Help find a few math contest questions, a math contest contest question

grams can weigh 1-6g.

Add 15g to call 15-21.

With 15 minus can be called 9-14.

Add 40 to weigh 40-46, 49-54, 55-61.

Subtract 40 to have 34-39, 25-19, 26-31 and then finish, except for can be weighed, so a total of 61-6 = 55 kinds.

ps: When you encounter this kind of question with a small number during the exam, the quickest way is to enumerate, don't think about any simple and fast way, it's better to do it if you have time.

A odd number of the number chain square divided by 4 remainder 1, pure grandson.

99 divided by 4 to the remainder of 3

So it's impossible to imitate it.

What is a perfect square number?

Let there be a sequence of positive integers (n>=0).

Then 2010*mn=2010*(1+n*10) makes 2010*mn=(n+1)!

Therefore, there is 2010*(1+n*10)=(n+1)!

n+1)!-20100*n=2010

n((n+1)(n-1)!-20100) = 2010 to get n = 9

The conclusion should be correct, I don't know if there is a mathematical approach. When the computer calculates that n=14136, the first four digits of an are 2010.

Obviously, any one person knows at least three people well.

And 3x7 21 is not divisible by 2, so at least one person knows 4 people well.

You might as well set him A, then the 6 people after excluding A will sit at the table according to the principle of acquaintance. Then two of A's four acquaintances must be next to each other, and A only needs to be among them.

Solution: Suppose you can see the x-level, the escalator upward speed a, Li Ming's upward speed b, and the downward speed 3b, then the downward time is 150 (3b)=50 b In this time, the elevator goes up 50 (b a) level Xiao Ming walks from the top to the end, so (x+50) (b a)=150 is the same, and there is (x-75) (b a)=75{( x+50) (b a)=150

x-75）÷（b÷a）=75

The solution is x=120, that is, the elevator can see 120 steps.

I don't understand at all that any six of them can sit down around a round table, and what that means is the key.

2.(x 2-1)m-2x+1<0 for -2 =m =2 constant set y=(x 2-1)m-2x+1

y is a primary or constant function of m, and its image is straight, so the two endpoints corresponding to y are less than 0, and the whole line segment is below the x-axis, and y is less than 02 (x 2-1)-2x+1 0

2（x^2-1）-2x+1＜0

After solving x, take the common part.