5

Part of 2014 JBMO Shortlist

Problems(3)

Source: AZE JBMO TST

x,y

z

be non-negative real numbers satisfying the equation

x+y+z=xyz

2(x^2+y^2+z^2)\geq3(x+y+z)

2014 JBMO Shortlist G5

Source: 2014 JBMO Shortlist G5

ABC

be a triangle with

{AB\ne BC}

{BD}

be the internal bisector of

\angle ABC,\

\left( D\in AC \right)

{M}

the midpoint of the arc

{AC}

which contains point

{B}

. The circumscribed circle of the triangle

{\vartriangle BDM}

intersects the segment

{AB}

{K\neq B}

{J}

be the reflection of

{A}

with respect to

{K}

{DJ\cap AM=\left\{O\right\}}

, prove that the points

{J, B, M, O}

belong to the same circle.

Source: AZE JBMO TST

Find all non-negative solutions to the equation

2013^x+2014^y=2015^z

number theoryAZE JBMO TST