 | In a class of 60 students, the mathematics scores $\xi$ follow a normal distribution $N(110, 10^2)$. If $P(100 \leq \xi \leq 110) = 0.35$, estimate the number of students who scored above 120 in mathematics. | 9 |
81,867 | Given the sets of points \( A = \left\{(x, y) \left| (x-3)^2 + (y-4)^2 \leq \left(\frac{5}{2}\right)^2 \right.\right\} \) and \( B = \left\{(x, y) \mid (x-4)^2 + (y-5)^2 > \left(\frac{5}{2}\right)^2 \right\} \), find the number of integer points in the intersection \( A \cap B \). | 7 |
81,884 | Convert the binary number $1010001011_{(2)}$ to base 7. | 1620 |
81,890 | 2. (3 points) On the Island of Misfortune, there live knights who always tell the truth, and liars who always lie. One day, $n$ islanders gathered in a room.
The first one said: "Exactly 1 percent of those present in this room are liars."
The second one said: "Exactly 2 percent of those present in this room are liars."
and so on
The person with number $n$ said: "Exactly $n$ percent of those present in this room are liars."
How many people could have been in the room, given that it is known for sure that at least one of them is a knight? | 100 |
81,901 | If $cos\frac{π}{5}$ is a real root of the equation $ax^{3}-bx-1=0$ (where $a$ and $b$ are positive integers), then $a+b=\_\_\_\_\_\_$. | 12 |
81,936 | Example 4: Arrange 5 white stones and 10 black stones in a horizontal row, such that the right neighbor of each white stone must be a black stone. How many arrangements are there?
(1996, Japan Mathematical Olympiad Preliminary) | 252 |
81,956 | Three girls \( A, B \) and \( C \), and nine boys are to be lined up in a row. Let \( n \) be the number of ways this can be done if \( B \) must lie between \( A \) and \( C \), and \( A \) and \( B \) must be separated by exactly 4 boys. Determine \( \lfloor n / 7! \rfloor \). | 3024 |
81,966 | How many integer solutions does the equation \(\sqrt{x} + \sqrt{y} = \sqrt{1960}\) have? | 15 |
81,977 | During training shooting, each of the soldiers shot 10 times. One of them successfully completed the task and scored 90 points. How many times did he score 9 points if he scored four 10s, and the results of the other shots were 7s, 8s, and 9s, with no misses? | 3 |
82,011 | Given \( P \) is the product of \( 3,659,893,456,789,325,678 \) and \( 342,973,489,379,256 \), find the number of digits of \( P \). | 34 |
82,052 | A quadrilateral and a pentagon are given on the plane (both may be non-convex), such that no vertex of one lies on the side of the other. What is the maximum possible number of points of intersection of their sides? | 20 |
82,069 | On a rectangular piece of paper, a picture in the form of a "cross" is drawn using two rectangles \(ABCD\) and \(EFGH\), the sides of which are parallel to the edges of the sheet. It is known that \(AB=9\), \(BC=5\), \(EF=3\), and \(FG=10\). Find the area of the quadrilateral \(AFCH\). | 52.5 |
82,090 | Task B-2.5. The difference in the lengths of the bases of an isosceles trapezoid is $10 \mathrm{~cm}$. The length of its leg is the arithmetic mean of the lengths of the bases. Calculate the lengths of the sides of the trapezoid if the ratio of the height to the larger base is $2: 3$. | 18 |
82,099 | Let $\{a_n\}$ be a geometric sequence and $\{b_n\}$ be an arithmetic sequence with $b_1 = 0$. Define $\{c_n\} = \{a_n + b_n\}$, and suppose that $\{c_n\}$ is the sequence 1, 1, 2, .... Find the sum of the first 10 terms of $\{c_n\}$. | 978 |
82,120 | Find \((11a + 2b, 18a + 5b)\) if \((a, b) = 1\).
Note: The notation \((x, y)\) represents the greatest common divisor (gcd) of \(x\) and \(y\). | 1 |
82,254 | The captain's assistant, who had been overseeing the loading of the ship, was smoking one pipe after another from the very start of the process. When $2 / 3$ of the number of loaded containers became equal to $4 / 9$ of the number of unloaded containers, and noon struck, the old sea wolf started smoking his next pipe. When he finished this pipe, the ratio of the number of loaded containers to the number of unloaded containers reversed the ratio that existed before he began smoking this pipe. How many pipes did the second assistant smoke during the loading period (assuming the loading rate and the smoking rate remained constant throughout)? | 5 |
82,333 | Problem 3. A bag of sunflower seeds was passed around a table. The first person took 1 seed, the second took 2, the third took 3, and so on: each subsequent person took one more seed than the previous one. It is known that in the second round, the total number of seeds taken was 100 more than in the first round. How many people were sitting at the table?
