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95,852 | 86. A three-digit number has a remainder of 2 when divided by $4,5,6$. If three digits are added to the end of this three-digit number to make it a six-digit number, and this six-digit number can be divided by $4,5,6$, then the smallest six-digit number that meets the condition is $\qquad$ . | 122040 |
95,860 | 7. (10 points) There are 10 cards each of the numbers “3”, “4”, and “5”. Select any 8 cards so that their sum is 33, then the maximum number of cards that can be “3” is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 3 |
95,883 | (7) Let $a, b, c$ be non-negative real numbers, then the minimum value of $\frac{c}{a}+\frac{a}{b+c}+\frac{b}{c}$ is | 2 |
95,886 | 52. A communication soldier rides a motorcycle for 3000 kilometers. In addition to the 2 tires on the motorcycle, there is also one spare tire. To ensure that the 3 tires wear out to the same extent, the 3 tires should be rotated. Then, upon reaching the destination, each tire would have traveled $\qquad$ kilometers.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 2000 |
95,909 | 56. As shown in the figure, $A D=D C, E B=3 C E$, if the area of quadrilateral $C D P E$ is 3, find the area of the figure enclosed by the broken line $A P B C A$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 9.6 |
95,914 | 3. Let there be a non-empty set $A \subseteq\{1,2,3,4,5,6,7\}$ and when $a \in A$, it must also be that $8-a \in A$. The number of such sets $A$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 15 |
95,915 | * Given a four-digit number $N$ is a perfect square, and each digit of $N$ is less than 7. Increasing each digit by 3 results in another four-digit number that is also a perfect square. Find $N$. | 1156 |
95,935 | 3. For a $5 \times 5$ grid colored as follows, place 5 different rooks in the black squares, such that no two rooks can attack each other (rooks in the same row or column will attack each other). How many different ways are there to do this?
Place the rooks in the black squares of the grid, ensuring that no two rooks are in the same row or column. | 1440 |
95,947 | 38. As shown in the figure, $P$ is a point inside $\triangle A B C$, and lines are drawn through $P$ parallel to the sides of $\triangle A B C$, forming smaller triangles $\triangle E P N$, $\triangle D P M$, and $\triangle T P R$ with areas $4$, $9$, and $49$, respectively. Then the area of $\triangle A B C$ is $\qquad$. | 144 |
95,960 | Example 3.3.5 Let $T$ be the set of all positive divisors of $2004^{100}$, and $S$ be a subset of $T$, where no element in $S$ is an integer multiple of any other element in $S$. Find the maximum value of $|S|$. | 10201 |
95,968 | 6. Xiaojun's mother sells fish at the market. By the end, there are still 5 fish left in the basin. One large fish can be sold for 10 yuan, one medium fish for 5 yuan, and the other 3 small fish can each be sold for 3 yuan. Later, a customer bought the fish, and the amount this customer should pay has $\qquad$ different possibilities. | 15 |
96,026 | 2. Calculate $1^{2}+2^{2}-3^{2}-4^{2}+5^{2}+6^{2}-7^{2}-8^{2}+\cdots+2021^{2}+2022^{2}=$ | 4090505 |
96,034 | (7) Let the function $f(x)=\left\{\begin{array}{l}\frac{1}{p}\left(x=\frac{q}{p}\right), \\ 0\left(x \neq \frac{q}{p}\right),\end{array}\right.$ where $p, q$ are coprime (prime), and $p \geqslant 2$. Then the number of $x$ values that satisfy $x \in[0,1]$, and $f(x)>\frac{1}{5}$ is $\qquad$ | 5 |
96,059 | 9. Given the sets $M=\{x, x y, \lg x y\}, N=\{0,|x|, y\}$, and $M=N$, then the value of $\left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\right.$ $\left.\frac{1}{y^{3}}\right)+\cdots+\left(x^{2 n+1}+\frac{1}{y^{2 n+1}}\right)$ is $\qquad$. | -2 |
96,079 | 8. The average score of Xiaoyangyang, Lazy Sheep, and Slow Sheep in a math exam is 82 points. Their scores are all two-digit numbers, and Slow Sheep scored 87 points. Then, the lowest score is at least $\qquad$ points.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 60 |
96,087 | For example, how many pairs of integer solutions $(x, y)$ does the equation $\sqrt{x}+\sqrt{y}=\sqrt{200300}$ have? (2003 Singapore Mathematical Olympiad) | 11 |
