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International Contests
Tournament Of Towns
2019 Tournament Of Towns
2019 Tournament Of Towns
Part of
Tournament Of Towns
Subcontests
(7)
4
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100 piles of 400 stones each, game with awarded no of points
There are
100
100
100
piles of
400
400
400
stones each. At every move, Pete chooses two piles, removes one stone from each of them, and is awarded the number of points, equal to the non- negative difference between the numbers of stones in two new piles. Pete has to remove all stones. What is the greatest total score Pete can get, if his initial score is
0
0
0
?(Maxim Didin)
a square 101 x101 with all cells except one corner cell are black
Peter has a wooden square stamp divided into a grid. He coated some
102
102
102
cells of this grid with black ink. After that, he pressed this stamp
100
100
100
times on a list of paper so that each time just those
102
102
102
cells left a black imprint on the paper. Is it possible that after his actions the imprint on the list is a square
101
×
101
101 \times 101
101
×
101
such that all the cells except one corner cell are black?(Alexsandr Gribalko)
Quantity phi(n)
We color some positive integers
(
1
,
2
,
.
.
.
,
n
)
(1,2,...,n)
(
1
,
2
,
...
,
n
)
with color red, such that any triple of red numbers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
(not necessarily distincts) if
a
(
b
−
c
)
a(b-c)
a
(
b
−
c
)
is multiple of
n
n
n
then
b
=
c
b=c
b
=
c
. Prove that the quantity of red numbers is less than or equal to
φ
(
n
)
\varphi(n)
φ
(
n
)
.
1
6
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3
5
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6
4
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2
6
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5
6
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