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72,395 | 5. Divide the six natural numbers $14, 20, 33, 117, 143, 175$ into groups, such that any two numbers in each group are coprime. Then, the minimum number of groups needed is $\qquad$. | 3 |
72,412 | 28. (5 points)
Today, the absent-minded teacher explained the Gauss notation in class, telling us that $[a]$ represents the greatest integer not greater than $a$, for example, $[1.1]=1, [3]=3$, and then calculated: $\left[\frac{1}{7}\right]+\left[\frac{3}{7}\right]+\left[\frac{5}{7}\right]+\cdots+\left[\frac{2019}{7}\right]+\left[\frac{2021}{7}\right]=$ | 145584 |
72,420 | 49. A two-digit number divided by the sum of its digits, what is the maximum remainder?
When translating the text into English, I've retained the original format and line breaks. | 15 |
72,435 | 43. Given $(x-2)^{5}=a x^{5}+b x^{4}+c x^{3}+d x^{2}+e x+f$, then $16(a+b)+4(c+d)+(e+f)=$ | -256 |
72,444 | 6. From $1,2, \cdots, 2005$ choose $n$ different numbers. If among these $n$ numbers, there always exist three numbers that can form the side lengths of a triangle, find the minimum value of $n$.
| 17 |
72,454 | 9. Given a positive integer $k$ that satisfies $\left(10^{3}-1\right)\left(10^{6}-1\right)$ divides $10^{k}-1$, then the minimum value of $k$ is | 5994 |
72,507 | 9. (6 points) The Red Scarf Spring Festival Consolation Group encountered the following problem when determining the performance program for the nursing home: On New Year's Eve, there are 4 performances: singing, dancing, acrobatics, and skit. If singing is not to be scheduled as the 4th item, dancing is not to be scheduled as the 3rd item, acrobatics is not to be scheduled as the 2nd item, and the skit is not to be scheduled as the 1st item. Then, the number of different arrangements that meet the above requirements is $\qquad$. | 9 |
72,516 | 4. The number of integer points within the triangle $O A B$ (where $O$ is the origin) bounded by the line $y=2 x$, the line $x=100$, and the $x$-axis is $\qquad$ . | 9801 |
72,566 | 4. Doraemon and Nobita are playing "Rock, Paper, Scissors," with the rule that the winner of each round gets two dorayaki, the loser gets no dorayaki, and if it's a tie, each gets one dorayaki. Nobita knows that Doraemon can only play rock, but he still wants to share dorayaki with Doraemon, so he decides to play scissors once every ten rounds and play rock several times. After 20 rounds, the dorayaki are all gone, and Nobita has 30 dorayaki. How many dorayaki did Doraemon get? $\qquad$ | 10 |
72,612 | 32. Six brothers are of different ages, the eldest is 8 years older than the youngest, this year the second oldest is twice as old as the fifth oldest, the sum of the ages of the eldest and the second oldest equals the sum of the ages of the other four brothers, then the third oldest is $\qquad$ years old. | 7 |
72,616 | 16. (10 points) If $48 a b=a b \times 65$, then $a b=$ | 0 |
72,654 | 8 A certain station has exactly one bus arriving during $8: 00-9: 00, 9: 00-10: 00$ every day, but the arrival time is random, and the arrival times of the two buses are independent, with the following distribution:
\begin{tabular}{|c|c|c|c|}
\hline Arrival time & \begin{tabular}{c}
$8: 10$ \\
$9: 10$
\end{tabular} & \begin{tabular}{l}
$8: 30$ \\
$9: 30$
\end{tabular} & \begin{tabular}{l}
$8: 50$ \\
$9: 50$
\end{tabular} \\
\hline Probability & $\frac{1}{6}$ & $\frac{1}{2}$ & $\frac{1}{3}$ \\
\hline
\end{tabular}
A passenger arrives at the station at $8: 20$, then the expected waiting time for the bus is $\qquad$ (to the nearest minute). | 27 |
72,679 | 2. Let $n$ be a positive integer, and satisfy $n^{5}=438427732293$, then $n=$ | 213 |
