1. Multiple choice questions (4 points for each question, ***40 points. Only one of the four options in each question below is correct. Please fill in the English letters indicating the correct answer in the brackets after each question.

1. Among the following four groups of radicals, the group that belongs to the similar quadratic radical is ().

2. In order to make the algebraic expression meaningful, the range of real number x is ().

3. Take line segments a= 13, b= 13, c= 10, d=6 as sides to form a trapezoid, where A and C are the two bases of the trapezoid, so this trapezoid ().

(1) You can make one. (b) You can make two. (c) it can make countless. (d) one cannot be made.

(English-Chinese dictionary: abbreviation of figure. Figure, figure; Quadrilateral quadrilateral; Diagonal diagonal; Value value; Variable variable; Dependence; Location location)

(a) Is it a perfect square number or an odd number? (b) Whether it is a perfect square number or an even number.

(c) Not a perfect square, but an odd number. (d) Not a perfect square, but even.

6. Fold any piece of paper in a convex quadrilateral in half to make its two non-adjacent vertices overlap, then cut off the non-overlapping part of the paper, unfold the paper, fold it in half again to make the other two vertices overlap, then cut off the non-overlapping part and unfold it. At this time, the shape of the paper is ().

(1) Square. (b) rectangular. Diamonds (d) isosceles trapezoid.

7. If a, b and c are all natural numbers greater than l and =252b, then the minimum value of n is ().

(A)42。 (B)24。 (C)2 1 (D) 15

(English-Chinese dictionary: two digits, two digits; Number, number; Satisfy; Perfect square (number); Grand total)

9. The following table is a list of popular songs of a radio station this week, in which the song J is a new song on the list, and the arrow "↑" or "↓" respectively indicates the change of the song compared with last week's ranking, "↑" indicates the increase, "↓" indicates the decrease, and if it is not marked, it indicates that the ranking has not changed. If it is known that the ranking of each song does not change by more than two places, then last week's ranking is 1, 5.

(A)D、E、h(B)C、F、I(C)C、E、I(D)C、F、h。

10. Let n(n≥2) positive integers,,, arbitrarily change their order, and write,,, if p = (-) (-)()...( 1), then ().

(A)P must be an odd number. (b) P must be an even number.

(c) When n is odd, p is even. (d) When ""is even, p is odd.

Fill in the blanks (4 points for each small question, ***40 points. )

1 1. The length of the fire escape is 34 meters. When performing a task, you can only stop at a distance of 0/6 meters from the building/kloc-,so the height that the ladder can reach the building is _ _ _ _ _ _ _.

15. All diagonals drawn from the vertices of the convex N polygon divide the convex N polygon into m small triangles. If m is equal to the diagonal number of the convex N-polygon, then the sum of the internal angles of the N-polygon is _ _ _ _.

16. A spherical virus, with a diameter of 0.0 1 nm, can reproduce 9 viruses like itself every minute. If this virus gathers to a certain number in human body and is arranged in a string according to this number, when the length reaches 1 decimeter, people will feel uncomfortable, so people will pass _ _

19. As shown in Figure 2, in isosceles △ABC, AB=AC, and the point P is on the high AD of BC, and,

The extension line of BP intersects with AC at point E. If = 10, then = _ _ _ _, = _ _ _ _.

20. There are 20 number plates on a circle (1, 2, 3, …, 20 * *). Pick a number plate at random (for example, 8), take it off first, and then pick one every other (for example, 8, 10, 12).

Third, the solution (this big question is ***3 small questions, ***40 points. ) Requirements: Write out the calculation process.

2 1. (Full score for this small question 10)

As shown in fig. 3, the side length of the square ABCD is a, and the points E, F, G and H are on the four sides of the square, and ef ‖ GH is known. EF = GH。

(1) If AE=AH=, find the perimeter and area of the quadrilateral EFGH;

(2) Find the minimum value of quadrilateral EFGH perimeter.

22. (The full score of this short question is 15)

It is known that Port A is upstream of Port B, and the ship left Port A for Port B at 3:00 a.m., and immediately returned after arrival, shuttling back and forth between Port A and Port B. If the speed of the ship in still water is 16 km/h and the current speed is 4 km/h, then the ship was seen driving to 80 km from Port A at 23: 00 that night to find the distance between Port A and Port B.

23. (The full score of this short question is 15)

Between 2 and 3, write for the first time, between 2, 5 and 5, 3, and write for the second time, as shown below:

The k-th operation is based on the last operation, and the sum of these two numbers is written between every two adjacent numbers.

