1

Part of 2009 JBMO Shortlist

Problems(2)

a + b + c = 15 , (a - 3)^3 + (b - 5)^3 + (c -7)^3 = 540

Source: JBMO 2009 Shortlist A1

Determine all integers

a, b, c

satisfying identities:

a + b + c = 15

(a - 3)^3 + (b - 5)^3 + (c -7)^3 = 540

JBMOalgebrasystem of equations

2009 JBMO Shortlist G1

Source: 2009 JBMO Shortlist G1

{ABCD}

{AC>BD}

{O}

intersection point of

{AC}

{BD}

. Circle with center at

{O}

{OA}

intersects extensions of

{AD}

{AB}

{G}

{L}

, respectively. Let

{Z}

be intersection point of lines

{BD}

{GL}

\angle ZCA={{90}^{{}^\circ }}