MathDB

Cauchy-Schwarz !

Source:

2/4/2015

Let

a

,

b

,

c

be positive real numbers such that

a+b+c=a^2+b^2+c^2

. Prove that :

\frac{a^2}{a^2+ab}+\frac{b^2}{b^2+bc}+\frac{c^2}{c^2+ca} \geq \frac{a+b+c}{2}

Circle !

Source:

2/4/2015

Let

AB

and

CD

be chords in a circle of center

O

with

A , B , C , D

distinct , and with the lines

AB

and

CD

meeting at a right angle at point

E

. Let also

M

and

N

be the midpoints of

AC

and

BD

respectively . If

MN \bot OE

, prove that

AD \parallel BC

.

Points in a circle!

Source: JBMO Shortlist 2012 C3

2/4/2015

In a circle of diameter

1

consider

65

points, no three of them collinear. Prove that there exist three among these points which are the vertices of a triangle with area less than or equal to

\frac{1}{72}

.

geometrycombinatorial geometrycombinatorics unsolvedcombinatorics

Equation with meaning !

Source:

2/4/2015

Decipher the equality :

(\overline{VER}-\overline{IA})=G^{R^E} (\overline {GRE}+\overline{ECE})

assuming that the number

\overline {GREECE}

has a maximum value .Each letter corresponds to a unique digit from

0

to

9

and different letters correspond to different digits . It's also supposed that all the letters

G

,

E

,

V

and

I

are different from

0

.

3

Problems(4)