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International Contests
Romanian Masters of Mathematics Collection
2010 Romanian Master of Mathematics
2010 Romanian Master of Mathematics
Part of
Romanian Masters of Mathematics Collection
Subcontests
(6)
6
1
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Romanian Masters in mathematics 2010 Day 2 Problem 3
Given a polynomial
f
(
x
)
f(x)
f
(
x
)
with rational coefficients, of degree
d
≥
2
d \ge 2
d
≥
2
, we define the sequence of sets
f
0
(
Q
)
,
f
1
(
Q
)
,
…
f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots
f
0
(
Q
)
,
f
1
(
Q
)
,
…
as
f
0
(
Q
)
=
Q
f^0(\mathbb{Q})=\mathbb{Q}
f
0
(
Q
)
=
Q
,
f
n
+
1
(
Q
)
=
f
(
f
n
(
Q
)
)
f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))
f
n
+
1
(
Q
)
=
f
(
f
n
(
Q
))
for
n
≥
0
n\ge 0
n
≥
0
. (Given a set
S
S
S
, we write
f
(
S
)
f(S)
f
(
S
)
for the set
{
f
(
x
)
∣
x
∈
S
}
)
\{f(x)\mid x\in S\})
{
f
(
x
)
∣
x
∈
S
})
. Let
f
ω
(
Q
)
=
⋂
n
=
0
∞
f
n
(
Q
)
f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})
f
ω
(
Q
)
=
⋂
n
=
0
∞
f
n
(
Q
)
be the set of numbers that are in all of the sets
f
n
(
Q
)
f^n(\mathbb{Q})
f
n
(
Q
)
,
n
≥
0
n\geq 0
n
≥
0
. Prove that
f
ω
(
Q
)
f^{\omega}(\mathbb{Q})
f
ω
(
Q
)
is a finite set.Dan Schwarz, Romania
5
1
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Romanian Masters in mathematics 2010 Day 2 Problem 2
Let
n
n
n
be a given positive integer. Say that a set
K
K
K
of points with integer coordinates in the plane is connected if for every pair of points
R
,
S
∈
K
R, S\in K
R
,
S
∈
K
, there exists a positive integer
ℓ
\ell
ℓ
and a sequence
R
=
T
0
,
T
1
,
T
2
,
…
,
T
ℓ
=
S
R=T_0,T_1, T_2,\ldots ,T_{\ell}=S
R
=
T
0
,
T
1
,
T
2
,
…
,
T
ℓ
=
S
of points in
K
K
K
, where each
T
i
T_i
T
i
is distance
1
1
1
away from
T
i
+
1
T_{i+1}
T
i
+
1
. For such a set
K
K
K
, we define the set of vectors
Δ
(
K
)
=
{
R
S
→
∣
R
,
S
∈
K
}
\Delta(K)=\{\overrightarrow{RS}\mid R, S\in K\}
Δ
(
K
)
=
{
RS
∣
R
,
S
∈
K
}
What is the maximum value of
∣
Δ
(
K
)
∣
|\Delta(K)|
∣Δ
(
K
)
∣
over all connected sets
K
K
K
of
2
n
+
1
2n+1
2
n
+
1
points with integer coordinates in the plane?Grigory Chelnokov, Russia
4
1
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Romanian Masters in mathematics 2010 Day 2 Problem 1
Determine whether there exists a polynomial
f
(
x
1
,
x
2
)
f(x_1, x_2)
f
(
x
1
,
x
2
)
with two variables, with integer coefficients, and two points
A
=
(
a
1
,
a
2
)
A=(a_1, a_2)
A
=
(
a
1
,
a
2
)
and
B
=
(
b
1
,
b
2
)
B=(b_1, b_2)
B
=
(
b
1
,
b
2
)
in the plane, satisfying the following conditions:(i)
A
A
A
is an integer point (i.e
a
1
a_1
a
1
and
a
2
a_2
a
2
are integers);(ii)
∣
a
1
−
b
1
∣
+
∣
a
2
−
b
2
∣
=
2010
|a_1-b_1|+|a_2-b_2|=2010
∣
a
1
−
b
1
∣
+
∣
a
2
−
b
2
∣
=
2010
;(iii)
f
(
n
