2. A vendor sold half a watermelon to the first customer and the remaining half to the second customer. In this way, he sold the remaining half watermelon to all customers. After seven customers bought it, the vendor ran out of watermelons.
3. Six classes in a school hold a chess match. In the competition, it is stipulated that three people in each class will participate in the single round robin of this class, and then the first person in each class will participate in the single round robin of the whole school on behalf of the class, so * * * needs to hold a competition to decide the ranking.
4. The teacher draws a number of axes on the blackboard. After taking the origin O, he uses a ring made of iron wire as a tool to draw the unit length 1 on the number axis with the diameter of the ring, and then straightens the ring into a line segment, and tangent point A from the origin O to the positive direction of the number axis, so the number represented by point A is.
Someone wants to find a house that can reach the company within 30 minutes' drive. It is known that cars can only travel at a speed of 30 kilometers per hour in places not more than 6 kilometers away from the company, and at a speed of 50 kilometers per hour in other places, so this person's house should be suitable in places not more than 6 kilometers away from the company.
6. 1996, due to the extraordinary flood, the villagers' Xiaojiang family suffered serious property losses. Because he was insured by an insurance company years ago and paid the insurance premium in 40 yuan for one year, the insurance company paid him a claim fee of 4,500 yuan afterwards, and told him that he would get a claim fee of13,500 yuan if he invested all the insurance premium at that time.
7. The overdraft of water source is worrying, so it is urgent to save water. In order to solve the problem of water waste for residents, Beijing will formulate the standard water consumption for residents, stipulate the monthly standard water consumption for buildings of a family of three, and increase the price for those exceeding the standard. Suppose that the water consumption that does not exceed the standard is 1.3 yuan per cubic meter, the water consumption that exceeds the standard is 2.9 yuan, the monthly water consumption for a family of three living in a building is 12 cubic meter, and the building water fee for a family of three in Beijing is 22 yuan.
8. To draw the curriculum on a rectangular ABCD paper with BC= 10cm long and AB=6cm wide, which can be placed in a pencil box, five vertical grid lines have been made, but three horizontal lines with equal distance will be made between A4D4 and BC. Please choose a method you are familiar with and draw it in the picture below (don't write, keep the drawing marks).
One two three four
1
2
three
Second, multiple-choice questions (6 points for each small question, ***48 points)
9. A passenger took 30 kilograms of luggage from Tianjin to Nanjing by plane. According to the civil aviation regulations, the passenger can carry up to 20 kilograms of luggage for free, and the overweight part can buy a baggage ticket at the rate of 0/.5% of the ticket fee per kilogram/kloc. If a passenger buys a baggage ticket for 120 yuan, his ticket price is ().
(A) 1000 yuan (B)800 yuan (C)600 yuan (D)400 yuan.
10. There are 196 apples in the basket. If you don't take it out at once, or take it one by one, and the number of apples taken out at a time is the same, and they are all just eaten, then the method of * * * is ().
(A)4 species (B)6 species (C)7 species (D)9 species.
1 1. An individual vendor sells two coats at the same time in a transaction, and each coat costs 135 yuan. If according to the cost, one party gains 25% and the other party loses 25%, then in this transaction, he ().
(a) Don't make a loss; (b) Make a loss in 9 yuan; (c) Make a loss 18 yuan; (d) Make a profit 18 yuan.
12. The three schools are marked as A, B and C respectively, and the stadium is marked as O, which is the intersection of the three corner lines of △ABC. There is a straight line connecting O, A, B and C. A long-distance running team starts from the stadium O and runs all the schools back to the O point, so the shortest distance of the running route is (called AC & gtBC & gtAB) ().
(A)OAB co(B)oac bo(C)oba co(D)ob Cao
13. School M is 30 northeast of Xiaoming's home N, so Xiaoming's route to school may be ().
14. After school at noon, Xiaoming rode home from school. As soon as he left school, he met Mr. Li. Teacher Li said, "You are going home against the wind today, and you will suffer." Xiao Ming said brightly, "It doesn't matter. It takes the same time to go home against the wind and back to school with the wind, and it is the same when there is no wind." Is that so? Please select ().
He's right, but save your strength when it's windy.
What he said is wrong. It takes less time when there is no wind.
