MathDB

p1. Sergio has at his disposal a jug containing two liters of tea and a jug containing two liters of milk. Sergio takes a graduated glass and removes

50

^3

of tea and places it in the jug with milk. After stirring the contents, he removes

50

^3

of the mixture and places it in the jug containing tea. Later of these operations, is there more tea in the milk jug or is there more milk in the tea jug?
p2. Prove that it is not possible to construct a triangle with an area greater than

\frac12 a^2

within a square of side

a

.
p3 Prove that there is no such sequence that fulfills simultaneously:

\bullet

a_{n + 1} + a_{n + 2} +... + a_{n + 2002}> 0

for all natural

n

\bullet

a_1 + a_2 +... + a_{2002n} + a_{2002n + 1} <0

for all natural

n

.
p4. A wedding party is organized for Valentine's Day.

K

couples join. The homeowner notices that each person, other than himself, health or a different number of participants. If a person does not greet himself or his spouse, prove that the homeowner is healthy or in the middle of the guests. (In particular, he is rude and so is his wife.).
p5. The number

3

has the following representations as the sum of positive numbers:

3 + 0

2 + 1

1 + 2

1 + 1 + 1

, considering important the order in which it is added, except in the case of

3

. Show that any positive integer n can be expressed in

2^{n-1}

different ways with these considerations of order.
p6. We consider a chessboard of

2002\times 2002

squares.

100.000

towers are placed in the board. Two towers are said to be neighboring if they are still in the same vertical or horizontal position and there is no other tower between them. Prove that the

100,000

towers can be painted using three colors such that two towers neighbors are not painted the same color.
p7. Let

C

be a circle, prove that the

n

-sided polygon inscribed in

C

with greater area is a regular polygon.
Hint: Treat first the case

n = 3

, then

n = 4

and lastly generalize.

Problem Statement