IT-CESENATICO 2013

A model car is tested on some closed circuit $600$ meters long, consisting of flat stretches, uphill and downhill. All uphill and downhill have the same slope. The test highlights the following facts:

[list] (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 $v_s, v_p$ and $v_d$ respectively, we have $v_s <v_p <v_d$ ;

(b) $v_s,v_p$ and $v_d$ , 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 $50$ seconds. [/list]

Find all possible values of $v_s, v_p$ and $v_d$ .

In triangle $ABC$ , suppose we have $a> b$ , where $a=BC$ and $b=AC$ . Let $M$ be the midpoint of $AB$ , and $\alpha, \beta$ are inscircles of the triangles $ACM$ and $BCM$ respectively. Let then $A'$ and $B'$ be the points of tangency of $\alpha$ and $\beta$ on $CM$ . Prove that $A'B'=\frac{a - b}{2}$ .

Each integer is colored with one of two colors, red or blue. It is known that, for every finite set $A$ of consecutive integers, the absolute value of the difference between the number of red and blue integers in the set $A$ is at most $1000$ . Prove that there exists a set of $2000$ consecutive integers in which there are exactly $1000$ red numbers and $1000$ numbers blue.

$\overline{5654}_b$ is a power of a prime number. Find $b$ if $b > 6$ .

$ABC$ is an isosceles triangle with $AB=AC$ and the angle in $A$ is less than $60^{\circ}$ . Let $D$ be a point on $AC$ such that $\angle{DBC}=\angle{BAC}$ . $E$ is the intersection between the perpendicular bisector of $BD$ and the line parallel to $BC$ passing through $A$ . $F$ is a point on the line $AC$ such that $FA=2AC$ ( $A$ is between $F$ and $C$ ). Show that $EB$ and $AC$ are parallel and that the perpendicular from $F$ to $AB$ , the perpendicular from $E$ to $AC$ and $BD$ are concurrent.

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 $n \times n$ , at his (Daniel's) discretion, a token black or white. Then the first magician asks Daniel to show him a square $C$ of his own choice. At this point, the first magician chooses a square $D$ (not necessarily different from $C$ ) and replaces the token that is on $D$ 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 $C$ . For what value of $n$ , the two magicians have a strategy such that the second magician makes a successful guess.