(c) He's right, but it's hard when it's windy.
What he said is wrong. It takes less time when there is wind.
15. Iron plate A is an isosceles triangle with a vertex angle of 45 and a waist length of 12cm. Iron plate B is a right-angled trapezoid with two base sides of 4cm and 10cm, respectively, with an angle of 60. Now we turn them over at will and try to pass through a round hole with a diameter of 8.5cm, and the result is ().
(a) A can pass B but not B. (b) A can't pass B.
(c) Neither A nor B can pass. (d) Both A and B can pass.
16.a commercial building sells goods with a value of1000000 yuan at a 10% discount, and B commercial building also sells goods with a value of100000000 yuan by way of prize sales, and stipulates that anyone who buys more than1000 yuan will be given a lottery ticket, with five first prizes for every 10,000 lottery tickets, and each prize will be/kloc-0. Second prize 10, each 500 yuan; 20 third prizes, each 200 yuan; 40 fourth prizes, each at 100 yuan; The fifth prize is 1 0,000, and each prize is 10 yuan, so the two commercial buildings will each sell 1 10,000 yuan ().
(a) A earns 72,000 yuan more than B ..
(b) B earns 72,000 yuan more than A.
(c) A earns at least 72,000 yuan more than B..
(d) B earns at least 72000 yuan more than a.
Third, solve the problem (each small question 10, ***40)
17. The following information is provided by various departments of the factory:
Personnel department: there will be no more than 800 production workers next year, and each person's working hours will be calculated according to 2400 working hours per year;
Marketing department: forecast the product sales volume 10000~ 12000 next year;
Technology Department: This product needs 120 man-hours on average, and each product needs to be equipped with four parts;
Supply Department: At the end of this year, there are 6,000 major spare parts in stock, and 60,000 can be purchased next year.
Please decide: ① How many pieces can the factory produce at most next year? (2) In order to reduce the backlog, how many workers can be laid off at most to develop other new products?
18. The admission price of swimming pool is as follows: (unit: RMB)
One-way ticket 12 card year family ticket
Adult 3.50 35.00 165.00 8.00
Children 2.00 65438+8.00 82.50
Provide family tickets for families with children.
Xiaoming and his parents went swimming. They bought a family ticket. How much cheaper than paying the bill
Mr. Luo took two children to swim. How to buy tickets?
Xiaofeng (a child) used an annual card last year. He has been there 35 times. Is it cost-effective for him to buy an annual card?
(4) During the summer vacation (* * * has a holiday of 42 days), Xiao Ming, Xiao Gang and Xiao Qiang want to go swimming often during the holidays. Xiao Ming once a day, Xiao Gang once every two days and Xiao Qiang once every three days. If they go swimming together on the first day of the holiday, how to buy tickets is most beneficial to them? (It is stipulated that each person can only buy one kind of ticket)
19. Mom took Xiaohua to the supermarket and bought two kilograms of candy. It happened that the electronic scale in the supermarket was broken. The salesman took an old balance and a one-kilogram weight, but the two arms of the balance were not equal. After discussing with Xiaohua's mother, the salesman agreed to weigh in the following way: the salesman put the weight of 1 kg on the left plate, and then put the candy on the right plate to balance the two sides.
On the way home, Xiaohua asked her mother, "Is this enough?" Mom said, "Is that enough? Swap positions twice, more or less even. " Xiaohua thinks what her mother said seems reasonable, but she still doesn't quite understand. After returning home, she studied them with what she had learned, and the result surprised her: the actual weight of candy exceeded two kilograms. Afraid that she had miscalculated, she tried to weigh it with her spare spring scale, and the result was consistent with the calculation. She can't help but sigh: the mathematical problems in life must be thought with a mathematical mind. Please write down your solution.
20. The public security department received a report that a container on a cargo ship about to set sail contained prohibited items and gave an unknown number. According to the investigation, all the goods on board are packed in boxes, and all the boxes are numbered. These numbers are continuous natural numbers starting from 1 After analysis and judgment, this is the average of all container numbers except those containing prohibited items. Accordingly, the case handlers passed.
