Junior High School Math! Bonus points! Middle School Math Problems Good Bonus Points!! ）

1) Proof that if x=0, then y = c is an integer.

Let x=1, then y = a + b + c is an integer.

Let x=-1, then y= a - b + c is an integer, and from c is an integer, a-b is an integer.

Add the above two integers y together, which is still an integer, that is, 2a+2c is an integer, and it can be seen from c being an integer.

2a is also an integer.

2) Inverse proposition: If 2a a-b c are all integers, then x is a quadratic function when taking any integer.

y=ax 2+bx+c total integer value.

True proposition. Proof: Let a=b+n and n be an integer.

Let 2a=m, m be an integer, then there is 2a=2b+2n=m

So 2b=m-2n must be an integer, let it be 2b= k, and k is an integer, i.e. b=k 2

y=(b+n)x^2 + bx + c

bx^2 + bx + n x^2 + c

The last two items are integers and are not considered, and the first two items are discussed.

bx^2 + bx = b x(x+1)

k/2 x (x+1）

x and x+1 must be odd and even, i.e. x(x+1) is divisible by 2.

So k 2 x (x+1) must be an integer.

To sum up, y must be an integer.

The first one is a b and subtract an equal trillion of liquid, and the equation remains the same.

The second x is equal to 0 when the sum of a number is equal to 0, then the two numbers must be equal in magnitude and value, and the sign is opposite.

The third can be thought of as -8*x=4 x is the multiplier, and the multiplier = product divided by another multiplier equals 4 divided by -8 equals -1 2

The fourth 1 3 * x=-2 x=-2 divided by 1 3=-2*3=6

1: Add 5 on both sides

2: Subtract 5 from both sides

3: Divide by -8 on both sides

4: Multiply both sides by 3

1.Let the first bag of corn be x kg, then the second bag of corn will be (x+40) kg, so:

x+5=1 3(x+40-5), the solution is x=10, so the first bag of corn has 10 kg, and the second bag of corn has 50 kg.

2.(1) It took X minutes for Xiaohong's father, because Xiaohong had already walked 5*80=400 (m), so:

400 + 80x = 180x, so x = 4, that is, Xiaohong's father took 4 minutes (2) 4 * 180 = 720 (m), that is, her father walked 720 meters (3) 1000-720 = 280 (m), that is, there are still 280 meters away from the school.

Question 1:

You first have to figure out the change in the quality of the two bags of corn before and after.

Initially, the first bag was X-40

The second bag is X, and after pouring, the first bag is X-35

The second bag is X-5

According to the meaning of the equation.

x-35=（x-5）/3

Get x=50, so the weight of the first bag is 10, and the weight of the second bag is 50

When his dad caught up with Red, they had traveled the same distance, and in five minutes, Red had already walked 400 meters, and according to the title, they met in x minutes.

180x=80x+400

Solution x=4

So his dad caught up with Xiaohong at the 4-minute mark.

His dad walked 180x4=720 meters from the school 1000-720=280 meters.

Let the first bag of corn weigh x kilograms and the second bag weigh y kilograms.

y-x=40

x+5=3(y-5)

The solution is x=10

y=502, (1) Assuming that Dad catches up with Xiaohong and uses t min, then 80(5+t)=180*t

The solution is t=4

2)s=180*t=180*4=720(m)(3)1000-720=280(m)

1. Set the first bag with x kg and the second bag with y kg, and get y -x=40

3(x+5）=y-5

So x=10, y=50

2. It took X minutes to catch up with Xiaohong.

So 180x=80*5+80x

Get x = 4 and his dad walks = 180 * 4 = 720

Distance from school = 1000 - 720 = 280

1 Let the first bag originally have xkg, according to the title, the second bag has (x+40) kg, take it out, put it in, the first has x 5, the second x 35 (x 5) (x 35) 1 3

The solution is x=10

The first original 10kg and the second 50kg

2 (1) set x minutes to catch up.

180x 80x 80 5 solutions x 4

1*3 1, should be burned to respect should be 1 (1*3), that is, 1*3 of the accumulation of simple and reserved blocks to destroy one.