First, multiple-choice questions (5 points for each small question, 25 points for * * *)
1. In a right triangle, the square of the hypotenuse is exactly twice the product of two right-angled sides. Then, the ratio of the lengths of the three sides of this triangle is ().
A, B, C, D,
2. There are () integers x * * satisfying the inequality.
a、9998 B、9999 C、 10000 D、 1000 1
3. Take out the number of k from *** 14 natural number and make sure there are two, so that one is twice as much as the other. Then the minimum value of k is ().
a、8 B、9 C、 10 D、 1 1
4, a natural number, take out two numbers at will, if the number on the left is greater than the number on the right, it is said that this number is in reverse order. The reverse number represented by (for example). The remainder of division by 4 is ().
a、0 B、 1 C、2 D、3
5. As shown in the figure, it is a point on the function image, and the straight line respectively intersects with the axis, axis on the point, axis on the point, intersection point on the point, axis on the point and intersection point on the point. Then this value is ().
a、2 B、C、 1 D、
Fill in the blanks (7 points for each small question, 35 points for * * *)
1. If each of five consecutive natural numbers is a composite number, this group of numbers is called a "twin 5-composite number". Then, among the natural numbers not exceeding 100, * * * has a twin 5- composite group.
2. In,,, are the two points on the side,
The area of.
There are two schemes for someone to go out from their residence: one is to ride a bike and the other is to take a bus. The bus is faster than the bike, but he has to wait (the waiting time can be regarded as fixed). In any case, he always chooses the scheme that takes the least time. The following table shows the time it takes him to reach A, B and C and adopt the best scheme.
The time needed to best solve the distance between the destination and the residence.
2 km 12 min
B 3 km 15.5 minutes
4 km 18 min
It takes him at least 20 minutes to get to the place 8 kilometers away from his residence.
4. As shown in the figure, in the rectangle,, is a moving point on the side, acting on the point, acting on the point.
5. There are black and white square papers with the same size and quantity. Xiao Zhang first made a rectangle with no gap in the middle with white paper, then continued to make a larger rectangle around the already made white rectangle with black paper, and then spelled it with white paper. Repeating spelling like this, when Xiao Zhang spelled it five times with black paper, the black and white paper was just used up. Then, there is at least one piece of black paper.
Three. (Full score 15)
In the five-pointed star, the intersecting letters of the intersecting line segments are shown in the figure. Known.
, , , .
Verification:
Four. (Full score 15)
Three mutually different positive integers, if the sum of the product of any two and 1 can be divisible by the third number, are called "exquisite triple arrays".
(1) Verification: Three positive integers in Linglong three arrays are pairwise coprime;
(2) Find all the delicate three arrays.
Verb (abbreviation of verb) (full mark 10)
As shown in the figure, there are 4m points in a △ABC, and some line segments are connected between these points and between these points and three points A, B and C. These line segments have no common points in the triangle except these M points, and all the small areas just divided into △ABC are small triangles. Please prove:
The total number of triangle areas divided by (1) must be odd;
(2) The number of connected line segments within △ABC is a multiple of 3.