Four, open questions (this topic 14 points)
2 1. Observe life, write an application test related to real life, and answer with what you have learned in mathematics.
also
The 10th "Wuyang Cup" Junior Middle School Mathematics Competition (1998)
(Examination time: 90 minutes; 100)
A, multiple-choice questions (4 choices 1 type, choose 5 points, otherwise get 0 points, this big question is out of 50 points. )
1.( 1-2- 1/32)- 1( 1-2- 1/ 16)( 1+2- 1/ 16)( 1+2- 1/8)( 1+2- 1/4)( 1+2- 1/2)=__________.
A.( 1-2- 1/32)- 1 b .( 1-2- 1/32)
C.d .( 1-2- 1/32)- 1
2. There are at most _ _ _ _ _ acute angles in the internal angle of convex N-polygon.
A.5b.4c.3d None of the above is true.
3. The positive integer coefficient quadratic equation ax2+bx+c=0 has a rational root, so in A, B, C _ _ _ _ _ _.
A. there must be at least one even number B. There is at least one prime number
C. there must be at least one odd d. There is at least one composite number.
4. The heights of the sides with lengths A, B and C in the triangle are HA, HB and HC respectively. If a ≤ ha and b ≤ HB, the triangle is _ _ _ _.
A. isosceles non-right triangle B. isosceles right triangle;
C. Right-angled non-isosceles triangle D. None of the above conclusions are correct
5. The integer solutions of the equation have _ _ _ _ _ _ _ groups.
A. Numerous B.4 C.2 D.0
6. Let A and B be natural numbers, a+b=33, and the least common multiple [a, b] = 90, then the greatest common divisor (A, B) = _ _ _ _ _.
a . 1 b . 3c . 1 1d . 9
7. If the sum of continuous positive integers A, B, C, D and E is a complete cubic number and the sum of B, C and D is a complete square number, then the minimum value of C is _ _.
100
8. the distance between two points a and b on the plane is a+b, where a, B >；; 0 is a constant value, so there are _ _ _ _ _ straight lines on the plane * *, which makes AB straight here.
The line distances are a and b respectively.
A. Infinitely multiple B.3 C.2 D. 1
9. A triangle with three sides A, B and C is suitable, then this triangle is _ _ _ _ _ _ _ _.
A. isosceles triangle with waist
C. equilateral triangle D. none of the above answers are correct
10.(x, y) is called a number pair, where x and y are arbitrary real numbers. The addition and multiplication operations of number pairs are defined as follows:
(x 1，y 1)+(x2，y2)=(x 1+x2，y 1+y2)
(x 1，y 1)？ (x2, y2) = (x 1x2-y 1y2, x 1y2+y 1x2), then _ _ _ _ _ _ _ does not hold.
A. Multiplicative commutative law: (x 1, y 1)? (x2，y2)=(x2，y2)？ (x 1，y 1)
B. Multiplicative associative law: (x 1, y 1)? (x2，y2)？ (x3，y3)=(x 1，y 1)？ [(x2，y2)，(x3，y3)]
C. the distribution law of multiplication to addition: (x, y)? [(x 1，y 1)+(x2，y2)]=[(x，y)？ (x 1，y 1))+((x，y)？ (x2，y2)]
The distribution law of D addition to multiplication: (x, y)+[(x 1, y 1)? (x2，y2)]=[(x，y)+(x 1，y 1)]？ [(x，y)+(x2，y2)]
2. Fill in the blanks (5 points for each item, 0 points for not filling, more filling, less filling, wrong filling, only part filling.
The score of the big question is 50 points. )
1. Let 0
2. Let |a|= 1, where b is an integer and the equation ax2-2x-b+5=0 has two negative real roots, then b = _ _ _ _ _ _ _ _ _ _ _ _
3. Let the real numbers X, Y, z y and Z apply to 9x3=8y3=7z3, then = _ _ _ _ _ _ _ _ _ _ _,
.
4. Let the real numbers x, y, z Y and z satisfy x+y+z=4 (), then X = _ _ _ _ _ _, y = _ _ _ _ _, and Z = _ _ _ _ _ _.
5. It is known that the three sides A, B and C of a triangle satisfy 9≥a≥8≥b≥4≥c≥3, then the maximum area of the triangle is = _ _ _ _ _ _ _.
6. Use the integer part of table X in [x], that is, the largest integer not greater than x, for example, [3.4] = 3, [-3.4] =-4. equation
All rational roots of 9x2-8[x]= 1 are _ _ _ _ _ _ _.
