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The 16th "Hope Cup" National Mathematics Invitational Tournament

The second exam of the second day of junior high school

April 2005, 17, 8: 30 am to 10: 30 am.

1. Multiple choice questions (5 points for each question, ***50 points) Only one of the following four options is correct. Please fill in the English letters indicating the correct answer in the brackets after each question.

1, if both a and b are positive integers, and m = ab (a+b), then ()

A.m. must be an odd number. B.M must be an even number.

C m is even only if a and b are even. D. M is even only when A and B are even and the other is odd.

2, set,, equal to ()

A.B.- C.-3 D.3

3. Given that A, B and C are positive integers, and A and B are prime numbers, then the value of ().

a . 14 b . 13 c . 12d . 1 1

(English-Chinese dictionary positive integer: positive integer. Prime number: prime number) _

4. Buy 7 pencils, 3 exercise books, 1 ballpoint pen * * * needs 3 yuan; 4 yuan bought 10 pencils, 4 exercise books, 1 ballpoint pens, and then bought * * * ()1pencils, 5 exercise books and 2 ballpoint pens.

A.4.5 yuan B. 5 yuan C. 6 yuan D. 6.5 yuan

5. The computer converts the information into binary numbers for processing. Binary numbers are "every binary number is one". For example, the binary number (11) 2 is converted into a decimal number, which is1× 23+1× 22+0× 265438.

22004+ 1

6. It is known that the ratio of the three internal angles of △ABC is m: (m+ 1): (m+2), where m is a positive integer greater than 1, then △ABC is ().

A. acute triangle B. right triangle C. obtuse triangle D. isosceles triangle

7. It is known that the ratio of three heights of △ABC is 3∶4∶5, and the lengths of three sides are integers, so the side length of △ABC may be ().

10 b . 12 c . 14d . 16

8. It is known that a two-digit number can be divisible by 3, the product of its ten digits and its one digit is equal to its one digit, and its one digit of any power is equal to its one digit. This two-digit number * * * has ()

A. 1 B.3 C.4 D.5

9.2005 boxes have 40 10 balls in a row, in which the A ball is placed in the leftmost box and the B ball is placed in the rightmost box. If there are 24 balls in any adjacent 65,438+02 boxes, then ().

a . a = b = 2b . a = b= 1 c . a = 1，b=2 D.a=2，b = 1

10, a known integer, satisfies ≤

A.2b. 14C.2 or 14D

Fill in the blanks (5 points for each small question, ***50 points. Including two empty questions, the first 3 points and the second 2 points. )

1 1. If |a|=3 and |b|=5, the absolute value of | a+b |-| a-b | is equal to.

12, if known, then =.

13. A car travels from A to B. If it travels one kilometer per minute, it will arrive at 1 1. If you drive one kilometer per minute, it will be 1 1: 20, and the distance b will be10km; if you change the departure time and drive one kilometer per minute, you will arrive at 1 1. If you drive one kilometer per minute, it will exceed 30 kilometers at B 1 1: 20. The distance between a and b is kilometers.

14. If it is a six-digit number, where A, B and C are three different numbers and none of them are equal to 0, 1, 2,3, and m is a multiple of 7, then the minimum value of m is.

15, decomposition factor:.

16. If the inner angle of a convex n(n is a natural number greater than 3) polygon has at most m acute angles and at least m acute angles, then m =；;

m=。

17, as shown in figure 1, the right-angle side length of isosceles Rt△ABC is 32, and the vertical line of BC, which is the hypotenuse from right-angle vertex A, intersects with D 1, then intersects with D 1D2⊥AC at D2, and then intersects with D2D3⊥BC at D2.

; d 1 D2+d3d 4+d5d 6+d7d 8+d9d 10 =。

18, as shown in Figure 2 and Figure 3 (in which EF ‖ BC) can be obtained by folding triangular paper ABC along EF. It is known that the ratio of the area of Figure 3 to the area of the original triangle is 3: 4, and the area of the shaded part is 8 square centimeters, so the area of the original triangle is square centimeters.

19, as shown in Figure 4, in △ABC, BC∶AC=3∶5, quadrilateral BDEC and ACFG are both squares. Given that the area ratio of △ABC to square BDEC is 3∶5, the area ratio of △CEF to the whole figure is equal to.

