MathDB

a^2 + b^2 + c^2 + a^2b^2 + b^2c^2 + c^2a^2 \geq 15/16 if 8abc=1

Source: JBMO Shortlist 2014 A2

4/24/2019

Let

a, b, c

be positive real numbers such that

abc = \dfrac {1} {8}

. Prove the inequality:

a ^ 2 + b ^ 2 + c ^ 2 + a ^ 2b ^ 2 + b ^ 2c ^ 2 + c ^ 2a ^ 2 \geq \dfrac {15} {16}

When the equality holds?

algebrainequalitiespositive realthree variable inequality

flights, rootes and two mins

Source: JBMO Shortlist 2014 C2

4/24/2019

In a country with

n

towns, all the direct flights are of double destinations (back and forth). There are

r>2014

rootes between different pairs of towns, that include no more than one intermediate stop (direction of each root matters). Find the minimum possible value of

n

and the minimum possible

r

for that value of

n

.

combinatoricsminimum valueminimum

2014 JBMO Shortlist G2

Source: 2014 JBMO Shortlist G2

10/8/2017

Acute-angled triangle

{ABC}

with

{AB<AC<BC}

and let be

{c(O,R)}

it’s circumcircle. Diameters

{BD}

and

{CE}

are drawn. Circle

{c_1(A,AE)}

interescts

{AC}

at

{K}

. Circle

{{c}_{2}(A,AD)}

intersects

{BA}

at

{L}

.(

{A}

lies between

{B}

and

{L}

). Prove that lines

{EK}

and

{DL}

intersect at circle

c

.
by Evangelos Psychas (Greece)

geometryJBMO

2

Problems(3)