72,776 | 3. [4] Find the smallest positive integer $n$ such that $\underbrace{2^{2 \cdot}}_{n}>3^{3^{3^{3}}}$. (The notation $\underbrace{2^{2 \cdot \cdot}}_{n}$ is used to denote a power tower with $n 2$ 's. For example, $\underbrace{2^{2^{2}}}_{n}$ with $n=4$ would equal $2^{2^{2^{2}}}$.) | 6 |
72,810 | I5.4 Let $f(1)=c+1$ and $f(n)=(n-1) f(n-1)$, where $n>1$. If $d=f(4)$, find $d$. | 6 |
72,960 | 4. Find the last 2 nonzero digits of 16 ! | 88 |
72,983 | G4.2 Amongst the seven numbers $3624,36024,360924,3609924,36099924,360999924$ and 3609999924 , there are $n$ of them that are divisible by 38 . Find the value of $n$. | 6 |
72,992 | 10. In $\triangle A B C, \angle A=80^{\circ}, \angle B=30^{\circ}$, and $\angle C=70^{\circ}$. Let $B H$ be an altitude of the triangle. Extend $B H$ to a point $D$ on the other side of $A C$ so that $B D=B C$. Find $\angle B D A$. | 70 |
73,017 | 18. A railway passes through four towns $A, B, C$, and $D$. The railway forms a complete loop, as shown on the right, and trains go in both directions. Suppose that a trip between two adjacent towns costs one ticket. Using exactly eight tickets, how many distinct ways are there of traveling from town $A$ and ending at town A? (Note that passing through A somewhere in the middle of the trip is allowed.) | 128 |
73,029 | 20. Let $a=444 \cdots 444$ and $b=999 \cdots 999$ (both have 2010 digits). What is the 2010 th digit of the product $a b$ ? | 3 |
73,050 | 13. How many positive integers, not having the digit 1 , can be formed if the product of all its digits is to be 33750 ? | 625 |
73,052 | 3. Let $N=\left(1+10^{2013}\right)+\left(1+10^{2012}\right)+\cdots+\left(1+10^{1}\right)+\left(1+10^{0}\right)$. Find the sum of the digits of $N$. | 2021 |
73,053 | 1.1. Find all integers $m$ for which the equation
$$
x^{3}-m x^{2}+m x-\left(m^{2}+1\right)=0
$$
has an integer solution. | -30 |
73,067 | 13. Let $S=\{1,2,3, \ldots, 12\}$. Find the number of nonempty subsets $T$ of $S$ such that if $x \in T$ and $3 x \in S$, then it follows that $3 x \in T$. | 1151 |
73,077 | 5. Let $x, y, z$ be three real numbers such that
- $y, x, z$ form a harmonic sequence; and
- $3 x y, 5 y z, 7 z x$ form a geometric sequence.
The numerical value of $\frac{y}{z}+\frac{z}{y}$ can be expressed in the form $p / q$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$ ? | 59 |
73,114 | 20. Find the maximum positive integer $n$ such that
$$
n^{2} \leq 160 \times 170 \times 180 \times 190
$$ | 30499 |
73,144 | 25. Let
$$
S=\sum_{r=0}^{n}\binom{3 n+r}{r}
$$
Evaluate $S \div(23 \times 38 \times 41 \times 43 \times 47)$ when $n=12$. | 1274 |
73,168 | 20 Find the number of ordered pairs of positive integers $(x, y)$ that satisfy the equation
$$
x \sqrt{y}+y \sqrt{x}+\sqrt{2009 x y}-\sqrt{2009 x}-\sqrt{2009 y}-2009=0 .
$$ | 6 |
73,429 | For real numbers $a, b$, it holds that $a^{2}+4 b^{2}=4$. How large can $3 a^{5} b-40 a^{3} b^{3}+48 a b^{5}$ be? | 16 |
73,467 | $\mathrm{Az}$
$$
x^{2}-5 x+q=0 \quad \text { and } \quad x^{2}-7 x+2 q=0
$$
in these equations, determine $q$ so that one of the roots of the second equation is twice as large as one of the roots of the first equation. | 6 |
73,548 | ## 75. Math Puzzle $8 / 71$
Fritz and Klaus are making metal parts and need 20 minutes for each part. They build a device in 2 hours that saves $50 \%$ of the production time. How many parts must be produced at least so that the time spent building the device is recovered and an additional 2 hours of free time is gained? | 24 |
73,549 | ## Task 6 - 040516
The sum of two natural numbers is 968. One of the addends ends with a zero. If you remove this zero, you get the other number.
