2011 Greece JBMO TST

Part of Greece JBMO TST

Subcontests

(4)

1

points are collinear, lying on a line parallel to a given one

ABC

be an acute and scalene triangle with

AB<AC

, inscribed in a circle

c(O,R)

O

R

c_1(A,AB)

intersects side

BC

E

c

F

EF

intersects for the second time circle

c

D

AC

M

AD

BC

K

. Circumcircle of triangle

BKD

AB

L

. Prove that points

K,L,M

lie on a line parallel to

BF

1

solve in Z: 8x^3-4=y(6x-y^2)

Find integer solutions of the equation

8x^3 - 4 = y(6x - y^2)

1

3 points on each side of a square, no of line segments and intersections

On every side of a square

ABCD

, we consider three points different (to each other). a) Find the number of line segments defined with endpoints those points , that do not lie on sides of the square. b) If there are no three of the previous line segments passing through the same point, find how many of the intersection points of those segmens line in the interior of the square.

1

2\sqrt {x-1} +4\sqrt {y-4} + 6\sqrt {z-9} = x+y+z

n

be a positive integer. Prove that

n\sqrt {x-n^2}\leq \frac {x}{2}

x\geq n^2

x,y,z

2\sqrt {x-1} +4\sqrt {y-4} + 6\sqrt {z-9} = x+y+z