MathDB

Just Hölder

Source: JBMO Shortlist 2009

5/12/2016

\boxed{\text{A5}}

Let

x,y,z

be positive reals. Prove that

(x^2+y+1)(x^2+z+1)(y^2+x+1)(y^2+z+1)(z^2+x+1)(z^2+y+1)\geq (x+y+z)^6

Inequalityalgebra

2009 JBMO Shortlist G5

Source: 2009 JBMO Shortlist G5

10/8/2017

Let

{A, B, C}

and

{O}

be four points in plane, such that

\angle ABC>{{90}^{{}^\circ }}

and

{OA=OB=OC}

.Define the point

{D\in AB}

and the line

{l}

such that

{D\in l, AC\perp DC}

and

{l\perp AO}

. Line

{l}

cuts

{AC}

at

{E}

and circumcircle of

{ABC}

at

{F}

. Prove that the circumcircles of triangles

{BEF}

and

{CFD}

are tangent at

{F}

.

geometryJBMO

(x^2 - c)(y^2 -c) = z^2 -c , (x^2 + c)(y^2 - c) = z^2 - c

Source: JBMO 2009 Shortlist N5

10/14/2017

Show that there are infinitely many positive integers

c

, such that the following equations both have solutions in positive integers:

(x^2 - c)(y^2 -c) = z^2 -c

and

(x^2 + c)(y^2 - c) = z^2 - c

.

JBMOnumber theorysystem of equations

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