MathDB

p1. You know a character who always lies from Monday to Wednesday and always tells the truth from Thursday to Sunday. What day of the week could he have said "yesterday I lied" and "in two more days I shall tell the truth "? Justify your answer.
p2. Sebastian plays the following solitaire number with numbers: start with

a

b

and

c

, then choose one of them, delete it and replace it with the sum of the other two remaining minus

1

. For example, if we eliminate the number

b

, is replaced by the number a

+ c -1

. After several moves we have the numbers

2001

2003

and

2005

. Say if it is possible that Sebastian started with the triple

(a, b, c) = (2, 2, 2)

.
p3. The following figure represents a river of constant horizontal width, and two houses

A

and

B

located on opposite sides of the river. There is a bridge of width equal to the width of the river that must be placed horizontally and in such a way that the distance from

A

B

is minimal. Determine the location of the bridge. https://cdn.artofproblemsolving.com/attachments/f/a/53e2af5e8dd4df3522014dc66556a276d90475.jpg
p4. Consider all

6

-digit natural numbers of the form

\overline{abcpqr}

, where the digits

\overline{pqr}

corresponds to a permutation of

\overline{abc}

. Let

A

be the sum of all the numbers mentioned above. Prove that

A

is divisible by

13

.
p5. In the constellation Ponce there is an odd number of planets. Observations made in the La Silla observatory determined that all of them are located at different distances in pairs. Also in, that on each planet only the planet that is closest to it is observed. Prove there is a planet in the constellation of Ponce that is not observed by any other planet of the constellation of Ponce.
p6. David and Claudio get together to play the rectangle game. The game consists of coloring integer coordinate points of the plane. David chooses a point and colors it blue, then Claudio chooses uncolored

2002

points and colors them red. Next, David chooses an uncolored point and paints it blue, and so on. David wins if he manages to make a rectangle with his

4

blue vertices, and Claudio wins if he manages to prevent it. Prove that David has a way of winning the game.

Problem Statement