MathDB

1

5 equal circles, intersections of 4 are vertices of a parallelogram

ABCD

inscribed in circle

(O,R)

(with center

O

and radius

R

). With centers the vertices of the quadrilateral and radii

R

, we consider the circles

C_A(A,R), C_B(B,R), C_C(C,R), C_D(D,R)

. Circles

C_A

and

C_B

intersect at point

K

, circles

C_B

and

C_C

intersect at point

L

, circles

C_C

and

C_D

intersect at point

M

and circles

C_D

and

C_A

intersect at point

N

(points

K,L,M,N

are the second common points of the circles given they all pass through point

O

). Prove that quadrilateral

KLMN

is a parallelogram.

1

7 cards, tears a few in 7 pieces, teare a few again in 7, ..., 2009 pieces ever?

One pupil has

7

cards of paper. He takes a few of them and tears each in

7

pieces. Then, he choses a few of the pieces of paper that he has and tears it again in

7

pieces. He continues the same procedure many times with the pieces he has every time. Will he be able to have sometime

2009

pieces of paper?

3

1

(a\sqrt2+b\sqrt3)/(b\sqrt2+c\sqrt3) \in Q => (a^2+b^2+c^2)/(a+b+c) in Z

Given are the non zero natural numbers

a,b,c

such that the number

\frac{a\sqrt2+b\sqrt3}{b\sqrt2+c\sqrt3}

is rational. Prove that the number

\frac{a^2+b^2+c^2}{a+b+c}

is an integer .

4

1

x+y+z=xy+yz+zx and xyz=1 , x,y,z>0 with min sum

Find positive real numbers

x,y,z

that are solutions of the system

x+y+z=xy+yz+zx

and

xyz=1

, and have the smallest possible sum.

2009 Greece JBMO TST

Subcontests

5 equal circles, intersections of 4 are vertices of a parallelogram

7 cards, tears a few in 7 pieces, teare a few again in 7, ..., 2009 pieces ever?

(a\sqrt2+b\sqrt3)/(b\sqrt2+c\sqrt3) \in Q => (a^2+b^2+c^2)/(a+b+c) in Z

x+y+z=xy+yz+zx and xyz=1 , x,y,z>0 with min sum

2009 Greece JBMO TST

Subcontests

5 equal circles, intersections of 4 are vertices of a parallelogram

7 cards, tears a few in 7 pieces, teare a few again in 7, ..., 2009 pieces ever?

(a\sqrt2+b\sqrt3)/(b\sqrt2+c\sqrt3) \in Q =&gt; (a^2+b^2+c^2)/(a+b+c) in Z

x+y+z=xy+yz+zx and xyz=1 , x,y,z&gt;0 with min sum

(a\sqrt2+b\sqrt3)/(b\sqrt2+c\sqrt3) \in Q => (a^2+b^2+c^2)/(a+b+c) in Z

x+y+z=xy+yz+zx and xyz=1 , x,y,z>0 with min sum