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International Mathematical Olympic Revenge
2017 International Olympic Revenge
2017 International Olympic Revenge
Part of
International Mathematical Olympic Revenge
Subcontests
(4)
4
1
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IMOR 2017 - Problem 4
Let
n
>
1
n>1
n
>
1
be a positive integer. Ana and Bob play a game with other
n
n
n
people. The group of
n
n
n
people form a circle, and Bob will put either a black hat or a white one on each person's head. Each person can see all the hats except for his own one. They will guess the color of his own hat individually.Before Bob distribute their hats, Ana gives
n
n
n
people a strategy which is the same for everyone. For example, it could be "guessing the color just on your left" or "if you see an odd number of black hats, then guess black; otherwise, guess white".Ana wants to maximize the number of people who guesses the right color, and Bob is on the contrary.Now, suppose Ana and Bob are clever enough, and everyone forms a strategy strictly. How many right guesses can Ana guarantee?Proposed by China.
3
1
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IMOR 2017 - Problem 3
Let
A
B
C
ABC
A
BC
be a triangle, and let
P
P
P
be a distinct point on the plane. Moreover, let
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
be a homothety of
A
B
C
ABC
A
BC
with ratio
2
2
2
and center
P
P
P
, and let
O
O
O
and
O
′
O'
O
′
be the circumcenters of
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
, respectively. The circumcircles of
A
B
′
C
′
AB'C'
A
B
′
C
′
,
A
′
B
C
′
A'BC'
A
′
B
C
′
, and
A
′
B
′
C
A'B'C
A
′
B
′
C
meet at points
X
X
X
,
Y
Y
Y
, and
Z
Z
Z
, different from
A
′
A'
A
′
,
B
′
B'
B
′
, and
C
′
C'
C
′
. In a similar way, the circumcircles of
A
′
B
C
A'BC
A
′
BC
,
A
B
′
C
AB'C
A
B
′
C
, and
A
B
C
′
ABC'
A
B
C
′
meet at
X
′
X'
X
′
,
Y
′
Y'
Y
′
, and
Z
′
Z'
Z
′
, different from
A
A
A
,
B
B
B
,
C
C
C
. Let
W
W
W
and
W
′
W'
W
′
be the circumcenters of
X
Y
Z
XYZ
X
Y
Z
and
X
′
Y
′
Z
′
X'Y'Z'
X
′
Y
′
Z
′
, respectively. Prove that
O
W
OW
O
W
is parallel to
O
′
W
′
O'W'
O
′
W
′
.Proposed by Mateus Thimóteo, Brazil.
2
1
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IMOR 2017 - Problem 2
A polynomial is good if it has integer coefficients, it is monic, all its roots are distinct, and there exists a disk with radius
0.99
0.99
0.99
on the complex plane that contains all the roots. Prove that there is no good polynomial for a sufficient large degree.Proposed by Rodrigo Sanches Angelo (rsa365), Brazil.
1
1
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IMOR 2017 - Problem 1
Let
f
(
x
)
f(x)
f
(
x
)
be the distance from
x
x
x
to the nearest perfect square. For example,
f
(
π
)
=
4
−
π
f(\pi) = 4 - \pi
f
(
π
)
=
4
−
π
. Let
α
=
3
+
5
2
\alpha = \frac{3 + \sqrt{5}}{2}
α
=
2
3
+
5
and let
m
m
m
be an integer such that the sequence
a
n
=
f
(
m
α
n
)
a_n = f(m \; \alpha^n)
a
n
=
f
(
m
α
n
)
is bounded. Prove that either
m
=
k
2
m=k^2
m
=
k
2
or
m
=
5
k
2
m = 5k^2
m
=
5
k
2
for some integer
k
k
k
.Proposed by Rodrigo Sanches Angelo (rsa365), Brazil.