Q:
What is the number 17?
The sum of the top 20?
Please answer with the formula given.
Question 2:
There is a column number: 2.4.6.8... 192.
Q:
Their sum?
Please decide which position in the series is 48. (Equation can be listed)
3. There is an integer, 70, 1 10 and 160 divided by which the sum of the three remainders is 50, so what is this integer?
4. Let M and N be natural numbers, remember that PM is the sum of all digits of natural number M and PN is the sum of all digits of natural number N. Also remember that M*N is the remainder of m divided by n, given that M+N=4084, what is the value of (PM+PN)*9?
5. As shown in the figure, it is known that CD=5, DE=7, EF= 15 and FG=6. The straight line AB divides the figure into two parts, the area of the left part is 38, and the area of the right part is 65. What is the area of triangle ADG?
6. A natural number can be expressed as the sum of 9 consecutive natural numbers, 10 consecutive natural numbers and 1 1 consecutive natural numbers. What is the minimum natural number that satisfies the above conditions?
7. It is known that the pure alcohol content of alcohol A is 72%, the pure alcohol content of alcohol B is 58%, and the pure alcohol content after mixing the two alcohols is 62%. If the amount of each alcohol is more than the original 15 liter, and the pure alcohol content after mixing is 63.25%, how many liters of methanol will be taken for the first time?
8. In the following formula, different Chinese characters represent different numbers, and the same Chinese characters represent the same numbers. So what are the three digits represented by "Happy New Year"?
9. There are two shopping centers. When the profit of the first mall decreases by 15% and the profit of the second mall increases by 18%, the profits of the two malls are the same. Then, how many times is the profit of the first shopping mall more than that of the second shopping mall?
10, take three from nine numbers 1~9, and these three numbers can form six different three digits. If the sum of six three digits is 3330, what is the largest of these six three digits?
1 1. There are five teams, A, B, C, D and E, and every two teams will play one game. When the game is about to end, the statistics are as follows:
Team name wins, draws, negative numbers, goals scored and goals conceded.
A piece of 2 1 0 4 1
B 1 2 0 4 2
C 1 1 1 2 3
D 1 0 3 5 5
E 0 2 1 1 5
It is known that A and E and B and C are tied, and the scores are all 1: 1. What's the score between b and d?
12, a bus and a van set off from A and B at the same time and walked in opposite directions. The bus speed is 32 kilometers per hour, and the van speed is 40 kilometers per hour. When the two cars reached the second place and the first place respectively, they immediately returned to the starting point. When returning, the bus speed increased by 8 kilometers per hour and the van speed decreased by 5 kilometers per hour. Given that the two encounters are 70 kilometers apart, how many hours before the bus, does the van return to the starting point?
A (1) and B (2) are walking in the escalator of a shopping mall. 1 from the floor of 1 to the second floor, and 2 from the second floor to the floor of 1. 1 Stand on the elevator and go up two steps every second (note: the elevator is also moving). It takes 50 seconds to walk to the second floor.
Take three of the nine numbers 1~9, and these three numbers can form six different three-digit numbers. If the sum of six three digits is 3330, what is the largest of these six three digits?
The question comes first and the answer comes last.
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+…+A1X+A0, and try to find the value of A0+A2+A4+A6.
5. Known equation
If there is a solution, find the value of k.
6. Solve equation 2 | x+ 1 |+x-3 | = 6.
7. Solving equations
8. Solve the inequality || x+3 |-x- 1 || > 2.
9. Compare the following two figures:
10.x, y and z are nonnegative real numbers and satisfy:
x+3y+2z=3,3x+3y+z=4,
Find the maximum and minimum values of u = 3x-2y+4z.
1 1. Find the quotient and remainder of x4-2x3+x2+2x- 1 divided by x2+x+ 1.
12. As shown in figure 1-88, Zhu Xiao lives in village A and grandma lives in village B. On Sunday, Zhu Xiao went to visit her grandmother. She first cut a bundle of grass on the north slope, and then cut a bundle of firewood on the south slope to send it to her grandmother. Excuse me, which route should Zhu Xiao take for the shortest journey?
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.
16. As shown in figure 1-92. In △ABC, ∠B=∠C, BD⊥AC is in D.
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?
2 1. If both positive integers p and p+2 are prime numbers greater than 3, then verify: 6 | (p+ 1).
22. Let n be the smallest positive integer satisfying the following conditions, which is a multiple of 75 and has exactly
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 a partner.
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 starts from a port 240 nautical miles apart, and its speed decreases by 65,438+00 nautical miles per hour before reaching its destination 48 nautical miles. The total time it takes after its arrival is equal to the time it takes for the whole voyage when its original speed is reduced by 4 nautical miles per hour, so that we can find out the original speed.
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. If a shopping mall sells goods with a unit price of 8 yuan per piece 12 yuan, 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 should each piece be reduced to get the best 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.
Answers to the review questions of junior one Olympic mathematics
Author: Anonymous article Source: Junior High School Mathematics Competition Counseling hits: 456 Update time: February 4, 2006
2. Because | A | =-A, a≤0, because | AB | = AB, b≤0, and because | C | = C, C ≥ 0. So A+B ≤ 0, c-b≥0 and A-C ≤ 0.
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。
5.②+③ Finishing
x=-6y,④
(k-5) y = 0 when substituting ①.
When k=5, y has infinite solutions, so the original equations have infinite groups of solutions; When k≠5, y=0. If it is substituted into ②, (1-k) x = 1+k is obtained. Because x=-6y=0, 1+k = 0, so k =-/kloc-0.
Therefore, when k=5 or k=- 1, the original equations have solutions.
When < x ≤ 3, 2 (x+ 1)-(x-3) = 6, so x =1; When x > 3, there are
, so you should give up.
7. From | x-y | = 2
X-y=2, or x-y=-2,
therefore
From the previous equations.
|2+y|+|y|=4。
When y
Similarly, it can be solved by the latter equations.
So the solution is
X of solution ① is ≤-3; Solve ②
-3 < x
③ x > 1 of the solution.
So the original inequality solution is x 0.9. Let a = 9999111,then
therefore
Obviously there is a > 1, so a-b > 0, that is, a > b.
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'. If the intersection points of the line segment connected by A' B' and the north hillside and the south hillside are A and B respectively, the route of A →A→B→ 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 ∠ Coe = 90.
Because ∠ COD = 55,
So ∠ 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,
So ∠ ABF = ∠ CFB
therefore
AB‖CD (internal dislocation angles are equal and two straight lines are parallel).
∠ABC divided by∠ CBF = 55 equals BE, so
∠ABC=2×55 = 1 10。 ①
AB‖CD is known by Shanghai Stock Exchange, so
∠EDF=∠A=70,②
Know from ① and ②
BC‖AE (the inner angles on the same side are complementary and the two straight lines are parallel).
15. As shown in figure 1- 100. EF ⊥ AB, CD⊥AB, so
∠EFB=∠CDB=90 degrees,
So EF‖CD (same angle, two straight lines are parallel). therefore
∠BEF=∠BCD (two straight lines are parallel and have the same angle). (1) also know ∠ CDG =∠ BEF. ②.
By ①, ② ∠ BCD = ∠ CDG.
therefore
BC‖DG (internal dislocation angles are equal and two straight lines are parallel).
therefore
∠AGD=∠ACB (two straight lines are parallel and have the same angle).
16. in △BCD,
∠ DBC+∠ C = 90 (because ∠ BDC = 90), ①
In △ABC again, ∠B=∠C, so
∠A+∠B+∠C=∠A+2∠C= 180,
therefore
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 on the side of BC, so
S△CEG=S△BCEE,
therefore
therefore
SEFDC=3x+2x=5x,
therefore
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, that is, 2(a+B+C) = 27, which is contradictory.
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. From the condition of n = 75k = 3× 52× k, in order to make n as small as possible, we can set n=2α3β5γ(β≥ 1, γ≥2) and have.
(α+ 1)(β+ 1)(γ+ 1)=75.
So α+ 1, β+ 1 and γ+ 1 are all odd numbers, and α, β and γ are even numbers. Therefore, γ = 2. At this time,
(α+ 1)(β+ 1)=25.
therefore
Therefore, (α, β) = (0,24), or (α, β) = (4,4), that is, n = 20.324.52.
23. There are X stools and Y chairs.
3x+4y+2(x+y)=43,
That is 5x+6y = 43.
So x=5 and y=3 are the only nonnegative integer solutions, so there are 8 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 ways to choose the first position and only 7 ways to choose the second position ... According to the principle of multiplication, men and women have different methods.
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 for pairing with male A, and 7 different situations for pairing with male B. …, the two columns are interchangeable, so * * * has.
2×8×7×6×5×4×3×2× 1=80640
Different situations.
26. Five ten thousandths.
4×3×2× 1=24 (pieces).
There are four tens of thousands.
4×3×2× 1=24 (pieces).
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, there is always a * * *
24+24+6+4=58
This number is greater than 34 152.
27. The distance traveled by two cars is the sum of the lengths of the two cars, namely
92+84 = 176 (m).
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 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, the maximum value of y = 1800 (yuan). That is, when the price of each piece is reduced by 1 yuan, the maximum profit is 1800 yuan. At this time, it sold more than 200 yuan, so the profit increased by 200 yuan.
34. Assuming that it takes X minutes for B to catch up with A, then A has to walk (25+x) minutes to the place where it is caught up, then the distances that A and B walk are 0.4 (25+x) km and 0.6x km respectively. Because they walk the same way, 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 x=0 and y=250, y is the smallest; When z=0, y=500 is the maximum, that is, 250≤y≤500, so the range of weight y of the second alloy in the new alloy is: minimum 250g, maximum 500g.
(3) In the new alloy, the weight of manganese is:
x 40%+y 10%+z 50%=400-0.3x,
And 0≤x≤500, so the weight range of manganese in the new alloy is: minimum 250g, maximum 400g.
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