(1) Please write down the 9 numbers obtained after the third operation and find their sum;

(2) Write the relationship between the sum of all numbers after k operations and the sum of all numbers after k+ 1 operations;

(3) the value.

A guide to the second-grade mathematics Olympics

Class name and student number

1, as shown in the figure, in trapezoidal ABCD, AD∨BC, DE = EC, EF∨AB intersects BC at point F, EF = EC, and connects DF.

(1) Try to explain that the trapezoid ABCD is an isosceles trapezoid;

(2) If AD = 1, BC = 3, DC =, try to judge the shape of △DCF;

(3) Under the condition (2), is there a point p on the ray BC, which makes △PCD an isosceles triangle? If yes, please write the length of PB directly; If it does not exist, please explain why.

2. In a rhombic ABCD with a side length of 6, the moving point M starts from point A, moves along A→B→C to the end point C, and connects DM and AC at point N. 。

(1) As shown in Figure 25- 1, when point M is on the AB side, connect BN.

① Verification: △ ABN △ ADN;

② If ∠ ABC = 60 and AM = 4, find the distance from point M to AD;

(2) As shown in Figure 25-2, if ∠ ABC = 90, remember that the distance traveled by point M is x(6≤x≤ 12). What is the value of x, and △ADN is an isosceles triangle.

3. For points O and M, point M moves towards MO until O turns left and continues to move towards N, so that OM = ON and OM⊥ON. This process is called that point M completes a "left turn" around point O. 。

Party ABCD and point P, point P turns left about A to P 1, P 1 turns left about B to P2, P2 turns left about C to P3, P3 turns left about D to P4, and P4 turns left about A to P5. ...

(1) Please use a ruler and compass to locate the point P 1 in the drawing;

(2) connect P 1A and P 1B to determine the relationship between △ABP 1 and △ADP. And explain why.

(3) Establish a rectangular coordinate system with D as the origin and straight line AD as the axis. It is known that point B is in the second quadrant, and the coordinates of point A and point P are (0,4) and (1, 1) respectively. Please infer the coordinates of points P4, P2009 and P20 10.

4. As shown in Figures 1 and 2, in 20×20 equidistant grids (the width and height of each grid are 1 unit length), Rt△ABC starts from the position where point A coincides with point M, and moves downward at the speed of 1 unit length per second. When the BC edge coincides with the bottom of the grid, it continues to move to the right at the same speed. When point c coincides with point p,

(1) as shown in figure 1, when Rt△ABC moves down to the position of rt △ a1b1,please draw rt △ a1b1c650 in the grid.

(2) As shown in Figure 2, during the downward translation of Rt△ABC, please find out the functional relationship between Y and X, and explain how to get the maximum and minimum values of Y when taking X respectively. What are the maximum and minimum values respectively?

(3) During the translation of RT △ ABC to the right, please explain what value X takes and what value Y takes to get the maximum and minimum values. What is the maximum value and the maximum value respectively? Why?

5. As shown in Figure ①, in △ABC, the bisectors of AB=AC, ∠B and ∠C intersect at point O, if passing through point O, EF∨BC intersects with AB, and AC intersects with E and F. 。

(1) How many isosceles triangles are there in the graph? Guess: what is the relationship BEtween EF, be and CF, and explain the reasons.

(2) As shown in Figure ②, if conditions such as AB≠AC remain unchanged, is there an isosceles triangle in the figure? If yes, please point out separately. Is there any doubt about the relationship between EF and BE and CF (1)?

(3) As shown in Figure ③, if the bisector BO of ∠B in △ABC and the bisector CO of the outer corner of the triangle intersect at O, then the intersection point O is OE∨BC, AB is in E, and AC is in F. Is there an isosceles triangle in the figure? What is the relationship between EF and be and CF? State your reasons.

6. As shown in the figure, in △ABC, ∠ BAC = 90, AB=AC, D is a point above AC, ∠ BDC = 124, extend BA to point E, make the extension lines of AE = AD and BD intersect CE at point F, and find the degree of ∠ E. ..

7. As shown in the figure, the diagonal AC and BD of the square ABCD intersect at the O point. Put the right-angled vertex of a triangular ruler at the O point and let it rotate around the O point. The right-angled edge of the triangular ruler intersects with the two sides of the square ABCD at the E point and the F point ... What did you find by observing or measuring the length of OE? Try to explain why.

