MathDB

1

This is insane

\Delta ABC

, the sum of the sides is

2s

and the radius of the incircle is

r

. Three semicircles with diameters

AB

,

BC

and

CA

are drawn on the outside of

ABC

. A circle with radius

t

touches all three semicircles. Prove that

\frac{s}{2} < t \leq \frac{s}{2} + \left(1 - \frac{\sqrt{3}}{2}\right)r.

5

1

Sum of the fourth powers of 2 consecutive integers

Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.

4

1

Pentagon and triangles with equal areas

In a given pentagon

ABCDE

, triangles

ABC

,

BCD

,

CDE

,

DEA

and

EAB

all have the same area. The lines

AC

and

AD

intersect

BE

at points

M

and

N

. Prove that

BM = EN

.

3

1

The first four digits are 1137

The first four digits of a certain positive integer

n

are

1137

. Prove that the digits of

n

can be shuffled in such a way that the new number is divisible by 7.

2

1

The diagonal of a parallelogram is trisected by...

Given a parallelogram

ABCD

, join

A

to the midpoints

E

and

F

of the opposite sides

BC

and

CD

.

AE

and

AF

intersect the diagonal

BD

in

M

and

N

. Prove that

M

and

N

divide

BD

into three equal parts.

1

Too easy

You have five pieces of paper. You pick one or more of them and cut each of them into five smaller pieces. Now you take one or more of the pieces from this lot and cut each of these into five smaller pieces. And so on. Prove that you will never have 2003 pieces.

2003 South africa National Olympiad

Subcontests

This is insane

Sum of the fourth powers of 2 consecutive integers

Pentagon and triangles with equal areas

The first four digits are 1137

The diagonal of a parallelogram is trisected by...

Too easy