Determine these two numbers! | 88880 |
73,604 | ## Task 2 - 280622
Mrs. Müller and her daughter Michaela, Mrs. Beyer and her sons Jan and Gerd, as well as Mrs. Schulz with her children Steffi and Jens are visiting an event together. Mrs. Müller buys the entrance tickets for everyone and pays 22 Marks.
How much money do Mrs. Beyer and Mrs. Schulz have to give to Mrs. Müller to pay for the entrance tickets for themselves and their children, if Michaela, Jan, Gerd, Steffi, and Jens each had to pay only half the entrance fee compared to an adult? | 8 |
73,626 | ## Task 2 - 100832
Pump $P_{1}$ fills a basin in exactly $4 \mathrm{~h} 30 \mathrm{~min}$. A second pump $P_{2}$ fills the same basin in exactly $6 \mathrm{~h} 45 \mathrm{~min}$. One day, when filling this basin, pump $P_{1}$ was initially used alone for exactly $30 \mathrm{~min}$. Subsequently, both pumps were used together until the basin was filled.
Calculate how long it took in total to fill the basin under these conditions! (It is assumed that both pumps operated at a constant performance during their operation.) | 2 |
73,628 | ## Task 4
How many numbers lie between the predecessor of 3 and the successor of 8?
Write them down! | 6 |
73,650 | ## Task 2 - 220712
The (unrelated) families Meier and Schmidt are going on a short vacation trip with their children and take a large supply of paper napkins. Each participant receives a napkin at each meal. Each participant had the same number of meals, and it was more than one.
At the end of the trip, it was found that exactly 121 napkins had been used.
How many children of these families participated in the trip in total? | 7 |
73,661 | ## Task 3
Pioneers from Magdeburg were in Berlin to fulfill an order of the Pioneer Relay "Red October." They are returning home in two groups.
Group A is traveling with the City Express "Börde." This train departs from Berlin at 15:46 and arrives in Magdeburg at 17:48.
Group B is traveling with the Express Train. The Express Train D 644 takes 2 hours and 21 minutes for this route. How much less travel time do the Pioneers in the Express "Börde" have compared to those in the Express Train? | 19 |
73,673 | ## Task 1
Pioneers of a 2nd grade bring rabbits and chickens to the exhibition. In the cage, five heads and fourteen legs can be seen.
How many rabbits and how many chickens are there? | 2 |
73,700 | 1. The price of one ticket is 50 denars. When the price is reduced, the number of visitors increases by $50\%$, and the revenue increases by $20\%$. By how much is the price of the entry ticket reduced? | 20 |
73,709 | \section*{Problem \(3-341013=340913\)}
Karin and Rolf collect tram tickets. Each ticket has a number consisting of 6 digits. If the sum of the first three digits is equal to the sum of the last three digits, the ticket is called a lucky ticket.
To estimate the chance of this, Karin and Rolf want to know what percentage of all tickets are lucky tickets. It is assumed that each number from 000000 to 999999 occurs equally often.
Karin writes a simple computer program to determine the desired percentage by running an instruction sequence 1,000,000 times. Since this takes a long time, Rolf writes a program in which a (different) instruction sequence only needs to run 1,000 times (and only a few additional instructions need to be executed).
Write such a program for each of them and explain why the desired percentage can be found this way! (The choice of programming language is of course free.) | 5.5252 |
73,732 | ## Task 5 - 171245
Let $f_{1}, f_{2}, \ldots$ be a sequence of functions defined for all real numbers $x$, given by
$$
\begin{gathered}
f_{1}(x)=\sqrt{x^{2}+48} \\
f_{k+1}(x)=\sqrt{x^{2}+6 f_{k}(x)}
\end{gathered}
$$
for $k=1,2,3, \ldots$
Determine for each $n=1,2,3, \ldots$ all real numbers $x$ that are solutions to the equation $f_{n}(x)=2 x$. | 4 |
73,741 | \section*{Problem \(3-340913=341013\)}
Karin and Rolf collect tram tickets. Each ticket has a number consisting of 6 digits. If the sum of the first three digits is equal to the sum of the last three digits, the ticket is called a lucky ticket.
