2013 ITAMO

Part of ITAMO

Subcontests

(6)

1

Strategic game involving guessing the right square

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.

1

Possible values of velocities

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:
(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

v_d

respectively, we have

v_s <v_p <v_d

v_s,v_p

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.
Find all possible values of

v_s, v_p

v_d

1

5654 is a power of a prime number in base b

\overline{5654}_b

is a power of a prime number. Find

b

b > 6

1

Difference between number of red & blue integers ≤ 1000

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

1000

. Prove that there exists a set of

2000

consecutive integers in which there are exactly

1000

red numbers and

1000

1

M is the midpoint of AB, incircles of ACM, BCM

ABC

, suppose we have

a> b

a=BC

b=AC

M

be the midpoint of

AB

\alpha, \beta

are inscircles of the triangles

ACM

BCM

respectively. Let then

A'

B'

be the points of tangency of

\alpha

\beta

CM

A'B'=\frac{a - b}{2}

1

parallelism and concurrency

ABC

is an isosceles triangle with

AB=AC

and the angle in

A

60^{\circ}

D

AC

\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

FA=2AC

A

F

C

EB

AC

are parallel and that the perpendicular from

F

AB

, the perpendicular from

E

AC

BD

are concurrent.