First, multiple-choice questions (65438+ 0 points for each question, ***5 points)
One and only one of the four conclusions A, B, C and D given in each question below is correct. Please fill in the English letter code of the conclusion you think is correct in brackets.
1. The total output value of a factory last year increased by% compared with the previous year, so the percentage of the previous year was less than that of last year (a).
Average grade%. B.( 1+a)%。 C. D。
2. Cup A contains 2m ml of red ink, and Cup B contains m ml of blue ink. Pour 1 ml from Cup A into Cup B,
0 < a < m, after stirring, pour one milliliter from the second cup into the first cup, and then (a)
There is less blue ink in Cup A than red ink in Cup B. 。
B there is more blue ink mixed in cup a than red ink mixed in cup B.
C the blue ink mixed in a cup is the same as the red ink mixed in b cup.
D The blue ink mixed in the A cup has nothing to do with the red ink mixed in the B cup.
3. The known number x= 100, then (a)
A.x is a complete square number. B. (x-50) is a complete square number.
C.(x-25) is a complete square number. D. (x+50) is a complete square number.
4. Observe the number axis in the figure 1: the letters A, B and C are used to indicate the numbers corresponding to points A, B and C in turn, and the size relationship is (C).
A.； B. & lt& lt； C. & lt& lt； D. & lt& lt。
5.X = 9 and Y =-4 are a set of integer solutions of binary quadratic equation 2x2+5.x=9+3y2 = 30, and different integer solutions of this equation * * * are ().
Group A.2, group B.6, group c12 and group d16.
Fill in the blanks (65438+ 0 point for each question, ***5 points)
1. The root of the equation |1990x-1990 | =1990 is _ _ _ _.
2. For any rational number x, y, define an operation *, and stipulate that x * y = ax+by-cxy, where a, b and c represent known numbers, and the right side of the equation is the usual addition, subtraction, multiplication and division operation. We also know that 1 * 2 = 3, 2 * 3 = 4, and x * m =
3. The new dormitory manager has got 20 keys and can open the doors of 20 rooms. He knows that each key can only open one of the doors, but he doesn't know which door each key can open. Now, to open all the 20 closed rooms, he has to try to open them _ _ _ _ times at most.
4. When m=______, binary quadratic sextuples 6x2+mxy-4y2-x+17y-15 can be decomposed into the product of two binary linear trinomials about x and y. 。
5. Sum of squares of three consecutive natural numbers (fill in "Yes" or "No" or "Possibility") _ _ _ _ _ Square of natural numbers.
Three. Solve problems (write down the process and final result of reasoning and operation. 5 points for each question, *** 15 points)
1. Two cars start from the same place at the same time and drive straight in the same direction at the same speed. Each car can only hold 24 barrels of gasoline at most, and no other oil can be used on the way. A barrel of oil can move a car 60 kilometers forward, and both cars must return to the starting point, but they can return at different times, and the two cars can borrow oil from each other. In order to keep one car as far away from the starting point as possible, how many kilometers should another car return from the starting point? How many kilometers has the car farthest from the starting point traveled?
2. As shown in Figure 2, four circles with the same size are drawn on the paper, the centers of which are A, B, C and D respectively. The straight line m passes through a and b, the straight line n passes through c and d, and the area of the circle is represented by S. If the total area covered by four circles on paper is 5 (S- 1), the area covered by the circle between straight lines m and n is 8, and the shadow area is 8.
3. Find the positive integer solution of the equation.
Junior high school mathematics competition counseling
2. Let a, b and c be real numbers, and | a |+a = 0, | ab | = ab, | c |-c = 0, and find the value of the algebraic expression | b |-| a+b |-c-b |+| a-c |.
3. If m < 0, n > 0, | m |
4. Let (3x-1) 7 = A7X7+A6X6+…+A1X1+A0, and try to find the value of A0+A2+A4+A6.
6. Solve equation 2 | x+ 1 |+x-3 | = 6.
8. Solve the inequality || x+3 |-x- 1 || > 2.
10.X, y and z are non-negative real numbers, which satisfy the following conditions: X+3Y+2Z = 3, 3x+3Y+Z = 4. Find the value and minimum value of U = 3x-2Y+4Z.
1 1. Find the quotient and remainder of x4-2x3+x2+2x- 1 divided by x2+x+ 1.
13. As shown in figure 1-89, AOB is a straight line, OC and OE are bisectors of ∠AOD and ∠DOB, respectively, and ∠ COD = 55. Find the complementary angle of ∠DOE.
14. As shown in figure 1-90, the bisected line ∠ABC, ∠ CBF = ∠ CFB = 55, ∠ EDF = 70. Verification: BC ∠ AE.
15. As shown in figure 1-9 1. In △ABC, EF⊥AB, CD⊥AB, ∠ CDG = ∠ BEF. Verification: ∠ AGD = ∠ ACB.
17. As shown in figure 1-93. In △ABC, e is the midpoint of AC, d is on BC, BD∶DC= 1∶2, and AD and BE intersect at F. Find the ratio of the area of △BDF to the area of quadrilateral FDCE.