96,091 | 15. A and B start from points A and B respectively at the same time and walk towards each other. A walks 60 meters per minute, and B walks 40 meters per minute. After a period of time, they meet at a point 300 meters away from the midpoint between A and B. If A stops for a while at some point during the journey, they will meet at a point 150 meters away from the midpoint. How long did A stop on the way? | 18.75 |
96,096 | 8. Use 2 colors to color the 4 small squares on a $2 \times 2$ chessboard, there are $\qquad$ different coloring schemes. | 6 |
96,105 | 26. $488 \square$ is a four-digit number, the math teacher says: “I fill in this blank with 3 digits in sequence, the resulting 3 four-digit numbers can be successively divisible by $9,11,7$.” What is the sum of the 3 digits the math teacher filled in? | 17 |
96,125 | 1. Four-cell Sudoku: Fill in the cells below with numbers $1 \sim 4$, so that each row, each column, and each block of four cells enclosed by thick lines contains no repeated numbers. The number represented by “?” is $\qquad$ | 4 |
96,200 | 19. (3 points) As shown in the figure, an isosceles triangle $A B C$ is folded along $E F$, with vertex $A$ coinciding with the midpoint $D$ of the base, if the perimeter of $\triangle$ $A B C$ is 16 cm, and the perimeter of quadrilateral $B C E F$ is 10 cm, then $B C=$ $\qquad$ cm. | 2 |
96,203 | As shown in the figure, fold a square piece of paper twice, then cut along the dotted line. Which of the following options is the unfolded shape?
Translate the above text into English, keep the line breaks and format of the source text, and output the translation result directly. | 4 |
96,212 | 13. As shown in the figure, in quadrilateral $A B C D$, $\angle A=\angle C=90^{\circ}, A B-A D=1$. If the area of this quadrilateral is 12, then $B C+C D=$ $\qquad$ . | 7 |
96,223 | 7. Let $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ be two points on the curve $C: x^{2}-y^{2}=2(x>0)$, then the minimum value of $f=\overrightarrow{O A} \cdot \overrightarrow{O B}$ is $\qquad$ . | 2 |
96,262 | 6. (6 points) The zodiac signs of the students in a school include Rat, Ox, Dragon, Snake, Horse, Sheep, Monkey, Rooster, Dog, and Pig. Then, at most how many students need to be selected to ensure that two students have the same zodiac sign.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
---
6. (6 points) The zodiac signs of the students in a school include Rat, Ox, Dragon, Snake, Horse, Sheep, Monkey, Rooster, Dog, and Pig. Then, at most how many students need to be selected to ensure that two students have the same zodiac sign. | 11 |
96,332 | 78. As shown in the figure, a cylindrical glass contains a conical iron block, with the base radius and height of the glass and the iron block being equal. Now water is poured into the glass. When the water level is $\frac{2}{3}$ of the glass height, an additional 260 cubic centimeters of water is needed to fill the glass. The volume of this conical iron block is $\qquad$ cubic centimeters. | 270 |
96,363 | 4. Four identical small rectangles are arranged to form the larger square shown below. The perimeter of each small rectangle is 20 cm. Therefore, the area of the larger square is $\qquad$ square centimeters. | 100 |
96,376 | 7. The teacher wrote 18 natural numbers on the blackboard and asked the students to find the average, with the result rounded to two decimal places. Li Jing's answer was 17.42, and the teacher said the last digit was wrong, but all other digits were correct. Then the correct answer is $\qquad$ | 17.44 |
96,399 | 22. (5 points)
2021 can be expressed as the sum of $\qquad$ consecutive natural numbers.
Express the above text in English, preserving the original text's line breaks and format, and output the translation result directly. | 47 |
96,440 | 【Question 7】
On Tree Planting Day, the students of Class 4(1) went to the park to plant trees. Along a 120-meter long road on both sides, they dug a hole every 3 meters. Later, due to the spacing being too small, they changed it to digging a hole every 5 meters. In this way, at most $\qquad$ holes can be retained. | 18 |
96,458 | 6. As shown in the figure, $AB=\sqrt{2}, CD=\sqrt{30}, AB // MN // DC$, and the area of quadrilateral $MNCD$ is 3 times the area of quadrilateral $ABNM$, then $MN=$ $\qquad$ . | 3 |
96,476 | 19. (6 points) A 30 cm long iron wire can form $\qquad$ different rectangles (with integer side lengths).
