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Problems
Contests
National and Regional Contests
Italy Contests
ITAMO
1994 ITAMO
1994 ITAMO
Part of
ITAMO
Subcontests
(6)
6
1
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1,2,...,100 in a a 10 x 10 chessboard , half negatives, 5 negatives in each row
The squares of a
10
×
10
10 \times 10
10
×
10
chessboard are labelled with
1
,
2
,
.
.
.
,
100
1,2,...,100
1
,
2
,
...
,
100
in the usual way: the
i
i
i
-th row contains the numbers
10
i
−
9
,
10
i
−
8
,
.
.
.
,
10
i
10i -9,10i - 8,...,10i
10
i
−
9
,
10
i
−
8
,
...
,
10
i
in increasing order. The signs of fifty numbers are changed so that each row and each column contains exactly five negative numbers. Show that after this change the sum of all numbers on the chessboard is zero.
5
1
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min, max area of the intersection of cube with plane passing through diagonal
Let
O
P
OP
OP
be a diagonal of a unit cube. Find the minimum and the maximum value of the area of the intersection of the cube with a plane through
O
P
OP
OP
.
4
1
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concurrency with reflection points and parallel
Let
A
B
C
ABC
A
BC
be a triangle contained in one of the halfplanes determined by a line
r
r
r
. Points
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
are the reflections of
A
,
B
,
C
A,B,C
A
,
B
,
C
in
r
,
r,
r
,
respectively. Consider the line through
A
′
A'
A
′
parallel to
B
C
BC
BC
, the line through
B
′
B'
B
′
parallel to
A
C
AC
A
C
and the line through
C
′
C'
C
′
parallel to
A
B
AB
A
B
. Show that these three lines have a common point.
3
1
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journalist on island of scoundrels and knights, true or false always
A journalist wants to report on the island of scoundrels and knights, where all inhabitants are either scoundrels (and they always lie) or knights (and they always tell the truth). The journalist interviews each inhabitant exactly once and gets the following answers:
A
1
A_1
A
1
: On this island there is at least one scoundrel,
A
2
A_2
A
2
: On this island there are at least two scoundrels,
.
.
.
...
...
A
n
−
1
A_{n-1}
A
n
−
1
: On this island there are at least
n
−
1
n-1
n
−
1
scoundrels,
A
n
A_n
A
n
: On this island everybody is a scoundrel. Can the journalist decide whether there are more scoundrels or more knights?
1
1
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exists N, such for all n>=N a square can be partitioned into n smaller squares.
Show that there exists an integer
N
N
N
such that for all
n
≥
N
n \ge N
n
≥
N
a square can be partitioned into
n
n
n
smaller squares.
2
1
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another problem
solve this diophantine equation y^2 = x^3 - 16