18. As shown in figure 1-94, two opposite sides of quadrilateral ABCD extend and intersect at K and L, and diagonal AC‖KL and BD extension lines intersect at F. Verification: KF = FL.
19. Can the sum of the number obtained by arbitrarily changing the order of a three-digit number and the original number be 999? Explain why.
20. There is a piece of paper with 8 rows and 8 columns, in which 32 squares are randomly painted black and the remaining 32 squares are painted white. Next, the color grid paper is operated, and each operation changes the color of each square in any horizontal or vertical column at the same time. Can you finally get a grid paper with only one black square?
23. There are several stools and chairs in the room. Each stool has three legs and each chair has four legs. When they are all seated, * * * has 43 legs (including everyone's two legs). How many people are there in the room?
24. Find the integer solution of the indefinite equation 49x-56y+ 14z=35.
25. Eight men and eight women dance in groups.
(1) If there are two substations, male and female;
(2) If men and women stand in two rows, in no particular order, only consider how men and women form partners. How many different situations are there?
26. 1, 2, 3, 4, 5, how many numbers are greater than 34 152?
27.A train is 92 meters long and B train is 84 meters long. If they travel in the opposite direction, they will miss each other after 1.5 seconds. If they travel in the same direction, they will miss each other in six seconds. Find the speed of two trains.
28. The two production teams of Party A and Party B grow the same vegetables. Four days later, Team A will finish the rest alone. It will take two more days. If Party A finishes all the tasks by itself three days faster than Party B, how many days does it take to ask Party A to finish it by itself?
29. A ship departs from a port 240 nautical miles apart, and its speed decreases by 10 nautical miles per hour 48 nautical miles before it reaches its destination. The total time it takes after its arrival is equal to the time it takes for the whole voyage when the original speed decreases by 4 nautical miles per hour, so that we can find the original speed. 16.
30. Last year, two workshops A and B of a factory planned to complete tax profits of 7.5 million yuan. As a result, workshop A exceeded the plan 15%, workshop B exceeded the plan 10%, and two workshops * * * completed tax profits of 8.45 million yuan. How many million yuan of tax profits did these two workshops complete last year?
3 1. It is known that the sum of the original prices of two commodities is 150 yuan. Due to market changes, the price of the first commodity decreased by 10%, and the price of the second commodity increased by 20%. After the price adjustment, the sum of the unit prices of the first and second commodities decreases by 1%. What are the original unit prices of the first and second commodities respectively?
Xiaohong bought two children's toothbrushes and three toothpastes in the shop last summer vacation, and just ran out of money with her. It is known that each toothpaste is more than each toothbrush 1 yuan. This summer, she took the same money to the store and bought the same toothbrush and toothpaste. Because each toothbrush rose to 1.68 yuan this year and the price of each toothpaste rose by 30%, Xiaohong had to buy two toothbrushes and two toothpastes, and she got back 40 cents. How much is each toothpaste?
33. A shopping mall sells goods with a unit price of 8 yuan at 12 yuan, and it can sell 400 pieces every day. According to experience, if each piece is sold less 1 yuan, you can sell more than 200 pieces every day. How much do I have to reduce each piece to get the benefit?
34. The distance from Town A to Town B is 28 kilometers. Today, A rode his bike at a speed of 0.4km/min, and set out from Town A to Town B. After 25 minutes, B rode his bike to catch up with A at a speed of 0.6km/min. How many minutes does it take to catch up with A?
35. There are three kinds of alloys: the first contains 60% copper and 40% manganese; The second type contains manganese 10% and nickel 90%; The third alloy contains 20% copper, 50% manganese and 30% nickel. Now a new alloy containing 45% nickel is composed of these three alloys, and its weight is 1 kg.
(1) Try to express the weight of the second alloy by the weight of the first alloy in the new alloy;
(2) Find out the weight range of the second alloy in the new alloy;
(3) Find out the weight range of manganese in the new alloy.
| =-A, so a≤0, b≤0 because | AB | = AB, C ≥ 0 because | C | = C So A+B ≤ 0, c-b≥0, A-C ≤ 0. therefore
The original formula =-b+(a+b)-(c-b)-(a-c) = b.
3. Because m < 0, n > 0, so | m | =-m, | n | = n So | m | 0. When x+m≥0, | x+m | = x+m; When x-n≤0, | x-n | = n-X. Therefore, when -m≤x≤n,
|x+m|+|x-n|=x+m-x+n=m+n。
4. Let x= 1 and x=- 1 respectively, and substitute them into the known equation to obtain.
a0+a2+a4+a6=-8 128。
10.y and z can be obtained by known.
