jmc

Question

For integers a and T, T 0, a parabola whose general equation is y = ax^2 + bx + c passes through the points A = (0,0), B = (2T,0), and C = (2T + 1,28). Let N be the sum of the coordinates of the vertex point. Determine the largest value of N.

Accepted Answer

Since the parabola passes through the points

(0, 0)

and

(2 T, 0),

the equation is of the form

y = a x (x - 2 T) .

For the vertex,

x = T,

and

y = a T (- T) = - a T^{2} .

The sum of the coordinates of the vertex is then

N = T - a T^{2} .

Setting

x = 2 T + 1,

we get

a (2 T + 1) = 28.

The possible values of

2 T + 1

are 7,

- 1,

and

- 7.

(We do not include 1, because

T \neq = 0.

) We compute the corresponding values of

T,

a,

and

T - a T^{2} .

2 T + 1 7 - 1 - 7 T 3 - 1 - 4 a 4 - 28 - 4 T - a T^{2} - 33 2760

Hence, the largest possible value of

N

is

60 .