MathDB

1

computational in ABCD, angles and midpoints related, area wanted

ABCD

be a quadrilateral with

\angle DAB=60^o

,

\angle ABC=60^o

and

\angle BCD=120^o

. Diagonals

AC

,

BD

intersect at point

M

and

BM=a, MD=2a

. Let

O

be the midpoint of side

AC

and draw

OH \perp BD, H \in BD

and

MN\perp OB, N \in OB

. Prove that i)

HM=MN=\frac{a}{2}

ii)

AD=DC

iii)

S_{ABCD}=\frac{9a^2}{2}

3

1

4 men, different speeds, 1 tunnel, 1 torch, max 2 at a time, min time

4

men stand at the entrance of a dark tunnel. Man

A

needs

10

minutes to pass through the tunnel, man

B

needs

5

minutes, man

C

needs

2

minutes and man

D

needs

1

minute. There is only one torch, that may be used from anyone that passes through the tunnel. Additionaly, at most

2

men can pass through at the same time using the existing torch. Determine the smallest possible time the four men need to reach the exit of the tunnel.

1

factorization and x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 in ZxZxZ

a) Factorize

A= x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2

b) Prove that there are no integers

x,y,z

such that

x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000

4

1

perfect cube, not divisible by 10, erasing 3 last digit remains a perfect cube

a) If positive integer

N

is a perfect cube and is not divisible by

10

, then

N=(10m+n)^2

where

m,n \in N

with

1\le n\le 9

b) Find all the positive integers

N

which are perfect cubes, are not divisible by

10

, such that the number obtained by erasing the last three digits to be also also a perfect cube.

2001 Greece JBMO TST

Subcontests

computational in ABCD, angles and midpoints related, area wanted

4 men, different speeds, 1 tunnel, 1 torch, max 2 at a time, min time

factorization and x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 in ZxZxZ

perfect cube, not divisible by 10, erasing 3 last digit remains a perfect cube