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
---
Note: The translation is provided as requested, but the note at the end is for clarification and should not be included in the final output if it is to strictly follow the instruction. Here is the final output without the note:
19. (6 points) A 30 cm long iron wire can form $\qquad$ different rectangles (with integer side lengths). | 7 |
96,480 | 9. A cubic wooden block, 1 is opposite to 6, 2 is opposite to 5, 3 is opposite to 4. It rolls forward 3 times, then rolls to the right 2 times, at this point the number facing up is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 2 |
96,519 | Example 16 (from the 2nd American Mathematical Invitational Competition) The function $f(x)$ is defined on the real number domain and satisfies the following conditions: for any real number $x$, $f(2+x)=f(2-x)$, and $f(7+x)=f(7-x)$. If $x=0$ is a root of the equation $f(x)=0$, how many roots should the equation $f(x)=0$ have at least in the interval $-1000 \leqslant x \leqslant 1000$? | 401 |
96,526 | 8. (3 points) Four volleyball teams are participating in a round-robin tournament, where each team plays against every other team exactly once. If the score of a match is 3:0 or 3:1, the winning team gets 3 points, and the losing team gets 0 points; if the score is 3:2, the winning team gets 2 points, and the losing team gets 1 point. The final scores of the teams are four consecutive natural numbers. What is the score of the first-place team?
The result of the translation is as follows:
8. (3 points) Four volleyball teams are participating in a round-robin tournament, where each team plays against every other team exactly once. If the score of a match is 3:0 or 3:1, the winning team gets 3 points, and the losing team gets 0 points; if the score is 3:2, the winning team gets 2 points, and the losing team gets 1 point. The final scores of the teams are four consecutive natural numbers. What is the score of the first-place team? | 6 |
96,540 | 7. Use 125 identical small cubes to form a large cube. Observing this large cube, the maximum number of small cubes that can be seen simultaneously is $\qquad$.
Translating the text into English while preserving the original formatting and line breaks:
7. Use 125 identical small cubes to form a large cube. Observing this large cube, the maximum number of small cubes that can be seen simultaneously is $\qquad$. | 61 |
96,542 | 42. The five-digit number $\overline{2} 73 a b$ is divisible by both 3 and 7. Find the number of five-digit numbers that satisfy the condition. | 5 |
96,579 | 7. A rectangle is divided into 4 different triangles (as shown in the right figure), the green triangle occupies $15 \%$ of the rectangle's area, and the area of the yellow triangle is 21 square centimeters. What is the area of the rectangle in square centimeters? | 60 |
96,598 | 3. In $\triangle A B C$, $\angle C=90^{\circ}, \angle A$ and $\angle B$ are bisected and intersect at point $P$, and $P E \perp A B$ at point $E$. If $B C=2, A C=3$,
then $A E \cdot E B=$ $\qquad$ | 3 |
96,610 | 9. Qiaohu wants to put 18 identical crystal balls into three bags, with the second bag having more crystal balls than the first, and the third bag having more crystal balls than the second. Each bag must contain at least one crystal ball. There are $\qquad$ different ways to do this. | 19 |
96,615 | 3. Given five numbers in sequence are $13,12,15,25,20$, the product of each pair of adjacent numbers yields four numbers, the product of each pair of adjacent numbers among these four numbers yields three numbers, the product of each pair of adjacent numbers among these three numbers yields two numbers, and the product of these two numbers yields one number. How many consecutive 0s can be counted from the units place to the left in this final number (refer to the figure)?