7. Let the real numbers x and y satisfy x2-2x|y|+y2-6x-4|y|+27=0, then the value range of y is _ _ _ _ _ _ _.
8. As shown in figure 1, the three line segments Mn, Ij and EF passing through point P are parallel to the three sides of Δ ABC, and Δ ABC is divided into three triangles and three.
A parallelogram, the areas of three of which are marked in the figure: S δ IMP = 9, S□ BFPM = 42, S□ CNPJ = 70, then
sδABC = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .
9. All the different real roots of the equation (x3-3x2+x-2) (x3-x2-4x+7)+6x2-15x+18 = 0 are _ _ _ _ _ _ _.
10. As shown in Figure 2, it is known that three straight lines aa', bb' and cc' passing through Q divide ABC into six small triangles.
Δ δaqb' =δbqa' = 4, cqa' = 3, then X = Δ AQC' = _ _ _ _ _, Y = Δ BQC' = _ _ _ _ _ _ _,
z = sδCQB .
Answers to the 10th "Wuyang Cup" Junior Middle School Mathematics Competition
1. Multiple choice questions: (5× 10=50 points)
1. An original formula = (1-2-1/32)-1(1-2-1/6) (/kloc-)
- 1( 1+2- 1/8)( 1+2- 1/8)( 1+2- 1/4)( 1+2- 1/2)=…=( 1-2- 1/32)- 1 ( 1-2- 1)= ( 1-2- 1/32)- 1。
2.c Because the sum of the external angles of the convex N-polygon is 360, the internal angles are no more than three obtuse angles and no more than three acute angles. In addition, an equilateral triangle has three acute angles.
3. Factorial discriminant δ = B2-4ac = m2, where m is an integer. If a, b and c are all odd numbers, so are ac and m. Let b=2n+ 1,
Ac=2k+ 1, then δ = 8 []-3, which contradicts that the square of odd m is a multiple of 8 plus 1. So answer a holds. In addition, the equations 2x2+4x+2 = 0, 4x2+8x+4 = 0, 3x2+5x+2 = 0 all have rational roots, and b, c, d, c and d can be negated. ..
4.c As shown in Figure 3, it is easy to see that ha≤b and HB ≤ a, therefore, a≤ha≤b≤hb≤a, so a=ha=b=hb. From a=hb, we know that ∠ C = 90.
5. The simplest root of B transformation is 3, and obviously it should be the same root. manufacture
, a and b are nonnegative rational numbers. If a is not an integer, it is easy to see that x is not an integer, which leads to contradictions.
. So a is a non-negative integer. Similarly, b is a non-negative integer. (a, b) = (0,3), (1, 2), (3,0), and there are only four groups of solutions.
6.b fortification (a, b)=x, then x can be divisible by a, b, a+b and [a, b], x is the common divisor of 33 and 90, and x= 1 or 3. If x= 1, A cannot be divisible by 3, otherwise B can also be divisible by 3, x ≠1; Therefore, neither A nor B can be divisible by 3, nor can [a, b] be divisible by 3, which creates a contradiction. So x=3, then A = 15, and B = 18 suit the meaning of the question.
7.D Because A+B+C+D+E = 5c and B+C+D = 3c, 5c = n3, 3c = m2, n and m are positive integers. So n=5p, m=3q, p and q are all integers, and c=52? P3 = 3q2, and the minimum value of c is 52? 33=675.
8.b as shown in figure 4, a circle with a and b as the center and a and b as the radius respectively. Let the straight line L be the tangent of ⊙A, because the distance from A to L is A; It is also the tangent of ⊙ B. Because the distance from B to L is B, it is the common tangent of two circles, ***3 (2 external tangents, 1 internal common tangent).
9. one,
A(b+c-a)=bc, (a-b)(c-a)=0, a=b or A = C.
