1。 In order to ensure the safety of people in a certain area in Wenchuan disaster area, it is necessary to dig a flood discharge trough. According to experts' calculation, the rescue team plans to excavate earthwork135,500m3 in 10 days. After one day of construction, considering the unpredictable factors such as weather, large-scale equipment was added to ensure foolproof and improve excavation efficiency. As a result, the task was completed four days ahead of schedule. After improving efficiency, how many cubic meters of earthwork will be dug every day on average?

2。 A pool is equipped with three water pipes, A, B and C, which are water injection pipes and C is drainage pipe. It takes 6 hours to open the A pipe alone to fill the pool, 8 hours to open the B pipe alone to fill the pool, and 12 hours to open the C pipe alone to fill the pool. Now open tube A and tube B for 2 hours, and then open tube C. How many hours can the pool be filled after opening tube C?

Answer; 1. Plan to dig in ten days 13.55. So the original plan was to dig 1.355 every day.

Because start one day, and then four days in advance. So after changing the efficiency, it took 10- 1-4=5 days.

So after changing the efficiency, dig (13.55-1.355)/10-1-4 = 2.439 every day.

2.439- 1.355= 1.084

Formula: (13.55-1.355)/(10-1-4)-1.355 =1.084.

2. 6 hours for Party A, 8 hours for Party B and 0/2 hour for Party C/kloc. ..

So A injects 1/6 of total water per hour.

B 1/8 of the total water injected every hour.

C112 of the total water released per hour.

Since Party A and Party B leave in two hours, the injection is (1/6+1/8) * 2 = 7/12.

The remaining space in the pool is1-7/12 = 5/12.

Finally, Party A, Party B and Party C worked together and spent X hours to form a * * *.

( 1/8+ 1/6- 1/ 12)* x = 5/ 12

The solution is x=2.

Formula: (1/6+1/8) * 2+(1/8+1/6-1) * x =1.

The solution is x=2.

3. It is proved that the sum of the products of four consecutive natural numbers and 1 must be the complete square of a certain number.

Such as 1*2*3*4+ 1=25=5 squared.

Problem solving process: proof: let the smallest of these four continuous natural numbers be a,

Then these four consecutive natural numbers are a, a+ 1, a+2 and a+3.

∫a(a+ 1)(a+2)(a+3)+ 1

= a(a+3)(a+ 1)(a+2)+ 1

=(a？ +3a)(a？ +3a+2)+ 1

=(a？ +3a)[(a？ +3a)+2]+ 1

=(a？ +3a)？ +2(a？ +3a)+ 1

=(a？ +3a+ 1)？

A is a natural number.

∴a？ +3a+ 1 is an integer.

∴ a (a+1) (a+2) (a+3)+1is the square of an integer.

That is, the product of four consecutive natural numbers plus 1 is the square of an integer.

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