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Problems
Contests
International Contests
International Olympiad of Metropolises
2017 IOM
2017 IOM
Part of
International Olympiad of Metropolises
Subcontests
(6)
6
1
Hide problems
Hexagon and a lot of circles
et
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon which has an inscribed circle and a circumcribed. Denote by
ω
A
,
ω
B
,
ω
C
,
ω
D
,
ω
E
\omega_{A}, \omega_{B},\omega_{C},\omega_{D},\omega_{E}
ω
A
,
ω
B
,
ω
C
,
ω
D
,
ω
E
and
ω
F
\omega_{F}
ω
F
the inscribed circles of the triangles
F
A
B
,
A
B
C
,
B
C
D
,
C
D
E
,
D
E
F
FAB, ABC, BCD, CDE, DEF
F
A
B
,
A
BC
,
BC
D
,
C
D
E
,
D
EF
and
E
F
A
EFA
EF
A
, respecitively. Let
l
A
B
l_{AB}
l
A
B
, be the external of
ω
A
\omega_{A}
ω
A
and
ω
B
\omega_{B}
ω
B
; lines
l
B
C
l_{BC}
l
BC
,
l
C
D
l_{CD}
l
C
D
,
l
D
E
l_{DE}
l
D
E
,
l
E
F
l_{EF}
l
EF
,
l
F
A
l_{FA}
l
F
A
are analoguosly defined. Let
A
1
A_1
A
1
be the intersection point of the lines
l
F
A
l_{FA}
l
F
A
and
l
A
B
l_{AB}
l
A
B
,
B
1
,
C
1
,
D
1
,
E
1
,
F
1
B_1, C_1, D_1, E_1, F_1
B
1
,
C
1
,
D
1
,
E
1
,
F
1
are analogously defined. Prove that
A
1
D
1
,
B
1
E
1
,
C
1
F
1
A_1D_1, B_1E_1, C_1F_1
A
1
D
1
,
B
1
E
1
,
C
1
F
1
are concurrent.
5
1
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LCM of numbers
Let
x
x
x
and
y
y
y
be positive integers such that
[
x
+
2
,
y
+
2
]
−
[
x
+
1
,
y
+
1
]
=
[
x
+
1
,
y
+
1
]
−
[
x
,
y
]
[x+2,y+2]-[x+1,y+1]=[x+1,y+1]-[x,y]
[
x
+
2
,
y
+
2
]
−
[
x
+
1
,
y
+
1
]
=
[
x
+
1
,
y
+
1
]
−
[
x
,
y
]
.Prove that one of the two numbers
x
x
x
and
y
y
y
divide the other.(Here
[
a
,
b
]
[a,b]
[
a
,
b
]
denote the least common multiple of
a
a
a
and
b
b
b
).Proposed by Dusan Djukic.
4
1
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Easy combo
Find the largest positive integer
N
N
N
for which one can choose
N
N
N
distinct numbers from the set
1
,
2
,
3
,
.
.
.
,
100
{1,2,3,...,100}
1
,
2
,
3
,
...
,
100
such that neither the sum nor the product of any two different chosen numbers is divisible by
100
100
100
.Proposed by Mikhail Evdokimov
1
1
Hide problems
Simple parallelogram geo
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram in which angle at
B
B
B
is obtuse and
A
D
>
A
B
AD>AB
A
D
>
A
B
. Points
K
K
K
and
L
L
L
on
A
C
AC
A
C
such that
∠
A
D
L
=
∠
K
B
A
\angle ADL=\angle KBA
∠
A
D
L
=
∠
K
B
A
(the points
A
,
K
,
C
,
L
A, K, C, L
A
,
K
,
C
,
L
are all different, with
K
K
K
between
A
A
A
and
L
L
L
). The line
B
K
BK
B
K
intersects the circumcircle
ω
\omega
ω
of
A
B
C
ABC
A
BC
at points
B
B
B
and
E
E
E
, and the line
E
L
EL
E
L
intersects
ω
\omega
ω
at points
E
E
E
and
F
F
F
. Prove that
B
F
∣
∣
A
C
BF||AC
BF
∣∣
A
C
.
2
1
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Even paths in graph
In a country there are two-way non-stopflights between some pairs of cities. Any city can be reached from any other by a sequence of at most
100
100
100
flights. Moreover, any city can be reached from any other by a sequence of an even number of flights. What is the smallest
d
d
d
for which one can always claim that any city can be reached from any other by a sequence of an even number of flights not exceeding
d
d
d
?
3
1
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Composition of polinomials
Let
Q
Q
Q
be a quadriatic polynomial having two different real zeros. Prove that there is a non-constant monic polynomial
P
P
P
such that all coefficients of the polynomial
Q
(
P
(
x
)
)
Q(P(x))
Q
(
P
(
x
))
except the leading one are (by absolute value) less than
0.001
0.001
0.001
.