To estimate the chance of this, Karin and Rolf want to know what percentage of all tickets are lucky tickets. It is assumed that each number from 000000 to 999999 occurs equally often.
Karin writes a simple computer program to determine the desired percentage by running an instruction sequence 1,000,000 times. Since this takes a long time, Rolf writes a program in which a (different) instruction sequence only needs to run 1,000 times (and only a few additional instructions need to be executed).
Write such a program for each of them and explain why the desired percentage can be found this way! (The choice of programming language is of course free.) | 5.5252 |
73,847 | The real numbers $x$, $y$, $z$, and $t$ satisfy the following equation:
\[2x^2 + 4xy + 3y^2 - 2xz -2 yz + z^2 + 1 = t + \sqrt{y + z - t} \]
Find 100 times the maximum possible value for $t$. | 125 |
73,882 | Find the smallest positive integer $n$ such that the decimal representation of $n!(n+1)!(2n+1)! - 1$ has its last $30$ digits all equal to $9$. | 34 |
73,885 | Alice's favorite number has the following properties:
[list]
[*] It has 8 distinct digits.
[*]The digits are decreasing when read from left to right.
[*]It is divisible by 180.[/list]
What is Alice's favorite number?
[i]Author: Anderson Wang[/i] | 97654320 |
73,998 | In triangle $ABC$, $AB = 28$, $AC = 36$, and $BC = 32$. Let $D$ be the point on segment $BC$ satisfying $\angle BAD = \angle DAC$, and let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $ABC$. Find the length of segment $AE$.
[i]Ray Li[/i] | 18 |
74,251 | At a concert $10$ singers will perform. For each singer $x$, either there is a singer $y$ such that $x$ wishes to perform right after $y$, or $x$ has no preferences at all. Suppose that there are $n$ ways to order the singers such that no singer has an unsatisfied preference, and let $p$ be the product of all possible nonzero values of $n$. Compute the largest nonnegative integer $k$ such that $2^k$ divides $p$.
[i]Proposed by Gopal Goel[/i] | 38 |
74,578 | What is the smallest positive integer $n$ such that $n=x^3+y^3$ for two different positive integer tuples $(x,y)$? | 1729 |
74,797 | Find the sum of all possible $n$ such that $n$ is a positive integer and there exist $a, b, c$ real numbers such that for every integer $m$, the quantity $\frac{2013m^3 + am^2 + bm + c}{n}$ is an integer. | 2976 |
74,886 | We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there?
[i]Proposed by Nathan Xiong[/i] | 16 |
74,933 | Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$. | 32 |
75,023 | The squares in a $7\times7$ grid are colored one of two colors: green and purple. The coloring has the property that no green square is directly above or to the right of a purple square. Find the total number of ways this can be done. | 3432 |
75,024 | Let $a$ and $b$ be the two possible values of $\tan\theta$ given that \[\sin\theta + \cos\theta = \dfrac{193}{137}.\] If $a+b=m/n$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$. | 28009 |
75,143 | For the four-digit number $\overline{abcd}$ (where $1 \leq a \leq 9$ and $0 \leq b, c, d \leq 9$):
- If $a > b$, $b < c$, and $c > d$, then $\overline{abcd}$ is called a $P$-type number;
- If $a < b$, $b > c$, and $c < d$, then $\overline{abcd}$ is called a $Q$-type number.
Let $N(P)$ and $N(Q)$ denote the number of $P$-type numbers and $Q$-type numbers, respectively. Find the value of $N(P) - N(Q)$. | 285 |
75,173 | Given $0 \leq a_k \leq 1$ for $k=1,2,\ldots,2020$, and defining $a_{2021}=a_1, a_{2022}=a_2$, find the maximum value of $\sum_{k=1}^{2020}\left(a_{k}-a_{k+1} a_{k+2}\right)$. | 1010 |
75,176 | 65. There is a cup of salt water. If 200 grams of water is added, its concentration becomes half of the original; if 25 grams of salt is added, its concentration becomes twice the original. The original concentration of this cup of salt water is $\qquad$ . | 10 |
75,244 | Find all odd numbers \( n \) that satisfy \( n \mid 3^n + 1 \). | 1 |
75,254 | Count the matches. A friend writes that he bought a small box of short matches, each one inch long. He found that he could arrange them in a triangle whose area in square inches was equal to the number of matches. Then he used 6 matches, and it turned out that from the remaining matches, he could form a new triangle with an area that contained as many square inches as there were matches left. After using another 6 matches, he was able to do the same thing again.
