2015 ITAMO

Part of ITAMO

Subcontests

(6)

1

10 musical genres, 17 songs by user, the next by an algorithm

A music streaming service proposes songs classified in

10

musical genres, so that each song belong to one and only one gender. The songs are played one after the other: the first

17

are chosen by the user, but starting from the eighteenth the service automatically determines which song to play. Elisabetta has noticed that, if one makes the classification of which genres they appear several times during the last

17

songs played, the new song always belongs to the genre at the top of the ranking or, in case of same merit, at one of the first genres. Prove that, however, the first

17

tracks are chosen, from a certain point onwards the songs proposed are all of the same kind.

1

All pairs of integers satisfying a^3+b^3+3ab=1

Determine all pairs of integers

(a, b)

that solve the equation

a^3 + b^3 + 3ab = 1

1

Problem 1, Itamo 2015

Let ABCDA'B'C'D' be a rectangular parallelipiped, where ABCD is the lower face and A, B, C and D' are below A', B', C' and D', respectively. The parallelipiped is divided into eight parts by three planes parallel to its faces. For each vertex P, let V P denote the volume of the part containing P. Given that V A= 40, V C = 300 , V B' = 360 and V C'= 90, find the volume of ABCDA'B'C'D'.

1

Itamo 2015, problem 6, hard combinatorics

Ada and Charles play the following game:at the beginning, an integer n>1 is written on the blackboard.In turn, Ada and Charles remove the number k that they find on the blackboard.In turn Ad and Charles remove the number k that they find on the blackboard and they replace it : 1 -either with a positive divisor k different from 1 and k 2- or with k+1 At the beginning each players have a thousand points each.When a player choses move 1, he/she gains one point;when a player choses move 2, he/she loses one point.The game ends when one of the tho players is left with zero points and this player loses the game.Ada moves first.For what values Chares has a winning strategy?

1

Itamo 2015, problem 3

Let ABC a triangle, let K be the foot of the bisector relative to BC and J be the foot of the trisectrix relative to BC closer to the side AC (3* m(JAC)=m(CAB) ). Let C' and B' be two point on the line AJ on the side of J with respect to A, such that AC'=AC and AB=AB'. Prove that ABB'C is cyclic if and only if lines C'K and BB' are parallel.

1

Tangent circles and equality of segments

AB

be a chord of a circle

\Gamma

C

be a point on the segment

AB

r

be a line through

C

which intersects

\Gamma

D,E

D,E

lie on different sides with respect to the perpendicular bisector of

AB

\Gamma_D

be the circumference which is externally tangent to

\Gamma

D

and touches the line

AB

F

\Gamma_E

be the circumference which is externally tangent to

\Gamma

E

and touches the line

AB

G

CA=CB

CF=CG