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Source: Belarus MO 2001, P.4

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#1 h Jun 14, 2017, 12:20 PM • 1 Y

Y by Adventure10

On the Cartesian coordinate plane, the graph of the parabola $y = x^2$ is drawn. Three distinct points $A$ , $B$ , and $C$ are marked on the graph with $A$ lying between $B$ and $C$ . Point $N$ is marked on $BC$ so that $AN$ is parallel to the y-axis. Let $K_1$ and $K_2$ are the areas of triangles $ABN$ and $ACN$ , respectively. Express $AN$ in terms of $K_1$ and $K_2$ .

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#2 h Dec 1, 2020, 10:27 PM

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$A=(a,a^2),B=(b,b^2),C=(c,c^2)$
$BC:y=(b+c)x-bc, N=(a,a(b+c)-bc),AN=|a(b+c)-bc-a^2|=|(a-b)(a-c)|$
$K_1=\frac{AN |a-b|}{2},K_2=\frac{AN|a-c|}{2} \to K_1K_2=\frac{AN^2|(a-b)(a-c)|}{4}=\frac{AN^3}{4}\to AN=\sqrt[3]{4K_1K_2}$

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