MathDB

[url=https://artofproblemsolving.com/community/c675547]Greece JBMO TST 2017
[url=http://artofproblemsolving.com/community/c6h1663730p10567608]Problem 1. Positive real numbers

a,b,c

satisfy

a+b+c=1

. Prove that

(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca).

Also, find the values of

a,b,c

for which the equality happens.
[url=http://artofproblemsolving.com/community/c6h1663731p10567619]Problem 2. Let

ABC

be an acute-angled triangle inscribed in a circle

\mathcal C (O, R)

and

F

a point on the side

AB

such that

AF < AB/2

. The circle

c_1(F, FA)

intersects the line

OA

at the point

A'

and the circle

\mathcal C

K

. Prove that the quadrilateral

BKFA'

is cyclic and its circumcircle contains point

O

.
[url=http://artofproblemsolving.com/community/c6h1663732p10567627]Problem 3. Prove that for every positive integer

n

, the number

A_n = 7^{2n} -48n - 1

is a multiple of

9

.
[url=http://artofproblemsolving.com/community/c6h1663734p10567640]Problem 4. Let

ABC

be an equilateral triangle of side length

a

, and consider

D

E

and

F

the midpoints of the sides

(AB), (BC)

, and

(CA)

, respectively. Let

H

be the the symmetrical of

D

with respect to the line

BC

. Color the points

A, B, C, D, E, F, H

with one of the two colors, red and blue.
[*] How many equilateral triangles with all the vertices in the set

\{A, B, C, D, E, F, H\}

are there? [*] Prove that if points

B

and

E

are painted with the same color, then for any coloring of the remaining points there is an equilateral triangle with vertices in the set

\{A, B, C, D, E, F, H\}

and having the same color. [*] Does the conclusion of the second part remain valid if

B

is blue and

E

is red?

Problem Statement