https://web2.0calc.com/questions/geometry_38746
n! / [ (n - 2)! *2] > 400
n! / (n -2)! > 800
n * ( n -1) > 800
n^2 - n > 800
n^2 - n - 800 = 0
n = [1 + sqrt [1 + 3200 ] ] / 2 ≈ 28.79
n = 29
Proof
C(28, 2) = 378
C (29, 2) = 406
(r -6)^2 +(r -6)^2 =r^2
2 (r -6)^2 =r^2
2[ r^2 -12r + 36 ]= r^2
r^2 - 24r + 72 = 0
r^2 - 24r = -72
r^2 - 24r + 144 = -72 + 144
(r - 12)^2 = 72 take the positive root
r - 12 = sqrt (72)
r = 12 + sqrt (72) = 12 + 6sqrt 2 = 6 ( 2 + sqrt 2)
There is no minimum for x > 1
The graph approaches y = 0 as x increases, but there is no true minimum
https://web2.0calc.com/questions/geometry_89864
Let O be the cemter of the red circle and r = its radius
Let N = the center od the right semi-circle
BN^2 + BO^2 = ON^2
1^2 + (2 -r)^2 = (1 + r)^2
1 + 4 - 4r +r^2 = r^2 + 2r + 1
5 - 4r = 2r + 1
4 = 6r
r = 4/6 = 2/3
c = 3
a(-5)^2 +b(-5) + 3 = 0
a(4)^2 + b(4) +3 = 0
25a - 5b = -3 → 100a - 20b = -12
16a + 4b = -3 → 80a + 20b = -15
180a = -27
a = -27/180 = -3/20
16 (-3/20) + 4b = -3
4b = -3 + 48/20
b = -3/4 + 48/80= -3/20
a + b + c = 2 * ( -3/20 ) + 3 = -6/20 + 60/20 = 54/20 = 27 / 10
A
10 15
B D C
AD^2 = 15^2 - 3BD^2
AD^2 = 10^2 - BD^2
AD^2 = AD^2
15^2 - 3BD^2 = 10^2 - BD^2
15^2 -10^2 = 2BD^2
150 = 2BD^2
75 = BD^2
AD^2 = 10^2 - 75
AD^2 = 100 - 75
AD^2 =25
AD = 5
8
B 2 K 3 C
AKC , AKB are right triangles
AK = sqrt [8^2 - 3^2] = sqrt [ 55]
AB = sqrt [2^2 + 55] = sqrt [59]
British Flag Theorem
AX^2 +CX^2 =BX^2 + DX^2
3^2 + 3^2 = 3^2 + DX^2
18 = 3^2 + DX^2
18 - 9 = DX^2
DX^2 =9
DX = 3