Translating the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 10 |
96,667 | 1. (8 points) The sum of the digits of the result of the expression $999999999-88888888+7777777-666666+55555-4444+333-22+1$ is $\qquad$ . | 45 |
96,716 | 4. (3 points) Given that $A, B, C, D$ and $A+C, B+C, B+D, D+A$ represent the eight natural numbers from 1 to 8, and are all distinct. If $A$ is the largest of the four numbers $A, B, C, D$, then $A$ is $\qquad$
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 6 |
96,746 | Example 5 Suppose among $n$ freshmen, any 3 people have 2 people who know each other, and any 4 people have 2 people who do not know each other. Try to find the maximum value of $n$.
| 8 |
96,748 | 96. The classmates in Xiaoming's class can be arranged in a square formation. If this square formation is reduced by 4 rows, and the students from these 4 rows are distributed to the remaining rows, it will increase the number of students in each of the remaining rows by 8. Xiaoming's class has a total of $\qquad$ students.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 64 |
96,758 | 9. The number of real roots of the equation $\left(\frac{3}{19}\right)^{x}+\left(\frac{5}{19}\right)^{2}+\left(\frac{11}{19}\right)^{x}=2 \sqrt{x-1}$ is | 1 |
96,776 | 5. (5 points) Xiaoming added the page numbers of a book starting from 1, and when he added them all up, he got 4979. Later, he found out that the book was missing one sheet (two consecutive page numbers). So, the book originally had $\qquad$ pages.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 100 |
96,778 | 10. (5 points) There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm, then starting from the right end, a line is drawn every 6 cm, and the stick is cut at the lines. Among the small sticks obtained, the number of 3 cm long sticks is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 12 |
96,797 | 9. (5 points) There are 4 distinct natural numbers, their average is 10. The largest number is at least
保留源文本的换行和格式，翻译结果如下：
9. (5 points) There are 4 distinct natural numbers, their average is 10. The largest number is at least | 12 |
96,798 | 6. Use 2 red beads, 2 blue, and 2 purple beads to string into a bracelet as shown in the figure below, you can make $\qquad$ different bracelets.
保留了源文本的换行和格式。 | 11 |
96,799 | 9. (15 points) The figure below consists of a large square grid made up of 9 smaller $2 \times 2$ grids, with the requirement that adjacent numbers in adjacent small grids must be the same (these small grids can be rotated, but not flipped). Now, one small grid has already been placed in the large grid. Please place the remaining 8 grids according to the requirements. The number in the bottom-right cell is $\qquad$
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 1 |
96,850 | 15. In $\triangle A B C$, the three sides $a$, $b$, and $c$ satisfy $2 b=a+c$. Find the value of $5 \cos A-4 \cos A \cos C+5 \cos C$. | 4 |
96,860 | 1. In $\triangle A B C$, it is known that $2 \sin A+3 \cos B=4,3 \sin B+2 \cos A=\sqrt{3}$, then the degree of $\angle C$ is | 30 |
96,872 | 4. Let $A_{1} A_{2} \cdots A_{21}$ be a regular 21-sided polygon inscribed in a circle. Choose $n$ different vertices from $A_{1}, A_{2}, \cdots, A_{21}$ and color them red, such that the distances between any two of these $n$ red points are all different. Then the maximum value of the positive integer $n$ is $\qquad$
$(2014$, Sichuan Province Junior High School Mathematics Competition) | 5 |
96,879 | 43. Two frogs start jumping towards each other from the two ends of a 10-meter log. One of the frogs jumps 20 centimeters every 2 seconds; the other frog jumps 15 centimeters every 3 seconds. When the distance between them is not enough for another jump, they stop. At this point, they are $\qquad$ centimeters apart. | 10 |
96,906 | 11. Lele's family raised some chicks and ducklings. If any 6 are caught, at least 2 of them are not ducklings; if any 9 are caught, at least 1 of them is a duckling. The maximum number of chicks and ducklings that Lele's family can have is $\qquad$ . | 12 |
96,928 | 1. When $n$ is a positive integer, the function $f$ satisfies:
$$
\begin{array}{l}
f(n+3)=\frac{f(n)-1}{f(n)+1}, \\
f(1) \neq 0 \text { and } f(1) \neq \pm 1 .
