MathDB

Inequality From JBMO

Source: JBMO 2016 Shortlist A5

6/25/2017

Let

x,y,z

be positive real numbers such that

x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.

Prove that

x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .

Proposed by Azerbaijan
[hide=Second Suggested Version]Let

x,y,z

be positive real numbers such that

x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.

Prove that

x+y+z\geq \sqrt{\frac{x^2+1}{2}}+\sqrt{\frac{y^2+1}{2}}+\sqrt{\frac{z^2+1}{2}} \ .

inequalitiesalgebra

2016 JBMO Shortlist G5

Source: 2016 JBMO Shortlist G5

10/8/2017

Let

ABC

be an acute angled triangle with orthocenter

{H}

and circumcenter

{O}

. Assume the circumcenter

{X}

of

{BHC}

lies on the circumcircle of

{ABC}

. Reflect

O

across

{X}

to obtain

{O'}

, and let the lines

{XH}

and

{O'A}

meet at

{K}

. Let

L,M

and

N

be the midpoints of

\left[ XB \right],\left[ XC \right]

and

\left[ BC \right]

, respectively. Prove that the points

K,L,M

and

{N}

are concyclic.

geometryJBMO

(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) =\overline{abcd}

Source: JBMO 2016 Shortlist N5

10/14/2017

Determine all four-digit numbers

\overline{abcd}

such that

(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) =\overline{abcd}

:

JBMOnumber theory

5