1. Multiple choice questions: (***5 small questions, 6 points for each small question, out of 30 points. Each of the following questions gives four options, code A, B, C and D, of which one and only one option is correct. Please put the code of the correct option in parentheses after the question. Don't fill in, fill in too much or fill in the wrong, get zero)
1, as shown in the figure, there is a rectangular piece of paper ABCD with AB=8 and AD=6. Fold the paper so that the edge of AD falls on the edge of AB, and the crease is AE. Then fold △AED to the right along DE, and the intersection of AE and BC is F, so the area of △CEF is ().
2 (B)4 (C)6 (D)8
2. If M= (x, y is a real number), the value of m must be ().
(a) positive number (b) negative number (c) zero (d) integer
3. It is known that point I is the center of acute triangle ABC, and it is the symmetrical point of point I about sides BC, CA and AB respectively. If point B is on the circumscribed circle of △, ∠ABC is equal to ().
30 (B)45 (C)60 (D)90
4, let A=, then the nearest positive integer to a is ().
18 20 24 25
5. Let a and b be positive integers and satisfy, then it is equal to ().
(A) 17 1(B) 177(C) 180(D) 182
Fill in the blanks: (***5 small questions, 6 points for each small question, out of 30 points)
6. On the surface of a circular clock, OA stands for the second hand and OB stands for the minute hand (O is the rotation center of the two hands). If the time is exactly 12, the area of △OAB will reach the maximum for the first time after two seconds.
7. In rectangular coordinate system, parabola (m >；; 0) intersects the X axis at points A and B. If the distances from point A and point B to the origin are OA and OB, respectively, and are satisfied, the value of m is equal to.
8. There are two decks of playing cards. The order of each deck is: the first deck is king, the second deck is Xiao Wang, and then the arrangement of spades, hearts, diamonds and clubs. The cards of each suit are arranged in the order of A, 2, 3, …, J, Q, K. Some people put two decks of playing cards arranged as above together, then throw away the first card from top to bottom, put the second card at the bottom, throw away the third card, put the fourth card at the bottom, and so on, until there is only one card left at last, and the remaining cards are.
9. It is known that D and E are points on the BC and CA sides of △ABC, respectively, BD=4, DC= 1, AE=5, and EC=2. Connect AD and BE, they intersect at point P, the passing points P are PQ‖CA and PR‖CB respectively, and they intersect with side AB at point Q and point R respectively, then the ratio of the area of △PQR to the area of △ABC.
The sum of 10 is known as a positive integer and =58. If the maximum value of is a and the minimum value is b, the value of a+b is equal to.
Iii. Answer: (***4 questions, each small question 15, out of 60).
1 1. When the school held the spring sports meeting, several students formed eight rectangular queues. If 120 people are added to the original queue, a square queue can be formed; If the original queue is reduced by 120 people, a square queue can also be formed. Ask how many students were in the original rectangular queue.
12. It is known that both p and q are prime numbers, and the quadratic equation about x has at least one positive integer root. Find all prime pairs (p, q).
13 As shown in the figure, take the sides AB, BC and CA of △ABC(△ABC is an acute triangle) as hypotenuse, and make isosceles right triangles DAB, EBC and FAC. Verification: (1) AE = df; (2)AE⊥DF。
14, from 1, 2, …, 205 * * 205 positive integers, how many numbers can be taken out at most, so that for any three numbers a, b, c (a
2005 National Junior Middle School Mathematics Competition
The total score of the first, second and third questions
1~5 6~ 10 1 1 12 13 14
score
1. Multiple choice question: (full score 30)
1. As shown in Figure A, ABCD is a rectangular piece of paper, AB=6cm, AD=8cm, E is a point above AD, AE=6cm. Operation: (1) Fold AB in the direction of AE, so that AB and AE overlap to obtain a crease AF, as shown in Figure B; ⑵ Fold △AFB to the right with BF as the crease to get Figure C, then the area of △GFC is ().
A.2 B.3 C.4 D.5
2. If m = 3x2-8xy+9y2-4x+6y+13 (x, y is a real number), then the value of m must be ().
A. positive number B. negative number C. zero D. integer
3. It is known that point I is the center of acute angle △ABC, and points A 1, B 1 and C 1 are the symmetrical points of point I with respect to edges BC, CA and AB respectively. If point B is on the circumscribed circle of △A 1B 1C 1, then ∠ABC is equal to ().
30 BC to 45 BC
4. If, the nearest positive integer to A is ()
A. 18
5. The number of integers whose function value of quadratic function is within the range of 59≤x≤60 of independent variable X is ().
a . 59 b . 120 c . 1 18d . 60
Two. Fill in the blanks (out of 30)
6. On the surface of a circular clock, OA stands for the second hand and OB stands for the minute hand (O stands for the rotation centers of the two hands). If the current time is exactly 12, the area of △OAB will reach the maximum value for the first time after _ _ _ seconds.
7. In the rectangular coordinate system, the parabola and the X axis intersect at two points A and B. If the distances from point A and point B to the origin are OA and OB, respectively, and they are satisfied, then m = _ _ _ _.
8. There are two playing cards. The order of each card is: the first card is king, the second card is Xiao Wang, and then there are four colors of spades, hearts, diamonds and clubs. The cards of each suit are arranged in the order of A, 2, 3, …, J, Q and K. Someone stacked the two cards arranged above together, then threw the first card from one to the bottom, put the second card at the bottom, put the third card at the bottom, and put the fourth card at the bottom ... and so on until there was only one card left, and the remaining cards were _ _ _ _ _ _ _.
9. It is known that D and E are points on the BC and CA sides of △ABC, respectively, BD=4, DC= 1, AE=5, and EC=2. Connect AD and BE, they intersect at point P, P is PQ‖CA and PR‖CB, and they intersect with side AB at points Q and R respectively, so the ratio of the area of △PQR to the area of △ABC is _ _ _ _ _ _ _ _.
10. It is known that x 1, x2, x3, …x 19 are all positive integers, and x1+x2+x3+…+x19 = 59, x12+.
Third, answer questions, (out of 60)
1 1.8 people take two cars with the same speed and rush to the railway station at the same time. There are four people in each car (excluding the driver). One of the cars broke down at a distance of 0/5 km from the railway station/kloc-and there were still 42 minutes to stop checking in. At this time, the only available means of transportation is another car. It is known that the car is limited to five people, including the driver, and the average speed of the car is 60km/h, and the average speed of people walking is 5 km/h. Two schemes are tried to design, and the calculation shows that these eight people can reach the railway station before stopping ticket checking.
12. When the school held the spring sports meeting, several students formed eight rectangular queues. If 120 people are added to the original queue, a square queue can be formed; If 120 people are reduced, a square queue can also be formed. Ask how many students were in the original rectangular queue.
13. It is known that both p and q are prime numbers. Let the quadratic equation x2-(8p- 10q) x+5pq = 0.
At least one positive integer root, find all prime pairs (p, q).
14. As shown in the figure, two circles with different radii intersect at point A and point B, and the line segment CD passes through point A and intersects at point C and point D respectively. Connect BC and BD, and let p, q and k be the midpoint of BC, BD and CD respectively. M and n are the midpoint of and respectively. Verification: