A few junior high school math Olympiad questions. Mathematics Olympiad questions in junior high scho

Question 1: It can be compared like this: 555 times for 3 and 444 times for 4 and 333 times for 5 can be seen as follows:

After the 5th power of 3, then take the 111th power, after the 4th power of 4, then take the 111th power, and then take the 111th power after the 3rd power of 5, so that there is 4 to the 4th power, 3 to the 5th power, 5 to the 3rd power, and then to the 111th power, respectively, to get: 4 to the 444th power, 3 to the 555th power, 533rd power of 5.

Question 2: This problem can be done by drawing a number line, marking the positions of a and b on the number line, b is to the left of a, |x-a|+|x-b|The meaning of =a-b is that the distance from x to a plus the distance to b on the number axis is equal to a-b, so that it is obvious that the value of x is b The third question: this can be explained that they are all negative numbers, so when you do it, you should do it according to the positive number, and after making the result according to the normal steps, you can take the opposite size of the sign

Question 4: This question is very simple, and the specific process is as follows:

by x:y=3:2

3y=2x....1

And x+3y=27....2

There are 1 and 2 formulas to solve.

x=9y=6

Therefore, the smallest is x=6

Question 5: This problem is a common sense problem, we know that the speed of the boat in the water = the hydrostatic velocity of the boat + the water velocity, the velocity of the ship in the reverse water = the hydrostatic velocity of the boat - the water velocity, so that you can set x, y, and set the distance to a fixed number 1, so that you can draw conclusions

Thank you, please correct any inaccuracies

Thank you!!!

1.(1) Solution: 3 555 = (3 5) 111 = 243 111

And 256>243>125, so 4 444>3 555> 5 333

1.(2) The question given by the landlord is wrong!!

The original question is: a=19997 19998-19996 (here) 19997 and b=19996 19997-19995 19996.

Solution: a=(19997 2-19998*19996) (19998*19997).

b=19996/19997-19995/19996

So when aa, then x>b, can be reduced to:

x-a + x-b = 2x-a-b is not equal to a-b

So it doesn't hold, she!

2): When x -100 212 >) 100 112

In the same way, -100 312*313 > 100 212*213 > 100 112*113

Note: The larger the denominator in a positive number, the smaller the molecule, and the larger the denominator in a negative number, the larger the molecule).

So so a b c

4 solution: x:y=3:2 takes x=3a, then y=2a

So x+3y=3a+6a=9a=27, then a=3

Therefore, x=9,y=6, then the smaller of x,y is (6).

Note: It is also possible to solve a system of equations to obtain the values of x and y, but it is not the simplest method).

5 Solution: Method 1:

Let the distance be s, the speed of the ship in still water is v1, the current velocity is v2, and v1 > v2, then.

The time required to sail along the water t1=s (v1+v2).

The time required to sail against the current t2=s (v1-v2).

Total time t=t1+t2=s (v1+v2)+s (v1-v2)=2s (v1 2-v2 2).

When v2 increases, apparently t also increases.

Method 2: Suppose the distance is 1, the speed of the boat is a, the original water velocity is b, and the current speed increases c

There is: the original time taken is 1 (a+b)+1 (a-b)=2a (a 2-b 2).

The time taken when the water velocity increases1 (a+b+c)+1 (a-b-c)=2a (a 2-b 2-2bc-c 2).

2a (a 2-b 2) 2a (a 2-b 2-2bc-c 2) so the time increases.

3 to the power of 5, to the power of 5, to the power of 333, to the power of 4, to the power of 4.

There are six radii of the 6 tracks, which are, ,42,0, and the starting line distance can be obtained by subtracting the perimeter of the latter from the previous one.

Because the first number is constantly added and subtracted, it will eventually be added and subtracted, so it is better to calculate whether 1 to 7 should be added or subtracted. 1+2+3+4+5+6+7=28, so the addition and subtraction should be 14, it is better to make a simple distribution, because 2+3+4+5 is equal to 14, so they can be assumed to be addition. Suppose the first number is x, the second is x-1, the third is (x-1)+2, the fourth is the third plus 3, and so on, you can get an equation 7x+32=1997, which is not an integer, what should I do?