How many matches did he originally have in the box? This number is less than 40. | 36 |
75,262 | A school has 160 teachers, 960 male students, and 800 female students. Now, using stratified sampling, a sample of size $n$ is drawn from all the teachers. Given that the number of female students drawn is 80, find the value of $n$. | 192 |
75,271 | The lines $x+(1+m)y=2-m$ and $mx+2y+8=0$ are parallel. Find the value of $m$. | 1 |
75,328 | Example 2 Let the odd function $f(x)$ have the domain $\mathbf{R}$. It satisfies $f(x)+f(x+2)=a, f(1)=0$, where $a$ is a constant. Try to determine how many roots the equation $f(x)=0$ has at least in the interval $(-3,7)$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 9 |
75,349 | 7. $[7]$ What is the minimum value of the product
$$
\prod_{i=1}^{6} \frac{a_{i}-a_{i+1}}{a_{i+2}-a_{i+3}}
$$
given that $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}\right)$ is a permutation of $(1,2,3,4,5,6)$ ? (note $a_{7}=a_{1}, a_{8}=a_{2} \cdots$ ) | 1 |
75,383 | Let \( f(x) = mx^2 + (2n + 1)x - m - 2 \) (where \( m, n \in \mathbb{R} \) and \( m \neq 0 \)) have at least one root in the interval \([3, 4]\). Find the minimum value of \( m^2 + n^2 \). | 0.01 |
75,451 | Define a function $f(x)$ on $\mathbb{R}$ satisfying: $f(-x) = -f(x)$, $f(x+2) = f(x)$, and when $x \in [0, 1]$, $f(x) = x$. Find the value of $f(2011.5)$. | -0.5 |
75,480 | If the tangent line of the curve $y=\ln x$ at point $P(x_{1}, y_{1})$ is tangent to the curve $y=e^{x}$ at point $Q(x_{2}, y_{2})$, then $\frac{2}{{x_1}-1}+x_{2}=$____. | -1 |
75,528 | Given an ellipse $C: \frac{x^2}{m} + y^2 = 1$, there is a proposition $P$: "If $m=4$, then the eccentricity of ellipse $C$ is $\frac{\sqrt{3}}{2}$". Let $f(P)$ denote the number of true propositions among the four forms: the proposition itself, its converse, its inverse, and its contrapositive. Then $f(P) =$ ______. | 2 |
75,561 | What is the maximum number of rooks that can be placed in an $8 \times 8 \times 8$ cube so that no two rooks attack each other? | 64 |
75,620 | 3. The quadratic function $f(x)$ satisfies
$$
f(-10)=9, f(-6)=7, f(2)=-9 \text {. }
$$
Then $f(2008)=$ $\qquad$ . | -509031.5 |
75,662 | Let $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ that have the following properties:
$\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$,
$\mathcal{A} \cap \mathcal{B} = \emptyset$,
The number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$,
The number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}$.
