MathDB
Problems
Contests
International Contests
EGMO
2013 EGMO
2013 EGMO
Part of
EGMO
Subcontests
(6)
6
1
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Snow White and the Seven Dwarves
Snow White and the Seven Dwarves are living in their house in the forest. On each of
16
16
16
consecutive days, some of the dwarves worked in the diamond mine while the remaining dwarves collected berries in the forest. No dwarf performed both types of work on the same day. On any two different (not necessarily consecutive) days, at least three dwarves each performed both types of work. Further, on the first day, all seven dwarves worked in the diamond mine.Prove that, on one of these
16
16
16
days, all seven dwarves were collecting berries.
5
1
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Tangency point of mixtilinear incircle is isogonal to...
Let
Ω
\Omega
Ω
be the circumcircle of the triangle
A
B
C
ABC
A
BC
. The circle
ω
\omega
ω
is tangent to the sides
A
C
AC
A
C
and
B
C
BC
BC
, and it is internally tangent to the circle
Ω
\Omega
Ω
at the point
P
P
P
. A line parallel to
A
B
AB
A
B
intersecting the interior of triangle
A
B
C
ABC
A
BC
is tangent to
ω
\omega
ω
at
Q
Q
Q
.Prove that
∠
A
C
P
=
∠
Q
C
B
\angle ACP = \angle QCB
∠
A
CP
=
∠
QCB
.
4
1
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P(n) takes integer values at three consecutive integers
Find all positive integers
a
a
a
and
b
b
b
for which there are three consecutive integers at which the polynomial
P
(
n
)
=
n
5
+
a
b
P(n) = \frac{n^5+a}{b}
P
(
n
)
=
b
n
5
+
a
takes integer values.
3
1
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Constructing sets of positive integers
Let
n
n
n
be a positive integer.(a) Prove that there exists a set
S
S
S
of
6
n
6n
6
n
pairwise different positive integers, such that the least common multiple of any two elements of
S
S
S
is no larger than
32
n
2
32n^2
32
n
2
.(b) Prove that every set
T
T
T
of
6
n
6n
6
n
pairwise different positive integers contains two elements the least common multiple of which is larger than
9
n
2
9n^2
9
n
2
.
2
1
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Dissecting a rectangle
Determine all integers
m
m
m
for which the
m
×
m
m \times m
m
×
m
square can be dissected into five rectangles, the side lengths of which are the integers
1
,
2
,
3
,
…
,
10
1,2,3,\ldots,10
1
,
2
,
3
,
…
,
10
in some order.
1
1
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AD=BE implies ABC right
The side
B
C
BC
BC
of the triangle
A
B
C
ABC
A
BC
is extended beyond
C
C
C
to
D
D
D
so that
C
D
=
B
C
CD = BC
C
D
=
BC
. The side
C
A
CA
C
A
is extended beyond
A
A
A
to
E
E
E
so that
A
E
=
2
C
A
AE = 2CA
A
E
=
2
C
A
. Prove that, if
A
D
=
B
E
AD=BE
A
D
=
BE
, then the triangle
A
B
C
ABC
A
BC
is right-angled.