$[4$ points
(A. V. Shapovalov) | 10 |
82,344 | By definition, a polygon is regular if all its angles and sides are equal. Points \( A, B, C, D \) are consecutive vertices of a regular polygon (in that order). It is known that the angle \( ABD = 135^\circ \). How many vertices does this polygon have? | 12 |
82,365 | Suppose you forgot one digit of the phone number you need and you are dialing it at random. What is the probability that you will need to make no more than two calls? | 0.2 |
82,395 | On side AC of triangle ABC with a 120-degree angle at vertex B, points D and E are marked such that AD = AB and CE = CB. A perpendicular DF is dropped from point D to line BE. Find the ratio BD / DF. | 2 |
82,483 | Volodya wants to make a set of same-sized cubes and write one digit on each face of each cube such that any 30-digit number can be formed using these cubes. What is the minimum number of cubes he needs for this? (The digits 6 and 9 do not convert into each other when rotated.) | 50 |
82,513 | After the announcement of the Sichuan Province's college entrance examination scores in 2023, Shishi High School continued its glory. The scores of 12 students in a base class are (unit: points): 673, 673, 677, 679, 682, 682, 684, 685, 687, 691, 697, 705. The upper quartile of the scores of these 12 students is ______. | 689 |
82,582 | Find the largest natural number that cannot be expressed as the sum of two composite numbers. | 11 |
82,659 | 126. Binomial coefficients. Find the maximum value of $y$ such that in the expansion of a certain binomial, $y$ consecutive coefficients are in the ratio $1: 2: 3: \ldots: y$. Determine this expansion and write out the corresponding coefficients. | 3 |
82,662 | In how many ways can 8 identical rooks be placed on an $8 \times 8$ chessboard symmetrically with respect to the diagonal that passes through the lower-left corner square? | 139448 |
82,667 | Given a set of data: 5, 6, 8, 6, 8, 8, 8, the mode of this set of data is ____, and the mean is ____. | 7 |
82,668 | For a positive integer \( n \), let \( p(n) \) denote the product of the positive integer factors of \( n \). Determine the number of factors \( n \) of 2310 for which \( p(n) \) is a perfect square. | 27 |
82,674 | Convert the largest three-digit number in base seven (666)7 to a trinary (base three) number. | 110200 |
82,685 | Dad's age this year is exactly four times that of Xiao Hong. Dad is 30 years older than Xiao Hong. How old are Dad and Xiao Hong this year, respectively? | 10 |
82,821 | Find the maximum distance between two points, where one point is on the surface of a sphere centered at $(-2,-10,5)$ with a radius of 19, and the other point is on the surface of a sphere centered at $(12,8,-16)$ with a radius of 87. | 137 |
82,823 | Problem 4. What is the maximum number of rooks that can be placed on a chessboard such that each rook attacks exactly two other rooks?
Alexandru Mihalcu, Oxford | 16 |
82,884 | Xiao Ming needs to do the following things after school in the evening: review lessons for 30 minutes, rest for 30 minutes, boil water for 15 minutes, and do homework for 25 minutes. The minimum time Xiao Ming needs to spend to complete these tasks is ___ minutes. | 85 |
82,908 | From the digits 0, 1, 2, 3, 4, 5, select any 3 digits to form a three-digit number without repeating any digit. Among these three-digit numbers, the number of odd numbers is ____ . (Answer with a number) | 48 |
83,012 | Form a four-digit number without repeating any digits using 1, 3, 4, 5, 7, 8, 9, and arrange these four-digit numbers in ascending order. What is the 117th number in this sequence? | 1983 |
83,014 | Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruit is the banana!'' The last fruit responds "No, one of the $\emph{last}$ forty fruit is the banana!'' The fruit in the middle yells "I'm the banana!'' In how many positions could the banana be? | 21 |
83,021 | Given the sample \\(3\\), \\(4\\), \\(5\\), \\(6\\), \\(7\\), the variance of these \\(5\\) numbers is ____. | 2 |
83,025 | In the geometric sequence $\{a_n\}$, $2a_1$, $\frac{3}{2}a_2$, $a_3$ form an arithmetic sequence. Find the common ratio of the geometric sequence $\{a_n\}$. | 2 |
83,056 | 4. We are looking at four-digit PIN codes. We say that a PIN code dominates another PIN code if each digit of the first PIN code is at least as large as the corresponding digit of the second PIN code. For example, 4961 dominates 0761, because $4 \geqslant 0, 9 \geqslant 7, 6 \geqslant 6$ and $1 \geqslant 1$. We want to assign a color to every possible PIN code from 0000 to 9999, but if one PIN code dominates another, they must not have the same color.