1
,
n
2
)
>
f
(
a
1
,
a
2
)
f(n_1, n_2)>f(a_1, a_2)
f
(
n
1
,
n
2
)
>
f
(
a
1
,
a
2
)
for all integer points
(
n
1
,
n
2
)
(n_1, n_2)
(
n
1
,
n
2
)
in the plane other than
A
A
A
;(iv)
f
(
x
1
,
x
2
)
>
f
(
b
1
,
b
2
)
f(x_1, x_2)>f(b_1, b_2)
f
(
x
1
,
x
2
)
>
f
(
b
1
,
b
2
)
for all integer points
(
x
1
,
x
2
)
(x_1, x_2)
(
x
1
,
x
2
)
in the plane other than
B
B
B
.Massimo Gobbino, Italy
3
1
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Romanian Masters in mathematics 2010 Day 1 Problem 3
Let
A
1
A
2
A
3
A
4
A_1A_2A_3A_4
A
1
A
2
A
3
A
4
be a quadrilateral with no pair of parallel sides. For each
i
=
1
,
2
,
3
,
4
i=1, 2, 3, 4
i
=
1
,
2
,
3
,
4
, define
ω
1
\omega_1
ω
1
to be the circle touching the quadrilateral externally, and which is tangent to the lines
A
i
−
1
A
i
,
A
i
A
i
+
1
A_{i-1}A_i, A_iA_{i+1}
A
i
−
1
A
i
,
A
i
A
i
+
1
and
A
i
+
1
A
i
+
2
A_{i+1}A_{i+2}
A
i
+
1
A
i
+
2
(indices are considered modulo
4
4
4
so
A
0
=
A
4
,
A
5
=
A
1
A_0=A_4, A_5=A_1
A
0
=
A
4
,
A
5
=
A
1
and
A
6
=
A
2
A_6=A_2
A
6
=
A
2
). Let
T
i
T_i
T
i
be the point of tangency of
ω
i
\omega_i
ω
i
with the side
A
i
A
i
+
1
A_iA_{i+1}
A
i
A
i
+
1
. Prove that the lines
A
1
A
2
,
A
3
A
4
A_1A_2, A_3A_4
A
1
A
2
,
A
3
A
4
and
T
2
T
4
T_2T_4
T
2
T
4
are concurrent if and only if the lines
A
2
A
3
,
A
4
A
1
A_2A_3, A_4A_1
A
2
A
3
,
A
4
A
1
and
T
1
T
3
T_1T_3
T
1
T
3
are concurrent.Pavel Kozhevnikov, Russia
2
1
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Romanian Masters in mathematics 2010 Day 1 Problem 2
For each positive integer
n
n
n
, find the largest real number
C
n
C_n
C
n
with the following property. Given any
n
n
n
real-valued functions
f
1
(
x
)
,
f
2
(
x
)
,
⋯
,
f
n
(
x
)
f_1(x), f_2(x), \cdots, f_n(x)
f
1
(
x
)
,
f
2
(
x
)
,
⋯
,
f
n
(
x
)
defined on the closed interval
0
≤
x
≤
1
0 \le x \le 1
0
≤
x
≤
1
, one can find numbers
x
1
,
x
2
,
⋯
x
n
x_1, x_2, \cdots x_n
x
1
,
x
2
,
⋯
x
n
, such that
0
≤
x
i
≤
1
0 \le x_i \le 1
0
≤
x
i
≤
1
satisfying
∣
f
1
(
x
1
)
+
f
2
(
x
2
)
+
⋯
f
n
(
x
n
)
−
x
1
x
2
⋯
x
n
∣
≥
C
n
|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n
∣
f
1
(
x
1
)
+
f
2
(
x
2
)
+
⋯
f
n
(
x
n
)
−
x
1
x
2
⋯
x
n
∣
≥
C
n
Marko Radovanović, Serbia
1
1
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Romanian Masters in mathematics 2010 Day 1 Problem 1
For a finite non empty set of primes
P
P
P
, let
m
(
P
)
m(P)
m
(
P
)
denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of
P
P
P
.(i) Show that
∣
P
∣
≤
m
(
P
)
|P|\le m(P)
∣
P
∣
≤
m
(
P
)
, with equality if and only if
min
(
P
)
>
∣
P
∣
\min(P)>|P|
min
(
P
)
>
∣
P
∣
.(ii) Show that
m
(
P
)
<
(
∣
P
∣
+
1
)
(
2
∣
P
∣
−
1
)
m(P)<(|P|+1)(2^{|P|}-1)
m
(
P
)
<
(
∣
P
∣
+
1
)
(
2
∣
P
∣
−
1
)
.(The number
∣
P
∣
|P|
∣
P
∣
is the size of set
P
P
P
)Dan Schwarz, Romania