It doesn't matter, there are many ways to divide 28 into 14, try each one yourself, and you will have such a set of answers in the end: 283 284 282 285 289 284 290

Let the first number be x

x+(x+1)+(x+3)+(x+6)+(x+10)+(x+15)+(x+21)=1997

7x=1941

x=1941/7

..I'm sorry to do this wrong.

Counter-evidence. Suppose the inside angle of each triangle is greater than 45 degrees.

Then each of the inner angles of the quadrilateral is greater than 90 degrees, and the internal angle is greater than 360

1.Let the first velocity be x km h then the second velocity is km h, and the inscription is: 50 x -2 = 50

The multiplication of both sides of the equation has 125-5x=50

x=15 is the solution of the original equation.

So v A=15km h vB=

2.Let the original single digit be x, then the original tens of digits are (5-x), which is solved by the inscription: 10(5-x)+x=10x+(5-x)+27, x=1

So the original two-digit number was 41

3.Let the single digit be x, then the number of tens is 2x the number of hundreds, and the meaning of the title is: x+2x+4x=14

x=2, so the original number is 842

Let A be x and B be.

50 ( 20 x = 50 x - 1 20 = 50-x x = 30 A is 30 km h, B is 75 km h

Let the ten digits be x and the single digit be 5-x

10x+（5-x) -27= 10(5-x)+x 9x+5-27=50-9x 18x=45+27=72

x=4 so 41

Let the single digit be x x + 2x + 4x = 14 x = 2, so this number is 842

1.If 1 a:1 b:1 c = 2:3:4, then a:b:c is equal to ?

2.Given that there are 9 positive integer solutions in inequality ax+2004》0, what is the range of values for a?

3.Let x-1 be a factor of x-5x+c, then c=?

4.If (x-1)(x+3)(x-4)(x-8)+m is a perfect flat, then m=?

5△.The side length of abc is a=m -1, b=m +1, c=2m (m 0), then what kind of triangle is abc?

6.If the length of all three sides of a triangle is such that the value of the algebraic formula x = 9x+18 is 0, then what is the circumference of the triangle?

7。Knowing a-b=4, ab+c +4=0, then a+b+c ?

Then b a+a b=?

9.A school participates in the eighth-grade mathematics competition, and there are a total of students in class A and B, of which class A scores an average of 71 points per person, class B scores an average of 69 points per person, and the total score of the two classes is 3480 points, so how many students are in class A and B?

10.The side length of an equilateral triangle abc is 12cm, and if you take any point on one side of it, then the sum of the distances from this point to the other two sides = ?

11.Knowing ab≠1, and 5a +2010a+8=0, 8b +2010b+5=0, then a b ?

There are a total of 143 calculators in a store, type A and type B, type A calculator is 60 yuan each, and type B calculator is each yuan A school purchased all the type B calculators and part of the type A calculator in the store, and after accounting, it was found that the total amount of payables had nothing to do with the total number of type A calculators Q: What percentage of the total number of type A calculators purchased in the store is the total number of type A calculators in the store? What is the total amount of the payable?

A group of passengers decided to divide the number of large cars, and to make each car have the same number of people, the period, each car carried 22 people, and found that one of them could not sit in the car, and if an empty car was driven, then all the passengers would be divided equally among the rest of the cars. Each car can not carry more than 32 passengers, how many cars are there in the branch? How many people are in this group?

1: Is it possible to find 16 positive integers so that the sum of any 9 of them is not divisible by 9? If you can, write them out and explain the characteristics of the numbers written; If not, please explain why. Can 17 positive integers be found to meet the above requirements? Why?

2: The system of equations about x,y is known 2x+3y x+7y=m, and when -18 m -10 has an integer solution, then the value of x2+xy+y2 is equal to

3: Define a new operation for any two positive integers x,y, i.e., x y=2 (2xy-x-y). If the positive integers a,b satisfy a b=888, then there are several pairs of such ordered pairs (a,b).

4: In the planar Cartesian coordinate system, there are a(0,5),b(5,0),c(0,3),d(3,0), and ad and bc intersect at the point e, find the area of abe.