1, A, B, C, D and E five students took part in a festival. Fortunately, they all got a beautiful gift. Here's the thing: there are two strings of gifts hanging on the wall (as shown in figure 1). You can only take one string from the bottom of one string at a time until all the gifts are taken away. A gets the gift first, and then B, C, D and E get the second to fifth gifts in turn. Of course, there are various ways to get it, so there are () different ways to get it. Afterwards, they opened these gifts and compared them carefully, and found that gift D was the most exquisite, so the most likely gift D was () and the most unlikely gift D was ().
2. J, Q and K in playing cards stand for 1 1, 12 and 13 respectively. A takes 13 hearts and b takes 13 flowers. Two people randomly play a card to make a pair, so a * * * can make a pair 13. If you sum each pair, multiply the sum of these 13. Judging from the parity of product, product should be () number.
3. The sum of all true fractions with denominator less than 2005 is ().
Three students, Xiao Zhao, Xiao Zhang and Xiao Wang, estimated the number of books in Max's schoolbag. Xiao Zhao said: "There are at least 10 books in the schoolbag and at most 15 books." Xiao Zhang said, "There are less than 10 books in the schoolbag." Xiao Wang said, "There are at least 1 book in the schoolbag and at most 15 books." But michaels said, "Only one of you three is right." In this way, there are () books in the wheat bag.
5. As shown in Figure 2, several small cubes with the same volume are stacked to form a big cube. If all diagonals of a big cube (the connecting line between the farthest two vertices of the eight vertices of a cube) pass through a small cube, all other small cubes are white, and there are even number of small cubes on each side of the big cube. After stacking, the volume of the white cube accounts for 96% of the total volume, so a * *.
6. Take eight different numbers from 0 to 9, and the difference is the four digits of 2005, so the maximum sum of the two four digits is () and the minimum is ().
7. All students in a class have a basketball shooting practice, and each student has to throw 10 balls, and each ball gets 1 point. The scores are shown in the table on the right: It is also known that among the students in this class, those who scored at least 3 points scored an average of 6 points, and those who scored less than 8 points scored an average of 3 points, so the students in this class had ().
8. One day, four pairs of painters went to the countryside to sketch, and they always drew 44 paintings. Among them, four female painters A, B, C and D painted 2, 3, 4 and 5 paintings respectively; The number of paintings drawn by four male painters is: A drew the same number as his wife; The width of B, C and D is twice, three times and four times that of the wife. Then the husbands of A, B, C and D are (), (), () and () respectively.
Second, the application problem
1. The unit prices of A, B and C in a certain store are 10 yuan respectively. 6 yuan, 4 yuan and Wang Mai bought several pieces of each of these three commodities, and * * * paid 40 yuan. Later, Xiao Wang thought that one of the goods was overbought and wanted to return two, but the clerk only had RMB with 20 yuan face value and no change to return. Xiao Wang had to adjust the purchase quantity of the other two commodities to keep the total price unchanged. How many products did Wang Mai buy?
The distance between a and b is 2400 kilometers. A starts from A and B at the same time and runs back and forth between A and B. A runs 300 meters per minute and B runs 240 meters per minute, and stops exercising after 35 minutes. When did A and B meet for the first time, the closest to A? What's the nearest distance?
Third, the operation questions
1. There is an electric toy. At point A, the front is connected by a small disk with a radius of 10 cm (with a baby face painted inside) and a large disk with a radius of 20 cm (as shown in Figure 3). If the small disc follows the circumference of the big disc, it starts from point A, keeps rolling counterclockwise (the big disc does not move), and finally returns to its original position. (1) Draw the correct positions of hair, eyes and mouth in the three empty circles in the picture.
(2) How many times did the small circle rotate?
2. As shown in Figure 4, a robot walked10m due east from point A to point B, then turned left by an angle of α 1 and walked10m to point C; Turn left at an angle of α2, take 10 meter, and reach point D; Then turn left at an angle of α3, walk 10 meters, and reach point e; Turn left at an angle of α 4, and then take 10 meter to reach point F; Then turn left at the angle of α 5, and at this time, it is just 10 meters to the west, reaching the G point; Then turn left at the angle of α 6, go due south 10 meter, and reach the robot's home-H point. Excuse me, how many meters did the robot walk? * * * How many degrees did you turn left?
3. Try to divide a square into 2 1 small non-overlapping squares. These small squares are not necessarily the same size. Please draw a picture.
Fourth, question and answer.
1. As shown in Figure 5, OAB is a right-angled sector, and the inside of the sector is semicircular, with diameters of OA and OB respectively. Do you think the shaded part in the picture looks like a leisurely goldfish with a big tail? So which area is larger, the body or the tail of a goldfish? Why?
There are five points on the plane, and the distance between each two points is different. Please connect the nearest two points with a straight line. How many triangles have these lines formed? Why?
3. The grocery store bought a batch of new folding fans at the price of 10 yuan, and the original inventory of three old fans was reduced to 10 yuan. On the first day, the store * * * sold 30 new fans and 30 old fans. Fresh air fans sell 150 yuan, and old fans buy 100 yuan, which is counted as 250 yuan. The next day, the shopkeeper put the old and new fans together for convenience, and 20 yuan sold one as five. Similarly, new and old fans sold 30 fans each, but only 20*(60/5)=240 yuan. Therefore, the shopkeeper thinks that the profit of individual sales is higher than that of joint sales.
From the above example, the owner's conclusion is correct. But is this conclusion also correct in general commodity trading? Why? Please give an example.