What is the minimum number of colors with which you can do this? | 37 |
83,071 | Given that $a$ and $b$ are positive integers satisfying $\frac{1}{a} - \frac{1}{b} = \frac{1}{2018}$, find the number of all positive integer pairs $(a, b)$. | 4 |
83,086 | 7. The commission consists of 11 people. The materials that the commission works on are stored in a safe. How many locks should the safe have, and how many keys should each member of the commission be provided with, so that access to the safe is possible when a majority of the commission members gather, but not possible if fewer than half of the members gather? | 252 |
83,092 | The opposite number of $-1 \frac{1}{2}$ is ______, the reciprocal is ______, and the absolute value is ______. | 1.5 |
83,105 | Let \(\mathrm{P}\) be the set \(\{1, 2, 3, \ldots, 14, 15\}\). Consider \(\mathrm{A} = \{a_{1}, a_{2}, a_{3}\}\), a subset of \(\mathrm{P}\) where \(a_{1} < a_{2} < a_{3}\) and \(a_{1} + 6 \leq a_{2} + 3 \leq a_{3}\). How many such subsets are there in \(\mathrm{P}\)? | 165 |
83,136 | $3 \in \{x+2, x^2+2x\}$, then $x=$ ? | -3 |
83,172 | 15. Let $0<\theta<\frac{\pi}{2}$, find the minimum value of $\frac{1}{\sin \theta}+\frac{3 \sqrt{3}}{\cos \theta}$, and the value of $\theta$ when the minimum value is obtained. | 8 |
83,259 | 6. A farmer presented 6 types of sour cream in barrels of $9,13,17,19,20,38$ liters at the market. On the first day, he sold sour cream from three barrels completely, and on the second day, from two more barrels completely. The volume of sour cream sold on the first day was twice the volume of sour cream sold on the second day. Which barrels were emptied on the first day? In your answer, indicate the largest possible sum of the volumes of sour cream sold on the first day.
# | 66 |
83,275 | If the solution set of the inequality system about $x$ is $\left\{{\begin{array}{l}{x+1≤\frac{{2x-5}}{3}}\\{a-x＞1}\end{array}}\right.$ is $x\leqslant -8$, and the solution of the fractional equation about $y$ is $4+\frac{y}{{y-3}}=\frac{{a-1}}{{3-y}}$ is a non-negative integer, then the sum of all integers $a$ that satisfy the conditions is ____. | 24 |
83,297 | Count the number $N$ of all sets $A:=\{x_1,x_2,x_3,x_4\}$ of non-negative integers satisfying $$ x_1+x_2+x_3+x_4=36 $$ in at least four different ways.
*Proposed by Eugene J. Ionaşcu* | 9139 |
83,315 | The cells of a $5\times5$ grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through $9$ cells on his path, including the start and end cells. Compute the number of colorings for which Sam is guaranteed to pass through a total of exactly $3$ red cells, exactly $3$ white cells, and exactly $3$ blue cells no matter which route he takes. | 1680 |
83,417 | Solve the equation: $x^2 + 2x - 15 = 0$. | -5 |
83,434 | 42. In 1900, a reader asked in 1930 the following question. (At first glance, one might think that there is not enough data to answer it, but that is not the case.) He knew a person who died at an age that was $\frac{1}{29}$ of the year of their birth. How old was this person in 1900? | 44 |
83,473 | At the rally commemorating the 60th anniversary of the Chinese people's victory in the War of Resistance against Japan, two schools each send 3 representatives to speak in turns, criticizing the heinous crimes committed by the Japanese aggressors and praising the heroic deeds of the Chinese people in their struggle against Japan. How many different speaking orders are possible? | 72 |
83,482 | A literary and art team went to a nursing home for a performance. Originally, there were 6 programs planned, but at the request of the elderly, they decided to add 3 more programs. However, the order of the original six programs remained unchanged, and the added 3 programs were neither at the beginning nor at the end. Thus, there are a total of different orders for this performance. | 210 |
83,511 | Let the operation $x*y$ be defined as $x*y = (x+1)(y+1)$, and let $x^{*2}$ be defined as $x^{*2} = x*x$. Then, the value of the polynomial $3*(x^{*2}) - 2*x + 1$ when $x=2$ is ____. | 32 |
83,531 | 8. Given 8 points $A_{1}, A_{2}, \cdots, A_{8}$ on a circle. Find the smallest positive integer $n$, such that among any $n$ triangles with these 8 points as vertices, there must be two triangles that share a common side.
(Tao Pingsheng, problem contributor) | 9 |
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