C2: Suppose that 
students are standing in a circle. Prove that there exists an integer 
with 
such that in this circle there exists a contiguous group of 
students, for which the first half contains the same number of girls as the second half.
students are standing in a circle. Prove that there exists an integer with such that in this circle there exists a contiguous group of students, for which the first half contains the same number of girls as the second half.
C4: Determine the greatest positive integer that satisfies the following property: The set of positive integers can be partitioned into subsets 
such that for all integers 
and all 
there exist two distinct elements of 
whose sum is 
.
that satisfies the following property: The set of positive integers can be partitioned into subsets such that for all integers and all there exist two distinct elements of whose sum is .
C5: Let 
be a positive integer, and consider a 
checkerboard consisting of unit squares. At the centre of some of these unit squares there is an ant. At time 
, each ant starts moving with speed 
parallel to some edge of the checkerboard. When two ants moving in the opposite directions meet, they both turn 
clockwise and continue moving with speed . When more than 
ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear.
be a positive integer, and consider a checkerboard consisting of unit squares. At the centre of some of these unit squares there is an ant. At time , each ant starts moving with speed parallel to some edge of the checkerboard. When two ants moving in the opposite directions meet, they both turn clockwise and continue moving with speed . When more than ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear.
Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard, or prove that such a moment does not necessarily exist.
C6: Let be a positive integer, and let 
be an infinite periodic word, consisting of just letters 
and/or 
. Suppose that the minimal period 
of 
is greater than 
.
be a positive integer, and let be an infinite periodic word, consisting of just letters and/or . Suppose that the minimal period of is greater than .
A finite nonempty word 
is said to appear in if there exist indices 
such that 
. A finite word is called ubiquitous if the four words 
, 
, 
, and 
all appear in . Prove that there are at least ubiquitous finite nonempty words.
is said to appear in if there exist indices such that . A finite word is called ubiquitous if the four words , , , and all appear in . Prove that there are at least ubiquitous finite nonempty words.
C7: On a square table of 
by cells we place a finite number of napkins that each cover a square of 
by cells. In each cell we write the number of napkins covering it, and we record the maximal number of cells that all contain the same nonzero number. Considering all possible napkin configurations, what is the largest value of ?
by cells we place a finite number of napkins that each cover a square of by cells. In each cell we write the number of napkins covering it, and we record the maximal number of cells that all contain the same nonzero number. Considering all possible napkin configurations, what is the largest value of ?
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