2006 Greece JBMO TST

Part of Greece JBMO TST

Subcontests

(4)

1

arranging no around a circle such 2 neighbours of 1,2,..,13 have prime sum

a) Is it possible to arrange numbers

1,2,...,13

in a circumference such that the sum of any two neighbouring numbers to be a prime number? b) Is the same problem possible for the numbers

1,2,...,16

1

min of K(x,y)=16 x^3/y +y^3/x - \sqrt{xy}

Find the minimum value of

K(x,y)=16\frac{x^3}{y}+\frac{y^3}{x}-\sqrt{xy}

x,y

are the real allowed values

1

k=b^c+a, l=a^b+c, m=c^a+b prime numbers

a,b,c

be positive integers such that the numbers

k=b^c+a, l=a^b+c, m=c^a+b

to be prime numbers. Prove that at least two of the numbers

k,l,m

1

find the angle , ratio of cevians related to orthocenter given

\angle A

ABC

, when we know it's altitudes

BD

CE

intersect in an interior point

H

of the triangle and

BH=2HD

CH=HE