1. A model car is tested on some closed circuit 
meters long, consisting of flat stretches, uphill and downhill. All uphill and downhill have the same slope. The test highlights the following facts:
meters long, consisting of flat stretches, uphill and downhill. All uphill and downhill have the same slope. The test highlights the following facts:
(a) The velocity of the car depends only on the fact that the car is driving along a stretch of uphill, plane or downhill; calling these three velocities 
and 
respectively, we have 
;
and respectively, we have ;
(b) 
and , expressed in meter per second, are integers.
and , expressed in meter per second, are integers.
(c) Whatever may be the structure of the circuit, the time taken to complete the circuit is always 
seconds.
seconds.
Find all possible values of and .
and .
2. In triangle 
, suppose we have 
, where 
and 
. Let 
be the midpoint of 
, and 
are inscircles of the triangles 
and 
respectively. Let then 
and 
be the points of angency of 
and 
on 
. Prove that
, suppose we have , where and . Let be the midpoint of , and are inscircles of the triangles and respectively. Let then and be the points of angency of and on . Prove that
3. Each integer is colored with one of two colors, red or blue. It is known that, for every finite set 
of consecutive integers, the absolute value of the difference between the number of red and blue integers in the set is at most 
. Prove that there exists a set of 
consecutive integers in which there are exactly red numbers and numbers blue.
of consecutive integers, the absolute value of the difference between the number of red and blue integers in the set is at most . Prove that there exists a set of consecutive integers in which there are exactly red numbers and numbers blue. 
5. is an isosceles triangle with 
and the angle in is less than 
. Let 
be a point on 
such that 
. 
is the intersection between the perpendicular bisector of 
and the line parallel to 
passing through . 
is a point on the line such that 
( is between and 
).
is an isosceles triangle with and the angle in is less than . Let be a point on such that . is the intersection between the perpendicular bisector of and the line parallel to passing through . is a point on the line such that ( is between and ).
Show that 
and are parallel and that the perpendicular from to , the perpendicular from to are concurrent.
and are parallel and that the perpendicular from to , the perpendicular from to are concurrent.
6 Two magicians are performing the following game. Initially the first magician encloses the second magician in a cabin where he can neither see nor hear anything. To start the game, the first magician invites Daniel, from the audience, to put on each square of a chessboard 
, at his (Daniel's) discretion, a token black or white. Then the first magician asks Daniel to show him a square of his own choice. At this point, the first magician chooses a square (not necessarily different from ) and replaces the token that is on with other color token (white with black or black with white).
, at his (Daniel's) discretion, a token black or white. Then the first magician asks Daniel to show him a square of his own choice. At this point, the first magician chooses a square (not necessarily different from ) and replaces the token that is on with other color token (white with black or black with white).
Then he opens the cabin in which the second magician was held. Looking at the chessboard, the second magician guesses what is the square . For what value of 
, the two magicians have a strategy such that the second magician makes a successful guess.
. For what value of , the two magicians have a strategy such that the second magician makes a successful guess.
Italian Mathematical Olympiad 2013
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