\end{array}
$$
Then the value of $f(11) f(2021)$ is $\qquad$ | -1 |
96,947 | 21. (10 points) When Xiao Hong was organizing her change purse, she found that there were a total of 25 coins with denominations of 1 cent, 2 cents, and 5 cents, with a total value of 0.60 yuan. Then, the maximum number of 5-cent coins is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Note: The instruction at the end is not part of the translation but is included to clarify the request. The actual translation is provided above. | 8 |
96,951 | 8. As shown in the figure, $D E$ is the midline of $\triangle A B C$, point $F$ is on $D E$, and $\angle A F B=90^{\circ}$. If $A B=6$, $B C=10$, then $E F=$ $\qquad$ | 2 |
96,965 | 4. On Sunday, Xiao Jun helped his mother do some housework. The time spent on each task was: making the bed 3 minutes, washing dishes 8 minutes, using the washing machine to wash clothes 30 minutes, hanging clothes 5 minutes, mopping the floor 10 minutes, peeling potatoes 12 minutes. With proper planning, Xiao Jun would need at least minutes to complete these chores. | 38 |
96,984 | 4. (10 points) As shown in the figure, quadrilateral $ABCD$ is a square with a side length of 11 cm, $G$ is on $CD$, quadrilateral $CEFG$ is a square with a side length of 9 cm, $H$ is on $AB$, $\angle EDH$ is a right angle, the area of triangle $EDH$ is $\qquad$ square centimeters. | 101 |
96,989 | 3. 2. 14 * The sequence $\left\{x_{n}\right\}$ is defined as follows: $x_{1}=\frac{1}{2}, x_{n+1}=x_{n}^{2}+x_{n}$, find the integer part of the following sum: $\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\cdots+\frac{1}{1+x_{2 \times 00}}$. | 1 |
96,998 | 5、In a certain examination room of the Central Loop Cup, there are a total of 45 students, among whom 35 are good at English, 31 are good at Chinese, and 24 are good at both subjects. How many students are not good at either subject? $\qquad$ people
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 3 |
97,005 | II. (40 points) Given that $m$ and $n$ are two different positive integers, the last four digits of $2019^{m}$ and $2019^{n}$ are equal. Find the minimum value of $m+n$.
保留源文本的换行和格式，直接输出翻译结果。 | 502 |
97,018 | 5. Elsa, Anna, and Kristoff are in a chess tournament. Elsa, who played the most games, played 7 games; Anna, who played the fewest games, played 5 games. Together, they played $\qquad$ games. | 9 |
97,022 | 5. For any non-zero real numbers $x, y, \frac{x}{|x|}+\frac{|y|}{y}+\frac{2|x| y}{x|y|}$ has different values.
保留源文本的换行和格式，翻译结果如下：
5. For any non-zero real numbers $x, y, \frac{x}{|x|}+\frac{|y|}{y}+\frac{2|x| y}{x|y|}$ has different values. | 3 |
97,068 | 2. The set $A=\left\{z \mid z^{18}=1\right\}$ and $B=\left\{w \mid w^{48}=1\right\}$ are both sets of complex roots of 1, and the set $C=$ $\{z w \mid z \in A, w \in B\}$ is also a set of complex roots of 1. How many distinct elements are there in the set $C$? | 144 |
97,075 | 7. (10 points) A convoy passes through a 298-meter-long bridge at a speed of 4 meters/second, taking a total of 115 seconds. It is known that each vehicle is 6 meters long, and the interval between adjacent vehicles is 20 meters. Then, this convoy has $\qquad$ vehicles.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 7 |
97,082 | 25. There are 2015 integers, and by taking any 2014 of them and adding them together, their sum can exactly result in the 2014 different integers 1, 2, , , 2014. Then the sum of these 2015 integers is $\qquad$ . | 1008 |
97,083 | 6. (3 points) Two differently sized cubic building blocks are glued together, forming the solid figure shown in the diagram. The four vertices of the smaller cube's glued face are the non-midpoint quarter points of the larger cube's glued face edges. If the edge length of the larger cube is 4, then the surface area of this solid figure is $\qquad$ .