Find $N$. | 772 |
75,745 | In a meeting of mathematicians, Carlos says to Frederico: "The double of the product of the two digits of the number of mathematicians in the meeting is exactly our quantity." What is the minimum number of mathematicians that must join us so that our total is a prime number? Help Frederico solve the problem. | 1 |
75,765 | Use the Horner's method (also known as Qin Jiushao algorithm) to compute the value of the polynomial: $f(x) = 2x^6 + 3x^5 + 5x^3 + 6x^2 + 7x + 1$ when $x = 0.5$. Determine the number of multiplication and addition operations required. | 6 |
75,822 | Determine the number of integers $2 \leq n \leq 2016$ such that $n^{n}-1$ is divisible by $2,3,5,7$. | 9 |
75,899 | Observe the following equations: $23=3+5$, $33=7+9+11$, $43=13+15+17+19$, $53=21+23+25+27+29$, ..., if a similar method is used to decompose $m^3$ and the last number on the right side of the equation is 131, then the positive integer $m$ equals \_\_\_\_\_\_\_\_. | 11 |
75,927 | A notebook sheet is painted in 23 colors, with each cell in the sheet painted in one of these colors. A pair of colors is called "good" if there exist two adjacent cells painted in these colors. What is the minimum number of good pairs? | 22 |
75,996 | The random variable $\xi$ follows a normal distribution $N(1, \sigma^2)$. Given that $P(\xi < 0) = 0.3$, find $P(\xi < 2)$. | 0.7 |
76,035 | In the Asia-Europe Table Tennis Challenge, each team has 5 players who participate in a knockout competition in a pre-arranged order. The competition starts with the No. 1 players from both sides, with the loser being eliminated and the winner facing the next player from the opposing team. This process continues until all players from one side are eliminated, declaring the other side as the winner. There are \_\_\_\_\_\_ possible sequences of matches. | 252 |
76,044 | The cells of a $9 \times 9$ board are painted in black and white in a checkerboard pattern. How many ways are there to place 9 rooks on cells of the same color on the board such that no two rooks attack each other? (A rook attacks any cell that is in the same row or column as it.) | 2880 |
76,117 | Shapovalov A.V.
Before the start of the school chess championship, each participant said which place they expected to take. Seventh-grader Vanya said he would take the last place. By the end of the championship, everyone took different places, and it turned out that everyone, except, of course, Vanya, took a place worse than they expected. What place did Vanya take? | 1 |
76,126 | Compute the circumradius of cyclic hexagon $A B C D E F$, which has side lengths $A B=B C=$ $2, C D=D E=9$, and $E F=F A=12$. | 8 |
76,135 | If a square is divided into acute-angled triangles, what is the minimum number of parts that can be created? | 8 |
76,185 | In a WeChat group, there are 5 individuals: A, B, C, D, and E, playing a game involving grabbing red envelopes. There are 4 red envelopes, each person may grab at most one, and all red envelopes must be grabbed. Among the 4 red envelopes, there are two 2-yuan envelopes, one 3-yuan envelope, and one 4-yuan envelope (envelopes with the same amount are considered the same). How many situations are there where both A and B grab a red envelope? (Answer with a numeral). | 36 |
76,187 | There are 5 students in a queue. The questions are as follows:
1) In how many ways can students A and B be adjacent to each other?
2) In how many ways can students A, B, and C be not adjacent to each other?
3) In how many ways can student B not be in front of student A, and student C not be in front of student B?
4) In how many ways can student A not be in the middle, and student B not be at either end? | 60 |
76,238 | Calculate: $\sqrt{8}\div 2{2}^{\frac{1}{2}}=\_\_\_\_\_\_$. | 2 |
76,299 | Gregor divides 2015 successively by 1, 2, 3, and so on up to and including 1000. He writes down the remainder for each division. What is the largest remainder he writes down? | 671 |
76,366 | N2.
Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$ $(d(n)$ is the number of divisors of the number $n$ including 1 and $n)$. | 9 |
76,375 | 1007. From a natural number, the sum of its digits was subtracted, and from the resulting difference, the sum of its digits was subtracted again. If this process continues, with what number will the calculations end? | 0 |
76,380 | A basket is called "*Stuff Basket*" if it includes $10$ kilograms of rice and $30$ number of eggs. A market is to distribute $100$ Stuff Baskets. We know that there is totally $1000$ kilograms of rice and $3000$ number of eggs in the baskets, but some of market's baskets include either more or less amount of rice or eggs. In each step, market workers can select two baskets and move an arbitrary amount of rice or eggs between selected baskets. Starting from an arbitrary situation, what's the minimum number of steps that workers provide $100$ Stuff Baskets? | 99 |
76,458 | 4. The sequence $\left\{a_{n}\right\}$ has 9 terms, where $a_{1}=a_{9}=1$, and for each $i \in\{1,2, \cdots, 8\}$, we have $\frac{a_{i+1}}{a_{i}} \in\left\{2,1,-\frac{1}{2}\right\}$. Find the number of such sequences.
