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Problems
Contests
National and Regional Contests
Italy Contests
ITAMO
1995 ITAMO
1995 ITAMO
Part of
ITAMO
Subcontests
(6)
1
1
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tiling a square of side n with an 9 piece 1-3-5 dominos
Determine for which values of
n
n
n
it is possible to tile a square of side
n
n
n
with figures of the type shown in the picture[asy] unitsize(0.4 cm);draw((0,0)--(5,0)); draw((0,1)--(5,1)); draw((1,2)--(4,2)); draw((2,3)--(3,3)); draw((0,0)--(0,1)); draw((1,0)--(1,2)); draw((2,0)--(2,3)); draw((3,0)--(3,3)); draw((4,0)--(4,2)); draw((5,0)--(5,1)); [/asy]
2
1
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20 students with integer scores in maths
No two of
20
20
20
students in a class have the same scores on both written and oral examinations in mathematics. We say that student
A
A
A
is better than
B
B
B
if his two scores are greater than or equal to the corresponding scores of
B
B
B
. The scores are integers between
1
1
1
and
10
10
10
. (a) Show that there exist three students
A
,
B
,
C
A,B,C
A
,
B
,
C
such that
A
A
A
is better than
B
B
B
and
B
B
B
is better than
C
C
C
. (b) Would the same be true for a class of
19
19
19
students?
3
1
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a drunkard wandering around 4 pubs
In a town there are four pubs,
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
, and any two of them are connected to each other except
A
A
A
and
D
D
D
. A drunkard wanders about the pubs starting with
A
A
A
and, after having a drink, goes to any of the pubs directly connected, with equal probability. (a) What is the probability that the drunkard is at
C
C
C
at its fifth drink? (b) Where is the drunkard most likely to be after
n
n
n
drinks (
n
>
5
n > 5
n
>
5
)?
4
1
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equal segments wanted, circumcircle and angle bisector related
An acute-angled triangle
A
B
C
ABC
A
BC
is inscribed in a circle with center
O
O
O
. The bisector of
∠
A
\angle A
∠
A
meets
B
C
BC
BC
at
D
D
D
, and the perpendicular to
A
O
AO
A
O
through
D
D
D
meets the segment
A
C
AC
A
C
in a point
P
P
P
. Show that
A
B
=
A
P
AB = AP
A
B
=
A
P
.
5
1
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tangent circles in space with same tangents at common point lie on sphere
Two non-coplanar circles in space are tangent at a point and have the same tangents at this point. Show that both circles lie on some sphere.
6
1
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diophantine x^2 +615 = 2^y
Find all pairs of positive integers
x
,
y
x,y
x
,
y
such that
x
2
+
615
=
2
y
x^2 +615 = 2^y
x
2
+
615
=
2
y