MathDB

1

Array of light bulbs and switches

100

light bulbs, arranged in a

10

by

10

square array. Some of them are on, the others are off. The electrical system is such that when the switch corresponding to a light bulb is pressed, all the light bulbs that are on the same row or column of it (including the bulb linked to the pressed switch) change their state (that is they are turned on or off).
[*] From which starting configurations, pressing the right sequence of switches, is it possible to achieve that all bulbs are on at the same time?
[*] What is the answer to the previous question if the bulbs are

81

, arranged in a

9

by

9

panel?

5

1

Incircle and length of segments

Let

ABC

be a triangle and

\gamma

the circle inscribed in

ABC

. The circle

\gamma

is tangent to side

AB

at the point

T

. Let

D

be the point of

\gamma

diametrically opposite to

T

, and

S

the intersection point of the line through

C

and

D

with side

AB

. Prove that

AT=SB

.

3

1

Two-variable exponential diophantine equation with 2001

Consider the equation

x^{2001}=y^x .

[*] Find all pairs

(x,y)

of solutions where

x

is a prime number and

y

is a positive integer.
[*] Find all pairs

(x,y)

of solutions where

x

and

y

are positive integers.
(Remember that

2001=3 \cdot 23 \cdot 29

.)

4

1

Italian Mathematical Olympiad 2001 - Problem 4

A positive integer is called monotone if has at least two digits and all its digits are nonzero and appear in a strictly increasing or strictly decreasing order.
(a) Compute the sum of all monotone five-digit numbers. (b) Find the number of final zeros in the least common multiple of all monotone numbers (with any number of digits).

2

1

How many teams in a tournament with certain point results?

In a basketball tournament every two teams play two matches. As usual, the winner of a match gets

2

points, the loser gets

0

, and there are no draws. A single team wins the tournament with

26

points and exactly two teams share the last position with

20

points. How many teams participated in the tournament?

1

Italian Mathematical Olympiad 2001 - Problem 1

A hexagon has all its angles equal, and the lengths of four consecutive sides are

5

,

3

,

6

and

7

, respectively. Find the lengths of the remaining two edges.

2001 ITAMO

Subcontests

Array of light bulbs and switches

Incircle and length of segments

Two-variable exponential diophantine equation with 2001

Italian Mathematical Olympiad 2001 - Problem 4

How many teams in a tournament with certain point results?

Italian Mathematical Olympiad 2001 - Problem 1