20. If a positive integer n has the following properties: one eighth of n is a square number, one ninth of n is a cubic number, and one fifth of n is a quintic number, then n is called a "hope number" and the minimum hope number is.

Third, the answer (each question 10, ***30) requirements: write out the calculation process.

2 1, Figure 5 is a circular runway with a length of 400 meters, where A and B are two points on the symmetry axis of the runway.

There is a 50-meter-long straight passage between A and B ... Both parties start from point A at the same time, and Party A presses.

Run along the runway at a speed of v 1 counterclockwise. When you run to point B, continue along the runway and press the B key.

Run clockwise along the runway at speed v2, and run back to point A along the straight line when you run to point B. ..

Suppose two people run long enough. Q:

(1) If v 1∶v2=3∶2, how many miles did A run before the first meeting at point A?

⑵ If v 1∶v2=5∶6, how many miles did B run before the first meeting at point B?

22.( 1) If A is a prime number less than 20 and can be converted into a cyclic decimal, what is the value of A?

⑵ If a is a composite number less than 20, which can be reduced to a cyclic decimal, what is the value of a?

23. As shown in Figure 6, the side length of the regular triangle ABC is A, D is the midpoint of BC, and P is the point on the side of AC. Add PB and PD to get △PBD. Q:

(1) When point P moves to the midpoint of AC, the circumference of δ △ PBD;

⑵ Minimum value of delta perimeter ⑵△PBD.

The 16th "Hope Cup" National Mathematics Invitational Tournament

Reference answers and grading standards

The second exam of the second day of junior high school

First, multiple-choice questions (5 points for each small question)

The title is 1 23455 6789 10.

Answer B C D B C A B C A A

Fill in the blanks (5 points for each small question, including two empty small questions, 3 points for the front and 2 points for the back)

The title is11213141516171819 20.

Answer 6 54 4683213; 0 3 1; 3 1 16 2 15 320 5 12

Third, answer questions.

2 1, (1) Suppose that after running n laps, the two meet at point A for the first time, and then suppose that the speeds of A and B are v 1=3m and v2=2m respectively.

When they met in A, the running time was (2 minutes).

Yes (3 points)

Because B runs back to point A, it should be an integer multiple of 250, so the minimum value of n is 15, (4 points).

So after running 15 laps in A, the two met at A for the first time (5 points).

(2) Let B run meters. When A runs meters, the two meet at point B for the first time. Let the speeds of A and B be v 1=5m and v2=6m respectively, which can be obtained from the meaning of the question, that is, (7 points).

So, that is (p, q are positive integers).

So the minimum values of p and q are q=2, p=4, (8 points).

At this point, the running distance of B is 250× 4+200 = 1200 (m). (9 points)

So after B ran 1200m, they met at B for the first time. (10)

22. The prime numbers with (1) less than 20 are 2, 3, 5, 7, 1 1, 13, 17, 19 (2 points).

Except for 2 and 5, the reciprocal of other numbers can be converted into a cyclic decimal (4 points).

So a can be taken as: 3, 5, 7, 1 1, 13, 17, 19. (5 points)

(2) According to (1), as long as the factor of the composite number A contains prime numbers other than 2 or 5, then the reciprocal of the number can become a cyclic decimal, (8 points).

So a can be: 6,9, 12, 14, 15, 18. (10)

23.( 1) As shown in figure 1, when point P moves to the midpoint of AC, BP⊥AC, DP‖AB, (2 points).

So,,,, (4 points)

That is, the circumference of △ABC is BP+DP+BD =. (5 points)

(2) As shown in Figure 2, if point B is the symmetrical point E about AC, and EP, EB, ed and EC are connected, then Pb+PD = PE+PD, so the length of ED is the minimum value of Pb+PD, that is, when point P moves to the intersection point G between ED and AC, the circumference of △ PB+PD is the minimum. (7 points)

Let point d be DF⊥BE and vertical foot be f, because BC=a, so.

Because DBF = 30,,,

,。 (9 points)

So the minimum value of the circumference of △△PBD is. (10)