10. It is easy to see that the multiplicative commutative law holds. By ((x 1，y 1)？ (x2，y2))？ (x3y3)=(x 1x2-y 1y2，x 1y2+y 1x2)？ (x3，y3)=(x 1x2x 3-y 1y2x 3-x 1y2y 3-y 1x2y 3-y 1y2x 3+x 1y2x 3+y 1x2x 3)=(x 1，y 1)？ (x2x3-y2y3，x2y3+y2x3)=(x 1，y 1)？ [(x2，y2)？ (x3y3)], knowing that the law of multiplicative association holds. By (x，y)？
[(x 1，y 1)+(x2，y2)]=(x，y)？ (x 1+x2，y 1+y2)=[x(x 1+x2)-y(y 1+y2)，x(y 1+y2)+y(x 1+x2)]
=(xx 1-yy 1，xy 1+yx 1)+(xx2-yy2，xy2+yx2)=[(x，y)？ (x 1，y 1)]+[(x，y)？ (x2, y2)], can multiply.
The law of additive distribution holds. By (1 0)+[( 1 0)? ( 1,0)]=( 1,0)+( 1,0)=(2,0)≠(2,0)? (2,0)=
[( 1,0)+( 1,0)? ((1, 0)+( 1, 0))], we know that the distribution law of addition and multiplication is not valid. (Note: If (x, y) is regarded as a complex number x+yi, then the addition and multiplication of number pairs defined in this question are the addition and multiplication operations of complex numbers, so it is easy to know that A, B and C are true and D is not).
Two. Fill in the blanks (5× 10=50 points)
1.- 1
2.6
Because the sum of two negative roots is =, a=- 1, b>5. Equation discriminant =4+4(5-b)≥0, so b≤6, and b=6 can be known from the fact that b is an integer.
3.
Let 9x3=8y3=7z3=k3, then
therefore
4.9,8,7
Derived from the original equation
,
Therefore, x = 9, y = 8 and z = 7.
5. 16
Let the diagonal of side A be A, then the triangle area
.
6.
Let m and n be integers, m >；; 0, (m, n)= 1, (that is, m, n is coprime), so because 9x2 is an integer, there must be m2|9, so m= 1 or 3. If m= 1, then x = n, 9N2-8n = 1, and n =1; If m=3, then x=. Let n = 3k+θ (1 ≤ θ≤ 2) and k be an integer, then n2-8 = 1.
That is, (3k+θ) 2-8k = 1, 9k2+(6θ-8) k+(θ 2- 1) = 0, and the discriminant δ = (6θ-8) 2-36 (θ 2-1) =
0, only θ = 1, δ = 4, (integer values are rounded off).
So the original equation has two rational roots:.
7.y≥ 1.8 or y≤- 1.8.
The original equation x2-(2|y|+6)x+(|y2|-4|y|+27)=0, and the discriminant δ = (2 | y |+6) 2-4 (| y2 |-4 | y |+27) ≥ 0, that is.
40 | y |-72 ≥ 0, | y | ≥ 1.8, y ≥ 1.8 or y≤- 1.8.
8.225
As shown in fig. 5, if s δ pfj = x, the similarity ratio is IP:PJ:IJ and the area ratio is IP2:PJ2:IJ2, so
X=49. Similarly, let sδEPN = y, then y=25. Then from Δ pfj ∽ ABC, the similarity ratio is FJ:PN:MP:BC, and the area ratio is FJ2:PN2:MP2:BC2.
9.
Suppose the original equation becomes (A-B)(A+B)+6B-9=0, that is
A2-B2+6B-9 = 0, A2-(B2-6B+9) = 0, A2-(B-3) 2 = 0, (A+B-3) (A-B+3) = 0, A+B-3 = 0 or A-B+3=0. if
A+B-3=0, that is, x3-x2-4x+4 = 0, (x2-4) (x- 1) = 0, x2-4 = 0 or x- 1 = 0, x = 2 or1; If A-B+3=0, that is
X3-3x2+x+ 1 = 0, (x- 1) (x2-2x- 1) = 0, x- 1 = 0 or x2-2x- 1=0, x =
(Note: The equation of degree 6 in this question has six roots in the complex number field, all of which are real roots, namely 1 (double roots), 2, 1. )
10., , 3
As shown in Figure 6, sδ AQB: aδ AQC = the distance from b to AA': the distance from c to AA' = sδ bqa: sδ cqa, that is.
(Note: If the area of a small triangle in the diagram is known, the areas of the other three can be deduced, but in some cases, it can be deduced.
The guiding process is complicated, so readers may wish to try it themselves.