The image of parabola y = AX2 + BX + C has two intersections m (x1,0), n (x2,0) with x-axis, and passes through point a (0,1), where 0 ＜ x1 ＜ x2. The line L passing through point a intersects point C with x-axis, and intersects point B with parabola (different from point a), satisfying that △ can is isosceles right triangle and s △ BMN= five 2 s △ amn

The image of parabola y = AX2 + BX + C has two intersections m (x1,0), n (x2,0) with x-axis, and passes through point a (0,1), where 0 ＜ x1 ＜ x2. The line L passing through point a intersects point C with x-axis, and intersects point B with parabola (different from point a), satisfying that △ can is isosceles right triangle and s △ BMN= five 2 s △ amn

"And s △ BMN=
5   
2   S△AMN”
If it is "and s △ BMN = 5 / 2 & nbsp; s △ amn", then it is as follows
analysis:
From the point a (0,1) and △ can are isosceles right triangle, we can know that C (- 1,0), n (1,0), we can find the straight line AB from the coordinates of a and C, and we can know that the ordinate of point B is 5 / 2 & nbsp; from S △ BMN = 5 / 2S △ amn, we can find the abscissa of point B by substituting the analytic formula of line AB, and we can find the parabola analytic formula by substituting the coordinates of a, B and n into y = AX2 + BX + C
 
As shown in the figure, the parabola passes through a (0,1), m (x1,0), n (x2,0),
Where 0 ＜ x1 ＜ X2,
It can be seen that the opening of the parabola is upward, and the intersection of the parabola and the x-axis is on the positive half axis,
∵ point a (0,1), △ can are isosceles right triangles,
∴C（-1,0）,N（1,0）,
Let the analytic expression of line AB be y = MX + n,
Substituting the coordinates of a and C into,
We obtain n = 1-m + n = 0,
The solution is m = 1n = 1,
The analytic formula of line AB is y = x + 1,
∵ s △ BMN = 5 / 2S △ amn, two triangles with the same base Mn, the height of △ amn is 1, & nbsp;
The height of Δ BMN is 5 / 2,
That is, the ordinate of point B is 5 / 2;,
Substituting y = 5 / 2 into y = x + 1, we get x = 3 / 2;,
That is, B (3 / 2 & nbsp;, 5 / 2),
Substituting the coordinates of a, B and n into y = AX2 + BX + C,
We get C = 194a + 32B + C = 52A + B + C = 0;,
The results show that a = 4B = - 5C = 1,
Therefore, the analytical formula of parabola is y = 4x2-5x + 1,
So the answer is: y = 4x2-5x + 1

Point a (m, n) is on the parabola y = x2 + BX + C and n

Substituting point a (m, n) into
m^2+bm+c=n
(m+b/2)^2-b^2/4+c-n=0
(m+b/2)^2=b^2/4-c+n
(m+b/2)^2=(b^2-4c+4n)/4>=0
Because n0
x^2+bx+c=0
△=b^2-4c>0
So the equation x ^ 2 + BX + C = 0 has two different real roots
So the parabola y = x2 + BX + C and the X axis have two intersections