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Contests
International Contests
International Olympiad of Metropolises
2016 IOM
2016 IOM
Part of
International Olympiad of Metropolises
Subcontests
(6)
3
1
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International Olympiad of metropolises 2016 P3
Let
A
1
A
2
.
.
.
A
n
A_1A_2 . . . A_n
A
1
A
2
...
A
n
be a cyclic convex polygon whose circumcenter is strictly in its interior. Let
B
1
,
B
2
,
.
.
.
,
B
n
B_1, B_2, ..., B_n
B
1
,
B
2
,
...
,
B
n
be arbitrary points on the sides
A
1
A
2
,
A
2
A
3
,
.
.
.
,
A
n
A
1
A_1A_2, A_2A_3, ..., A_nA_1
A
1
A
2
,
A
2
A
3
,
...
,
A
n
A
1
, respectively, other than the vertices. Prove that
B
1
B
2
A
1
A
3
+
B
2
B
3
A
2
A
4
+
.
.
.
+
B
n
B
1
A
n
A
2
>
1
\frac{B_1B_2}{A_1A_3}+ \frac{B_2B_3}{A_2A_4}+...+\frac{B_nB_1}{A_nA_2}>1
A
1
A
3
B
1
B
2
+
A
2
A
4
B
2
B
3
+
...
+
A
n
A
2
B
n
B
1
>
1
.
6
1
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International Olympiad of metropolises 2016 P6
In a country with
n
n
n
cities, some pairs of cities are connected by one-way flights operated by one of two companies
A
A
A
and
B
B
B
. Two cities can be connected by more than one flight in either direction. An
A
B
AB
A
B
-word
w
w
w
is called implementable if there is a sequence of connected flights whose companies’ names form the word
w
w
w
. Given that every
A
B
AB
A
B
-word of length
2
n
2^n
2
n
is implementable, prove that every finite
A
B
AB
A
B
-word is implementable. (An
A
B
AB
A
B
-word of length
k
k
k
is an arbitrary sequence of
k
k
k
letters
A
A
A
or
B
B
B
; e.g.
A
A
B
A
AABA
AA
B
A
is a word of length
4
4
4
.)
5
1
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International Olympiad of metropolises 2016 P5
Let
r
(
x
)
r(x)
r
(
x
)
be a polynomial of odd degree with real coefficients. Prove that there exist only finitely many (or none at all) pairs of polynomials
p
(
x
)
p(x)
p
(
x
)
and
q
(
x
)
q(x)
q
(
x
)
with real coefficients satisfying the equation
(
p
(
x
)
)
3
+
q
(
x
2
)
=
r
(
x
)
(p(x))^3 + q(x^2) = r(x)
(
p
(
x
)
)
3
+
q
(
x
2
)
=
r
(
x
)
.
4
1
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International Olympiad of metropolises 2016 P4
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
has right angles at
A
A
A
and
C
C
C
. A point
E
E
E
lies on the extension of the side
A
D
AD
A
D
beyond
D
D
D
so that
∠
A
B
E
=
∠
A
D
C
\angle ABE =\angle ADC
∠
A
BE
=
∠
A
D
C
. The point
K
K
K
is symmetric to the point
C
C
C
with respect to point
A
A
A
. Prove that
∠
A
D
B
=
∠
A
K
E
\angle ADB =\angle AKE
∠
A
D
B
=
∠
A
K
E
.
2
1
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International olympiad of metropolises 2016 P2
Let
a
1
,
.
.
.
,
a
n
a_1, . . . , a_n
a
1
,
...
,
a
n
be positive integers satisfying the inequality
∑
i
=
1
n
1
a
n
≤
1
2
\sum_{i=1}^{n}\frac{1}{a_n}\le \frac{1}{2}
∑
i
=
1
n
a
n
1
≤
2
1
. Every year, the government of Optimistica publishes its Annual Report with n economic indicators. For each
i
=
1
,
.
.
.
,
n
i = 1, . . . , n
i
=
1
,
...
,
n
,the possible values of the
i
−
t
h
i-th
i
−
t
h
indicator are
1
,
2
,
.
.
.
,
a
i
1, 2, . . . , a_i
1
,
2
,
...
,
a
i
. The Annual Report is said to be optimistic if at least
n
−
1
n - 1
n
−
1
indicators have higher values than in the previous report. Prove that the government can publish optimistic Annual Reports in an infinitely long sequence.
1
1
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International olympiad of metropolises 2016 P1
Find all positive integers
n
n
n
such that there exist
n
n
n
consecutive positive integers whose sum is a perfect square.