| 136 |
97,098 | 11. At point A upstream, there is a large ship, and next to it is a patrol boat, which continuously moves from the bow to the stern of the large ship and then from the stern back to the bow (the length of the patrol boat is negligible). Meanwhile, at point B downstream, there is a small boat (the length of the small boat is negligible). The large ship and the small boat start moving towards each other at the same time, with the patrol boat and the large ship's bow both starting at point A. When the patrol boat returns to the bow of the large ship for the first time, it meets the small boat; when the patrol boat returns to the bow of the large ship for the 7th time, the bow of the large ship reaches point B. If the water speed doubles when the large ship starts, then when the patrol boat returns to the bow of the large ship for the 6th time, the bow of the large ship reaches point B. What is the ratio of the small boat's speed in still water to the original water speed? | 37 |
97,107 | 12. (5 points)
The amusement park "Forest River Adventure" has a circular river, as shown in the figure. At 8:00, Feifei sets off downstream in a small boat. Feifei's rowing speed in still water is 4 kilometers per hour, and the current speed is 2 kilometers per hour. Feifei rests for 5 minutes after every half hour of rowing, during which the boat drifts with the current. If Feifei returns to the starting point exactly at 10:00, then the length of the river is \qquad kilometers. | 11 |
97,156 | 1. If $n$ is an integer, and $72 \times\left(\frac{3}{2}\right)^{n}$ is an integer, then the number of $n$ that satisfy the condition is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Note: The note is not part of the translation but is provided to clarify the instruction. The actual translation is above. | 6 |
97,222 | 4. (10 points) A sequence of numbers: $8,3,1,4, . \cdots$, starting from the third number, each number is the unit digit of the sum of the two closest preceding numbers. What is the 2011th number? $\qquad$ . | 2 |
97,224 | 5. If real numbers $x, y$ satisfy
$$
2^{x}+4 x+12=\log _{2}(y-1)^{3}+3 y+12=0 \text {, }
$$
then $x+y=$ $\qquad$ . | -2 |
97,239 | Three, (25 points) Given that $3 n^{3}+2013(n>1)$ is divisible by 2016. Find the smallest positive integer $n$.
---
Translation:
Three, (25 points) Given that $3 n^{3}+2013(n>1)$ is divisible by 2016. Find the smallest positive integer $n$. | 193 |
97,244 | 31. (5 points)
The careless Pigsy failed to notice the dots above the repeating decimals when calculating $2.0 \ddot{2} 1 \times 165000$. His result was less than the correct result by $\qquad$ . | 35 |
97,297 | 3. Given the function $f(x)=\frac{2^{|2 x|}-2018 x^{3}+1}{4^{|x|}+1}$ with its maximum value $M$ and minimum value $m$ on $\mathbf{R}$, then $M+m=$ | 2 |
97,322 | 7. A string has a total of 85 green, red, and yellow beads, arranged in the pattern “three greens, four reds, one yellow, three greens, four reds, one yellow, ......”. How many $\qquad$ red beads are there.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 42 |
97,327 | 1. Calculate $\frac{2}{1} \times \frac{2}{3} \times \frac{4}{3} \times \frac{4}{5} \times \frac{6}{5} \times \frac{6}{7} \times \frac{8}{7}=$ $\qquad$.(Write in decimal form, accurate to two decimal places.) | 1.67 |
97,368 | 9. The figure below shows a glass in the shape of a right circular cylinder. A straight, thin straw (neglecting its thickness) of length 12 cm is placed inside the glass. When one end of the straw touches the bottom of the cylinder, the other end can protrude from the top edge of the cylinder by a minimum of 2 cm and a maximum of 4 cm. The volume of this glass is $\qquad$ cubic centimeters. (Take $\pi=314$) (Hint: In a right triangle, “leg 6, leg 8, hypotenuse 10”)
| 226.08 |
97,455 | 9.16 It is known that the number 32※35717※ can be divided by 72, find the digits represented by these two asterisks.
(Kyiv Mathematical Olympiad, 1963) | 26 |
97,462 | 71. A three-digit number leaves a remainder of 2 when divided by 4, 5, and 6. If three digits are added to the end of this number to make it a six-digit number, and this six-digit number is divisible by $4, 5, 6$, then the smallest six-digit number that meets the condition is | 122040 |
97,467 | (4) Given that line $l$ forms a $45^{\circ}$ angle with plane $\alpha$, and line $m \subset \alpha$, if the projection of line $l$ on $\alpha$ also forms a $45^{\circ}$ angle with line $m$, then the angle formed by line $l$ and $m$ is $\qquad$ | 60 |
97,479 | 4. (8 points) As shown in the figure, there are four circles of different sizes, with diameters from smallest to largest being $5$, $10$, $15$, and $20$ cm, respectively. Therefore, the sum of the areas of the shaded parts in the figure is $\qquad$ square cm. (Take $\pi$ as 3.14) | 314 |
97,571 | 22. The last three digits of $9^{2022}$ are
The translation maintains the original text's line breaks and format. | 881 |
97,584 | 5. Thomas and Edward are playing a three-digit number chain game, with the rules being:
(1) The first digit of a number must be the same as the last digit of the previous number;
(2) The tens digits of adjacent numbers cannot be the same.
