One. ? Multiple choice questions (each question? Points, * * *? Integral)

, set? Is a prime number. And then what? It's also a prime number, remember? ,

Then it must be () in the following cases.

、? Is a prime number; ? 、? Are plural numbers; ?

、? One is a prime number and the other is a composite number; ? , because of different? All of the above situations can happen.

Answer:? .

Solution: When? What time? With what? All prime numbers, and? ,

Is a prime number; ?

When is the prime number? Different from? When was that? Is it? Besides? , set? And then what? ,

None of them are prime numbers, which contradicts the conditions!

Simplify? The result is (? ).

、? ; ? 、? ; 、? ; 、? .

Answer:? .

Solution:? ;

,

So, prototype? .

、? The last digit of is ().

、? ; ? 、? ; ? 、? ; 、? .

Answer:?

Solution:? Press the last number? The order of the cycle, and? Press the last number? Continuous cycle,

Because? what's up The number of shapes, so? What is the last digit of? And then what? What is the last digit of? ,

So what? What is the last digit of? .

, equation? The solution is ().

, no solution; ? There is only one solution; There are exactly two solutions; There are infinite solutions.

Answer:? .

Solution: Transform the equation into …? (1), divided into three situations,

If, then ① becomes, what is that? , too? ;

If, then ① becomes, what is that? , too? ;

If so, what is it? , then (1) become, that is? This is an identity. Are you satisfied? Really? Are solutions to equations. Combined with the above discussion, we can see that the solutions of the equation are all satisfied real numbers, that is, there are infinite solutions.

A regular hexagon is divided into small regular triangles by three groups of parallel lines, so the number of all regular triangles in the graph is (? ).

、? ; ? 、? ; ? 、? ; 、? .

Answer:? .

Solution: classified calculation: let the side length of a regular hexagon be? So, what is the side length? Do you have a regular triangle? A, what is the side length? Do you have a regular triangle? A, what is the side length? Do you have a regular triangle? One,

* * * meter? Answer.

, set? Is an integer and a quadratic equation. Have equal roots? ,

There is also a quadratic equation with one variable? Have equal roots? ;

So, use? The quadratic equation with integral coefficients with roots is ().

、? ; ? 、? ; ?

、? ; 、? .

Answer:? .

Solution: What is the discriminant of the two equations? , is it? , and

, namely:

And then what? , eliminated? Have to? , its whole root is? ,

So what? ; So the two equations are:? And then what? ,

What is the equiroot of the last equation? What is the equiroot of the latter equation? , easy to get, to? What is a quadratic equation with integral coefficients with roots? .

Second,? Fill in the blanks (every little question? Points, * * *? Integral)

Right triangle? What are the lengths of the three sides? If the inscribed circle is removed, the area of the remaining part is equal to.

Answer:? .

Solution:? What is the area of? Let the radius of its inscribed circle be. , and then by

, so? So what is the inscribed circle area? So what's the remaining area? .

What if? ,

then what (? ).

Answer: (? ).

Solution:

,

By who? ,? ,? , solution,? ;

So what? .

As shown in the figure, square? What is the side length of? ,? what's up A little outside the edge, satisfied: hey? ,? ,

Then.

Answer:? .

Solution: Settings? And then what? ,? ,

By who? ∽? , too? ,

Really? , so? ,? And then what? ,

Again? , that is? , so? .

12 positive integers are filled on the circumference. Where does each number come from? In (each number can appear many times on the circumference), if the sum of the numbers in any three adjacent positions on the circumference is? Multiples of, with? Represents the sum of all twelve numbers on the circumference, so what is it? What are all possible values? Species.

Answer:? Species.

Solution: For three adjacent numbers on a circle? ,? Could it be? , or? , or? For example, what is the sum of three numbers? What time? Can I take it away? Or? Or? ; For any adjacent four numbers on the circumference, if the sequence is? Because? And then what? Both? A multiple of, then there must be? And then what? With what? Equal or different? ;

On the circumference again? With what? Interchangeable,? With what? Interchangeable; Now the circumference is divided into four sections, and the sum of three numbers in each section can be? , or? , or? , so the sum of four paragraphs can be taken? Any value goes in, a total of nine cases.

One of the filling methods is: first, fill in twelve numbers in turn on the circumference:? , and its sum is? And play it one at a time? Change to? , or will there be one? Change to? Every operation increases the sum? And this operation can be done eight times).

First one? Two? attempt

First, (? Try to determine, for what positive integer? , equation? Is there a positive integer solution? Find all positive integer solutions of the equation.

Solution: Rewrite the equation to

Because? As the sum of squares of two positive integers, there are only two different forms: ... 10'

So,

…? ①, or? …? ②

…? 3, or? …? ④? ………… 15'

From (1)? (when? Or? ); From 2? (when? Or? );

From 3 (when or? ); ? Or (when? Or? );

From 4? (when? ); Or (when? Or? ).？ …………20'

Second, (? Points) acute triangle? What is the heart outside? What is the radius of the circumscribed circle? , expand? , respectively with the opposite? Hand it in? ; Prove:

Certificate:? Extension? Turn it in Yu? Because? * * * point? ,

…………5'

Therefore ... ① ...10'

And then what? ,………… 15'

Similarly,

,…………20'

Substitute into ① to get, ...? ②

So ... ? …………25'

Third, (? Sub) set? Is a positive integer, which proves that:

1, if? Is the product of two consecutive positive integers. It is also the product of two consecutive positive integers;

2. What if? Is the product of two consecutive positive integers. It is also the product of two consecutive positive integers.

Proof: 1, if? Is the product of two consecutive positive integers, let. , among them? Is a positive integer, ... 5'? Is the product of two consecutive positive integers; ………… 10'

2. What if? Is the product of two consecutive positive integers, let. , among them? Is a positive integer, so ... ①………… 15'

So? what's up Multiples of sum? Is odd; Settings? , from (1),

…? ②…………20'

Therefore,

, that is? , which is the product of two consecutive positive integers ... 25'