Because y and z are non-negative real numbers, there are
u=3x-2y+4z
1 1. So the quotient is x2-3x+3 and the remainder is 2x-4.
12. The route of the small cylinder is a broken line consisting of three line segments (as shown in Figure 1-97).
We use the method of "symmetry" to transform the line of this broken line of a small cylinder into a "connecting line" (a line segment) between two points. The symmetry point of the north hillside of Shijiacun (the hillside is regarded as a straight line) is a'; The symmetry point of village B on the south hillside is B', which connects A' B'. Let the intersection points of the line segment connected by A' B' and the north and south slopes be A →A→B→ B respectively, then the route from A to A to B to B is the best choice (that is, the shortest route).
Obviously, the length of route A →A→B→ B is exactly equal to the length of line segment A ′ B ′. Using the above symmetry method, any other route from village A to village B can be transformed into a broken line connecting A' and B'. They are all longer than the line segment A'B'. So the distance from A to A → B → B is the shortest.
13. As shown in figure 1-98. Because OC and OE are bisectors of ∠AOD and ∠DOB, respectively, and ∠ AOD+∠ DOB = ∠ AOB = 180, so
Because ∠ COD = 55, ∠ DOE = 90-55 = 35.
Therefore, the complementary angle of ∠DOE is 180-35 = 145.
14. As shown in figure 1-99. Because Be divides ABC equally, so
∠CBF=∠ABF，
Because ∠CBF=∠CFB and ∠ ABF = ∠ CFB.
So AB‖CD (internal dislocation angles are equal and two straight lines are parallel).
∠ABC is divided into∠ CBF = 55 and BE, so∠ ABC = 2× 55 =110.
AB‖CD is known from Shanghai Stock Exchange, so ∠ EDF = ∠ A = 70, ②.
BC‖AE is known from ① and ②.
15. As shown in figure 1- 100. EF ⊥ AB，CD⊥AB，so ∠ EFB = ∠ CDB = 90，
So EF‖CD (isosceles angle is equal, two straight lines are parallel). So ∠BEF=∠BCD (two straight lines are parallel and the isosceles angles are equal).
① It is known that ∠ CDG = ∠ BEF. ② Known ∠ BCD = ∠ CDG.
So BC‖DG (internal dislocation angles are equal and two straight lines are parallel).
So ∠AGD=∠ACB (two straight lines are parallel and have the same angle).
16. in △BCD,
∠ DBC+∠ C = 90 (because ∠ BDC = 90), ① In △ABC, ∠B=∠C, so
∠A+∠B+∠C=∠A+2∠C= 180，
So when it comes to ①, ②
17. As shown in figure1-1kloc-0/,let the midpoint of DC be G and connect with GE. In △ADC, G and E are the midpoint of CD and CA, respectively. So GE‖AD, that is, in △BEG, DF ‖ GE.
And s△EFD = s△BFG- Seyford = 4s△BFD- Seyford,
So s △ efgd = 3s △ BFD.
Let S△BFD=x, then SEFDG=3x ... In △BCE, G is the bisector of BC, so S△CEG=S△BCEE.
So sefdc = 3x+2x = 5x,
So s △ BFD ∶ SEFDC = 1 ∶ 5.
18. As shown in figure 1- 102.
Since AC‖KL is known, S△ACK=S△ACL, so
That is, KF = FL. +B 1 = 9, a+a 1=9, so A+B+C+A 1+B 1 = 9+9, which is 2(a+B+C).
20. The answer is no. Let a horizontal or vertical column contain k black squares and 8k white squares, where 0 ≤ k ≤ 8. When the colors of squares change, 8k black squares and k white squares are obtained. Therefore, after one operation, the number of black squares "increases" (8-k)-k=8-2k, that is, one is added.
2 1. The prime number p greater than 3 can only be in the form of 6k+ 1 and 6k+5. If p = 6k+ 1 (k ≥ 1), then p+2 = 3 (2k+ 1) is not a prime number, so p
22. According to the condition N = 75k = 3× 52× K, in order to make n as small as possible, n=2α3β5γ(β≥ 1, γ≥2) and (α+ 1) (β+ 1) (γ
So α+ 1, β+ 1 and γ+ 1 are all odd numbers, and α, β and γ are even numbers, so γ = 2. At this time, (α+ 1) (β+ 1) = 25.
So (α, β) = (0,24), or (α, β) = (4,4), that is, n=20? 324? Fifty two
23. There are X stools and Y chairs, which means 3x+4y+2 (x+y) = 43.
That is 5x+6y = 43.
So x=5 and y=3 are all nonnegative integer solutions. So there are eight people in the room.