$(2013$, National High School Mathematics League Competition) | 491 |
76,460 | 1. A tractor is pulling a very long pipe on sled runners. Gavrila walked along the entire pipe at a constant speed in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction at the same speed, the number of steps was 100. What is the length of the pipe if Gavrila's step is 80 cm? Round the answer to the nearest whole number of meters. The speed of the tractor is constant. | 108 |
76,504 | 5. In a convex quadrilateral $A B C D$, the areas of triangles $A B D$ and $B C D$ are equal, and the area of $A C D$ is half the area of $A B D$. Find the length of the segment $C M$, where $M$ is the midpoint of side $A B$, if it is known that $A D=12$. | 18 |
76,578 | A bag contains nine blue marbles, ten ugly marbles, and one special marble. Ryan picks marbles randomly from this bag with replacement until he draws the special marble. He notices that none of the marbles he drew were ugly. Given this information, what is the expected value of the number of total marbles he drew? | 10 |
76,610 | Suppose that $a,$ $b,$ $c,$ $d,$ $e,$ $f$ are real numbers such that
\begin{align*}
a + b + c + d + e + f &= 0, \\
a + 2b + 3c + 4d + 2e + 2f &= 0, \\
a + 3b + 6c + 9d + 4e + 6f &= 0, \\
a + 4b + 10c + 16d + 8e + 24f &= 0, \\
a + 5b + 15c + 25d + 16e + 120f &= 42.
\end{align*}Compute $a + 6b + 21c + 36d + 32e + 720f$. | 508 |
76,677 | Four different positive integers are to be chosen so that they have a mean of 2017. What is the smallest possible range of the chosen integers? | 4 |
76,700 | Given that $O$ is any point in space, and $A$, $B$, $C$, $D$ are four points such that no three of them are collinear, but they are coplanar, and $\overrightarrow{OA}=2x\cdot \overrightarrow{BO}+3y\cdot \overrightarrow{CO}+4z\cdot \overrightarrow{DO}$, find the value of $2x+3y+4z$. | -1 |
76,771 | Let $a,$ $b,$ $c$ be complex numbers such that
\begin{align*}
ab + 4b &= -16, \\
bc + 4c &= -16, \\
ca + 4a &= -16.
\end{align*}Enter all possible values of $abc,$ separated by commas. | 64 |
76,772 | Xiaoming constructed a sequence using the four digits $2, 0, 1, 6$ by continuously appending these digits in order: 2, 20, 201, 2016, 20162, 201620, 2016201, 20162016, 201620162, … In this sequence, how many prime numbers are there? | 1 |
76,801 | In $\triangle ABC$, it is known that $\overrightarrow {CD}=2 \overrightarrow {DB}$. $P$ is a point on line segment $AD$ and satisfies $\overrightarrow {CP}= \frac {1}{2} \overrightarrow {CA}+m \overrightarrow {CB}$. If the area of $\triangle ABC$ is $2 \sqrt {3}$, and $\angle ACB= \frac {\pi}{3}$, then the minimum value of $| \overrightarrow {CP}|$ is $\_\_\_\_\_\_$. | 2 |
76,868 | The average life expectancy in the country of Gondor is 64 years. The average life expectancy in the country of Numenor is 92 years. The average life expectancy in both countries combined is 85 years. By what factor does the population of Gondor differ from the population of Numenor? | 3 |
76,938 | Given a function $f(x)$ that satisfies: ① For any $x\in (0,+\infty)$, $f(2x) = 2f(x)$ always holds; ② When $x\in (1,2]$, $f(x) = 2-x$. If $f(a) = f(2020)$, then the smallest positive real number $a$ that satisfies the conditions is ____. | 36 |
76,969 | If $\{a_{n}\}$ is an arithmetic sequence, with the first term $a_{1} \gt 0$, $a_{2022}+a_{2023} \gt 0$, and $a_{2022}\cdot a_{2023} \lt 0$, then the smallest natural number $n$ for which the sum of the first $n$ terms $S_{n} \lt 0$ is ____. | 4045 |
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