Thus, the combination of $\mathrm{X}$ and $\mathrm{Y}$ has $\qquad$ different possibilities
$$
398 \rightarrow 804 \rightarrow 447 \rightarrow 792 \rightarrow \mathrm{X} \rightarrow \mathrm{Y} \rightarrow 516
$$ | 657 |
97,621 | 4. In a charity donation event, a total of 72 people donated, and the total donation amount is a five-digit number, with the ten-thousands digit being 5, the hundreds digit being 3, the tens digit being 7, and the average donation amount per person is an integer. Therefore, the average donation per person is $\qquad$ yuan.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
---
Note: The last sentence is a repetition of the instruction and should not be included in the translated text. Here is the corrected version:
4. In a charity donation event, a total of 72 people donated, and the total donation amount is a five-digit number, with the ten-thousands digit being 5, the hundreds digit being 3, the tens digit being 7, and the average donation amount per person is an integer. Therefore, the average donation per person is $\qquad$ yuan. | 783 |
97,627 | 16. The teacher distributed 60 storybooks to all the students in the class. If each student gets 1 book, there are still some left; if the remaining books are distributed with 2 students sharing 1 book, they are just enough. The class has ( ) students.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 40 |
97,634 | 5. Given that $x, y$ satisfy $2 x+5 y \geq 7, 7 x-3 y \leq 2$, then, the minimum value of $-27 x+35 y$ is
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 18 |
97,657 | 8. A basketball team currently has a win rate of $40 \%$. If they win the next 10 games, their win rate will increase to $50 \%$. The team has already won $\qquad$ games.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Note: The last sentence is a repetition of the instruction and should not be included in the final translation. Here is the correct format:
8. A basketball team currently has a win rate of $40 \%$. If they win the next 10 games, their win rate will increase to $50 \%$. The team has already won $\qquad$ games. | 20 |
97,658 | 16. A four-digit number, when split in the middle, yields two two-digit numbers, whose sum equals 42. For example, 2022 is such a four-digit number. Besides 2022, there are $\qquad$ such four-digit numbers. | 22 |
97,660 | Example 7.10 Let $A_{6}$ be a regular hexagon. Now we use red and blue to color the 6 vertices of $A_{6}$, with each vertex being one color. Find the number of type II patterns of the vertex-colored regular hexagon. | 13 |
97,663 | 9. Positive integers $x, y$ satisfy $\frac{2}{5}<\frac{x}{y}<\frac{3}{7}$, then the minimum value of $y$ is | 12 |
97,665 | 1. Given that $a$ and $b$ are positive integers, satisfying $\frac{1}{a}-\frac{1}{b}=\frac{1}{2018}$. Then the number of all positive integer pairs $(a, b)$ is $\qquad$ . | 4 |
97,675 | 5. Among the three-digit numbers whose digits sum to 10, the numbers where the hundreds digit is no less than 3 and the units digit is no less than 2 total $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Note: The blank space represented by $\qquad$ is kept as is in the translation. | 21 |
97,680 | 67. A box of apples, Xiao Ying takes 3 the first time, 6 the second time, and each subsequent time she takes 3 more than the previous time, finishing in 10 times, the box of apples has $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Note: The $\qquad$ is kept as is, since it seems to be a placeholder for the answer in the original text. | 165 |
97,688 | Example 1-16 Find the sum of integers formed by the digits $1,3,5,7$ without repetition.
The digits $1,3,5,7$ can form various integers where each digit appears only once in each integer. The task is to find the sum of all such possible integers. | 117856 |
97,696 | 20. Piplu is learning long division, and he finds that when a three-digit number is divided by 19, the quotient is $a$, and the remainder is $b$ ($a, b$ are natural numbers), then the maximum value of $a+b$ is . $\qquad$ | 69 |
97,713 | 8. A rectangular garden is divided into 5 smaller rectangular areas of the same shape and size, each with an area of 24 square meters. The perimeter of this rectangular garden is $\qquad$ meters. | 44 |
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