24. The original equation can be simplified as follows
7x-8y+2z=5。
Let 7x-8y=t, t+2z = 5. It is easy to see that x=7t and y=6t are sets of integer solutions of 7x-8y = t, so all its integer solutions are.
And t= 1 and z=2 are a set of integer solutions of t+2z = 5. All its integer solutions are
Substituting the expression of T into the expressions of X and Y, we get all integer solutions of the original equation as follows.
25.( 1) There are 8 selection methods for the first position and only 7 selection methods for the second position ... According to the principle of multiplication, the male and female are 8× 7× 6× 5× 4× 3× 2× 1 = 40320.
There are two different arrangements. There is a relative positional relationship between the two columns, so there are different situations of 2×403202 * *.
(2) Consider the pairing problem one by one.
There are 8 possible situations when paired with male A, and 7 different situations when paired with male B. …, the two columns are interchangeable, so * * * has 2×8×7×6×5×4×3×2× 1=80640 different situations.
26. There are 4×3×2× 1=24 (pieces) with 5 digits.
There are 4×3×2× 1=24 digits.
The number of thousands is 3, the number of thousands can only be 5 or 4, the number of thousands is 3×2× 1=6, and the number of thousands is 4 as follows:
342 15,3425 1,345 12,3452 1.
So the total * * * is 24+24+6+4 = 58.
This number is greater than 34 152.
27. The distance missed by two cars is the sum of the lengths, that is, 92+84 = 176 (meters).
Let the speed of train A be x m/s, the speed of train B be y m/s, and the speed of two cars traveling in opposite directions be x+y; The speed of two cars traveling in the same direction is x-y.
Get a solution
X=9 (days), x+3 = 12 (days).
X= 16 (nautical mile/hour).
Upon inspection, x= 16 knots/hour is the original speed.
30. Last year, Workshop A and Workshop B planned to complete tax profits of RMB X million and RMB Y million respectively.
Get a solution
Therefore, Workshop A exceeded the tax benefits.
B workshop overfulfilled taxes and profits.
Therefore, A * * * completed the tax benefit of 400+60=460 (ten thousand yuan), and B * * * completed the tax benefit of 350+35=385 (ten thousand yuan).
3 1. Assume that the original unit prices of the two commodities are X yuan and Y yuan respectively, which can be obtained according to the meaning of the question.
By owning
0.9x+ 1.2y= 148.5，③
Get X= 150-y from ① and substitute it into ③.
0.9( 150-y)+ 1.2y = 148。 5,
The result of the solution is y=45 (yuan), so x= 105 (yuan).
32. Suppose each toothbrush cost X yuan last year, depending on the meaning of the question.
2× 1.68+2(x+ 1)( 1+30%)=[2x+3(x+ 1)]-0.4，
That is 2×1.68+2×1.3+2×1.3x = 5x+2.6,
That is 2.4x = 2.4x=2× 1.68,
So x= 1.4 (yuan).
If y is the price of each toothpaste last year, then y = 1.4+ 1 = 2.4 (yuan).
33. The original profit was 4×400= 1600 yuan. If the price of each piece is reduced by X yuan, then each piece can still make a profit of (4-x) yuan, of which 0 < x < 4. Since you can sell (400+200x) pieces every day after the price reduction, if you set the daily profit as Y yuan, then
y=(4-x)(400+200x)
=200(4-x)(2+x)
=200(8+2x-x2)
=-200(x2-2x+ 1)+200+ 1600
=-200(x- 1)2+ 1800。
Therefore, when x= 1, y= 1800 (yuan). That is, when each piece is reduced by 1 yuan, the profit is 1800 yuan. At this time, I sold 200 yuan more than before, so 200 yuan is more profitable.
34. If it takes X minutes for Party B to catch up with Party A, then Party A has to walk (25+x) minutes to the place where it is caught up, so the walking distances of Party A and Party B are 0.4 (25+X) km and 0.6xkm respectively. Because they walk the same distance, so
0.4(25+x)=0.6x，
X=50 minutes. therefore
Left = 0.4 (25+50) = 30 (km),
Right = 0.6×50=30 (km),
That is to say, it took B 50 minutes to walk 30 kilometers to catch up with A. But there is only 28 kilometers between A and B. Therefore, until B town, B can't catch up with A.
35.( 1) According to the meaning of the question, it is assumed that the new alloy contains the first alloy x (g), the second alloy Y and the third alloy Z.
(2) When 2)x = 0, it is 500 grams larger.
(3) In the new alloy, the weight of manganese is:
x？ 40%+y？ 10%+z？ 50%=400-0.3x，
Y=250, at this time, y is the smallest; When z=0, y=500 is, that is, 250≤y≤500, so the range of the second alloy weight y in the new alloy is: minimum 250g, maximum.
And 0≤x≤500, so the weight range of manganese in the new alloy is: minimum 250g, 400g.