1η Ημέρα
1. Let 
be a parallelogram and 
be a point on diagonal 
such that 
. Circumcircle of triangle 
intersects line 
at points 
and 
. Prove that
be a parallelogram and be a point on diagonal such that . Circumcircle of triangle intersects line at points and . Prove that 
3. A fly and 
spiders are placed in some vertices of 
lattice. One move consists of following: firstly, fly goes to some adjacent vertex or stays where it is and then every spider goes to some adjacent vertex or stays where it is (more than one spider can be in the same vertex). Spiders and fly knows where are the others all the time.
spiders are placed in some vertices of lattice. One move consists of following: firstly, fly goes to some adjacent vertex or stays where it is and then every spider goes to some adjacent vertex or stays where it is (more than one spider can be in the same vertex). Spiders and fly knows where are the others all the time. a) Find the smallest so that spiders can catch the fly in finite number of moves, regardless of their initial position.
so that spiders can catch the fly in finite number of moves, regardless of their initial position. b) Answer the same question for three-dimensional lattice 
.
.(Vertices in lattice are adjacent if exactly one coordinate of one vertex is different from the same coordinate of the other vertex, and their difference is equal to 
. Spider catches a fly if they are in the same vertex.)
. Spider catches a fly if they are in the same vertex.) 1. Find all natural numbers 
for which there is a permutation 
of numbers 
such that sets 
and 
are complete residue systems 
.
for which there is a permutation of numbers such that sets and are complete residue systems . 2. Let 
be two-dimensional integer lattice. Is there a bijection 
, such that for every distinct 
we have: 
be two-dimensional integer lattice. Is there a bijection , such that for every distinct we have: 3. We are given 
piles of coins. There are two different types of coins: real and fake coins; they all look alike, but coins of the same type have the same mass, while the coins from different types have different masses. Coins that belong to the same pile are of the same type. We know the mass of real coin.
piles of coins. There are two different types of coins: real and fake coins; they all look alike, but coins of the same type have the same mass, while the coins from different types have different masses. Coins that belong to the same pile are of the same type. We know the mass of real coin.Find the minimal number of weightings on digital scale that we need in order to conclude: which piles consists of which type of coins and also the mass of fake coin. (We assume that every pile consists from infinite number of coins.)
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