For 3 - 1 - 4 three numbers, the following operations are carried out: take two arbitrary numbers, divide their sum by root 2, their difference is also divided by root 2, add a number that was not taken before, and then generate three numbers. Can the sum of squares of three numbers be equal to 2008 after repeating the above operation several times?

For 3 - 1 - 4 three numbers, the following operations are carried out: take two arbitrary numbers, divide their sum by root 2, their difference is also divided by root 2, add a number that was not taken before, and then generate three numbers. Can the sum of squares of three numbers be equal to 2008 after repeating the above operation several times?

No
For three numbers a, B, C
Suppose a B
Then (a + b) / √ 2 (a-b) / √ 2 C after transformation
The sum of the squares of these three numbers = (a 2 + 2 ab + B 2) / 2 + (a 2 + 2 ab + B 2) / 2 + C 2 = a 2 + B 2 + C 2
So the sum of the squares of the three numbers does not change after each transformation
But 3 2 + (- 1) 2 + (- 4) 2 ≠ 2008
So it's impossible to get

For a three digit number, the number in the hundred is one more than the number in the ten, and the number in the one digit is three times less than that in the ten digit number. If the sequence of the three digits is reversed, the sum of the three digits obtained and the original three digit number is 1171, calculate the three digit number

If the number in the ten digit is x, then the number in the individual digit is 3x-2, and the digit in the hundred digit is x + 1,
Therefore, 100 (x + 1) + 10x + (3x-2) + 100 (3x-2) + 10x + (x + 1) = 1171
The solution is: x = 3
A: the original three digit number was 437

Ask geometry questions Given the trapezoid ABCD, ab = DC, AD / / BC, diagonal AC ⊥ BD, ad = 3cm, BC = 7cm, calculate the area of trapezoidal ABCD

Let AC, BD intersect with O, pass o as trapezoidal ABCD high crossing ad to e and BC to F
In trapezoidal ABCD, ab = DC, ad is parallel to BC,
So the trapezoid ABCD is isosceles trapezoid
So the angle oad = angle ODA = 45 degrees
Because ad = 3
So OA = od = (3 √ 2) / 2
OE=3/2
Similarly, the angle OBC = angle OCB = 45 degrees
Because BC = 7
So of = 7 / 2
So EF = 5
In conclusion, the area of trapezoidal ABCD = (1 / 2) * (3 + 7) * 5
=25

Ladies and gentlemen, ask mathematical geometry questions In rectangular ABCD, BC = 8, ab = 6, DP vertical AC, PM vertical AB, PN vertical BC Find PM;: PN Tell me what to ask for first and then, brothers and sisters,

It is known that both sides of the slope AC = 10, so sin angle DAC = 6 / 10; because DP is perpendicular to AC, so 6 / 10 = x / 8, x = 24 / 5, that is DP = 24 / 5. Similarly, AP, PC, PM, PN

As shown in the figure, ▱ ABCD, AE bisects ▱ bad intersects BC at e, EF ∥ AB intersects ad at F, What is the quadrilateral abef? Please state the reasons

The quadrilateral abef is a diamond
Reason: ∵ quadrilateral ABCD is a parallelogram,
∴AD∥BC，
∵EF∥AB，
The quadrilateral abef is a parallelogram,
∵ AE bisection ∵ bad,
∴∠BAE=∠FAE，
∵AD∥BC，
∴∠FAE=∠AEB，
∴∠BAE=∠AEB，
∴AB=BE，
▱ abef is a diamond

As shown in the figure, it is known that ∠ 1 = ∠ 2, ∠ 3 = ∠ 4, ∠ C = 32 ° and ∠ d = 28 ° are known to calculate the degree of ∠ P

Let AP and BC intersect with K,
∵ in ᙽ ACK and △ bpk, ∵ AKC = ∠ PKB (equal vertex angle),
∴∠P+∠3=∠1+∠C，
That is ∠ P = ∠ 1 - ∠ 3 + ∠ C, ①
Let AD and BP cross to F,
Similarly, there are ∠ P = ∠ 4 - ∠ 2 + ∠ D, ②
Because ∠ 1 = ∠ 2, ∠ 3 = ∠ 4,
Then ① + ② is obtained,
2∠P=∠C+∠D=32°+28°=60°，
∴∠P=30°
So the answer is: 30 degrees

Write the 2001 natural numbers from 1 to 2001 into a line to form a new natural number. The remainder of the new natural number divided by 9 is______ .

If the first of the nine adjacent numbers is n, then the others are n + 1, N + 2, until n + 8,
∴n+n+1+n+2+… N + 8 = 9N + 36 is divisible by 9,
The sum of every nine adjacent numbers must be divisible by 9,
∵2001
9 = 222 + 3,
The remainder can only be determined by the number composed of the last three numbers, that is, 19992002001,
The remainder of 19992002001 divided by 9 is 6,
The remainder of the new natural number divided by 9 is 6
So the answer is 6

Calculate 1 / [a (a + 1)] + 1 / [(a + 1) (a + 2)] + +1/[(a+2003)(a+2004)]

The original formula = 1 / A-1 / (a + 1) + 1 / (a + 1) - 1 / (a + 2) +1/(a+2003)-1/(a+2004)
=1/a-1/(a+2004)
=2004/a(a+2004)

For the three warehouses a, B and C, it is known that the ratio of grain stored in warehouse A to the sum of the two warehouses is 1:5, and the ratio of grain stored in warehouse B to the sum of warehouse A and warehouse C is 1:2, then the ratio of warehouse A to warehouse B is (): ()

A accounts for 1 / 6 1 ^ (1 + 5) of the total
B accounts for 1 / 3 1 ^ (1 + 2) of the total
Then C accounts for 1-1 / 6-1 / 3 of the sum = 1 / 2
So the ratio of warehouse A to warehouse B is 1 / 6:1 / 3 = 1:2

90 ° is known

The minimum of X-Y is equal
But it can't be equal, so X-Y > 0
Because x Max is close to 135 y and min is close to 90
﹤ less than 45
∴0°＜X-Y＜45°