Baidu knows:

2. The ternary linear equation 7x+3y-4z = 1 is expressed by an algebraic expression containing xy, and z = _ _ _ _.

3. In the ternary linear equation X+Y+Z = 3, if X =- 1 and Y = 2, then Z = _ _.

4. If the equation 2x-y-5zn-2 = 3 is a ternary linear equation, then n = _ _ _ _

5. If the equation -3x-my+4z = 6 is a ternary linear equation, the value range of m is _ _ _ _.

6. The ternary linear equation 2x-my+z = 3 has a set of solutions, then m = _ _ _.

7. It is known that the binary linear equations that eliminate z from the ternary linear equations are _ _ _ _ _.

8. The values of x+y+z satisfying the equation (2x-6) 2+2 (y+3) 2+7 = 0 are _ _ respectively.

9. When x = 0, 1 and-1, and the values of quadratic trinomial ax2+bx+c are 5,6 and 10 respectively, then A = _, B _ _, and C = _ _ _.

10. Choose carefully, you must be very accurate.

1 1, when solving the equation, the best way to eliminate the unknown for the first time is ().

Add and subtract X, and compare ①-③× 3 and ②-③× 2.

Add and subtract Y, and mix ①+③ with ①×3+②.

Add, subtract and eliminate Z, and add ①+② and ③+②.

Eliminate any one of x, y and z by substitution.

12. If the values of x and y in the solution of the equation are reciprocal, then the value of m is ().

a， 1 B，- 1 C，2 D，-2

13, if the solution of the equations is also the solution of the equation MX-2Y+Z = 0, then the value of m is ().

a，B，- C，D，-

15. Among the following four groups of numbers, the solution applicable to equation 2x-y+z = 0 is ().

A, B, C, D,

16, given that the equation 3x-y-7 = 0, 2x+3y = 1, y = kx-9 has a common * * * solution, then the value of k is ().

a，3 B，4 C，D，

18, known, xyz≠0, find x: y: z. (3 points)

19, what is the value of k, and the solution of is suitable for y = x-2? (3 points)

20. The following is the elimination process of Xiao Ming's solution to ternary linear equations. In the third step, he found that he still couldn't find the solution of the equations. Please help Xiao Ming analyze the cause of the problem and correct it.

solve an equation

The first step of solution: ①+②: (eliminate Y) 7x+3z = 2④.

Step 2: ①+③ Get: (eliminate Z) 6x+6y =-3⑤.

Step 3: ④ and ⑤ form an equation.

Why can't I find the values of x, y and z? (4 points)

2 1. You fill in (2 points for each question, *** 12 points).

-ax+y-zb5cx-y+z and a 1 1b-x+y+ZC are similar terms, so x = _, y = _, and z = _.

The solution of ternary linear equations is _ _.

X and y in the equation satisfy the condition that x+y = 6, then the value of z is equal to _ _.

22. if 4a-3b-3c = 0, a-3b+c = 0 (a ≠ 0, b≠0, c≠0), then a ∶ b ∶ c = _.

23 if x: y: z = 1: 2: 3 and x+y+z = 12, XYZ = _ _ _ _

24. the positive integer solution of the equation x+2y+3z = 14(x < y < z is _ _ _ _.

25. algebraic expression ax2+bx+c, when x = 1, the value is 0, when x = 2, the value is 3 and when x =-3, the algebraic expression is ().

a，2x2-3x+ 1 B，x2- 1 C，-x2+6x-5 D，x2+x+ 1-2

26. If (a+b+c-6) 2+(3b-2c) 2+= 0, then the value of-is ().

a，- 1 B，-2 C， 1 D，2

27. The following statement is true ()

The equation 3x+2y+z = 20 has a unique solution.

If x, y and z are nonnegative, the ternary linear equation 3x+5y+2z = 0 has only one set of solutions.

Equation 4x++2z = 7 is a ternary linear equation.

The system of linear equations with three variables has one and only one solution.

28. If the solution of the equations makes the algebraic expression KX+2Y-Z 10, then the value of k is ().

a，B，3 C，- D，-3

29, given 3x+y+2z = 28, 5x-3y+z = 7, find the value of x+y+z.

30. The values of 2x-3y-z = 0, x, 3y- 14z = 0, xyz≠0 are known.

The two equations of 3 1.X and y: sum has the same solution, so find the values of a and b .. (10)