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y=x^2 and 1998 points

Source: VI Soros Olympiad 1990-00 R3 9.3 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 28, 2024
analytic geometryparabolaalgebra

Problem Statement

On the coordinate plane, the parabola y=x2y = x^2 and the points A(x1,x12)A(x_1, x_1^2), B(x2,x22)B(x_2, x_2^2) are set such that x1=−998x_1=-998, x2=1999x_2 =1999 The segments BX1BX_1, AX2AX_2, BX3BX_3, AX4AX_4,..., BX1997BX_{1997}, AX1998AX_{1998} and XkX_k are constructed succesively with (xk,0)(x_k,0), 1≤k≤19981 \le k \le 1998 and x3x_3, x4x_4,..., x1998x_{1998} are abscissas of the points of intersection of the parabola with segments BX1BX_1, AX2AX_2, BX3BX_3, AX4AX_4,..., BX1997BX_{1997}, AX1998AX_{1998}. Find the value 1x1999+1x2000\frac{1}{x_{1999}}+\frac{1}{x_{2000}}
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