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Contests
International Contests
Romanian Masters of Mathematics Collection
2011 Romanian Master of Mathematics
2011 Romanian Master of Mathematics
Part of
Romanian Masters of Mathematics Collection
Subcontests
(6)
6
1
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RMM2011, P 6, Day 2 - An array on the torus
The cells of a square
2011
×
2011
2011 \times 2011
2011
×
2011
array are labelled with the integers
1
,
2
,
…
,
201
1
2
1,2,\ldots, 2011^2
1
,
2
,
…
,
201
1
2
, in such a way that every label is used exactly once. We then identify the left-hand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut). Determine the largest positive integer
M
M
M
such that, no matter which labelling we choose, there exist two neighbouring cells with the difference of their labels at least
M
M
M
. (Cells with coordinates
(
x
,
y
)
(x,y)
(
x
,
y
)
and
(
x
′
,
y
′
)
(x',y')
(
x
′
,
y
′
)
are considered to be neighbours if
x
=
x
′
x=x'
x
=
x
′
and
y
−
y
′
≡
±
1
(
m
o
d
2011
)
y-y'\equiv\pm1\pmod{2011}
y
−
y
′
≡
±
1
(
mod
2011
)
, or if
y
=
y
′
y=y'
y
=
y
′
and
x
−
x
′
≡
±
1
(
m
o
d
2011
)
x-x'\equiv\pm1\pmod{2011}
x
−
x
′
≡
±
1
(
mod
2011
)
.)(Romania) Dan Schwarz
5
1
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RMM2011, P 5, Day 2 - Find all configurations of points
For every
n
≥
3
n\geq 3
n
≥
3
, determine all the configurations of
n
n
n
distinct points
X
1
,
X
2
,
…
,
X
n
X_1,X_2,\ldots,X_n
X
1
,
X
2
,
…
,
X
n
in the plane, with the property that for any pair of distinct points
X
i
X_i
X
i
,
X
j
X_j
X
j
there exists a permutation
σ
\sigma
σ
of the integers
{
1
,
…
,
n
}
\{1,\ldots,n\}
{
1
,
…
,
n
}
, such that
d
(
X
i
,
X
k
)
=
d
(
X
j
,
X
σ
(
k
)
)
\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})
d
(
X
i
,
X
k
)
=
d
(
X
j
,
X
σ
(
k
)
)
for all
1
≤
k
≤
n
1\leq k \leq n
1
≤
k
≤
n
. (We write
d
(
X
,
Y
)
\textrm{d}(X,Y)
d
(
X
,
Y
)
to denote the distance between points
X
X
X
and
Y
Y
Y
.)(United Kingdom) Luke Betts
4
1
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RMM2011, P 4, Day 2 - Arithmetic function
Given a positive integer
n
=
∏
i
=
1
s
p
i
α
i
\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}
n
=
i
=
1
∏
s
p
i
α
i
, we write
Ω
(
n
)
\Omega(n)
Ω
(
n
)
for the total number
∑
i
=
1
s
α
i
\displaystyle \sum_{i=1}^s \alpha_i
i
=
1
∑
s
α
i
of prime factors of
n
n
n
, counted with multiplicity. Let
λ
(
n
)
=
(
−
1
)
Ω
(
n
)
\lambda(n) = (-1)^{\Omega(n)}
λ
(
n
)
=
(
−
1
)
Ω
(
n
)
(so, for example,
λ
(
12
)
=
λ
(
2
2
⋅
3
1
)
=
(
−
1
)
2
+
1
=
−
1
\lambda(12)=\lambda(2^2\cdot3^1)=(-1)^{2+1}=-1
λ
(
12
)
=
λ
(
2
2
⋅
3
1
)
=
(
−
1
)
2
+
1
=
−
1
). Prove the following two claims:i) There are infinitely many positive integers
n
n
n
such that
λ
(
n
)
=
λ
(
n
+
1
)
=
+
1
\lambda(n) = \lambda(n+1) = +1
λ
(
n
)
=
λ
(
n
+
1
)
=
+
1
; ii) There are infinitely many positive integers
n
n
n
such that
λ
(
n
)
=
λ
(
n
+
1
)
=
−
1
\lambda(n) = \lambda(n+1) = -1
λ
(
n
)
=
λ
(
n
+
1
)
=
−
1
.(Romania) Dan Schwarz
3
1
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RMM2011, P 3, Day 1 - Determine the locus as line varies
A triangle
A
B
C
ABC
A
BC
is inscribed in a circle
ω
\omega
ω
. A variable line
ℓ
\ell
ℓ
chosen parallel to
B
C
BC
BC
meets segments
A
B
AB
A
B
,
A
C
AC
A
C
at points
D
D
D
,
E
E
E
respectively, and meets
ω
\omega
ω
at points
K
K
K
,
L
L
L
(where
D
D
D
lies between
K
K
K
and
E
E
E
). Circle
γ
1
\gamma_1
γ
1
is tangent to the segments
K
D
KD
KD
and
B
D
BD
B
D
and also tangent to
ω
\omega
ω
, while circle
γ
2
\gamma_2
γ
2
is tangent to the segments
L
E
LE
L
E
and
C
E
CE
CE
and also tangent to
ω
\omega
ω
. Determine the locus, as
ℓ
\ell
ℓ
varies, of the meeting point of the common inner tangents to
γ
1
\gamma_1
γ
1
and
γ
2
\gamma_2
γ
2
.(Russia) Vasily Mokin and Fedor Ivlev
2
1
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RMM2011, P 2, Day 1 - Find polynomials with integer values
Determine all positive integers
n
n
n
for which there exists a polynomial
f
(
x
)
f(x)
f
(
x
)
with real coefficients, with the following properties:(1) for each integer
k
k
k
, the number
f
(
k
)
f(k)
f
(
k
)
is an integer if and only if
k
k
k
is not divisible by
n
n
n
; (2) the degree of
f
f
f
is less than
n
n
n
.(Hungary) Géza Kós
1
1
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RMM2011, P 1, Day 1 - Find monotonous functions
Prove that there exist two functions
f
,
g
:
R
→
R
f,g \colon \mathbb{R} \to \mathbb{R}
f
,
g
:
R
→
R
, such that
f
∘
g
f\circ g
f
∘
g
is strictly decreasing and
g
∘
f
g\circ f
g
∘
f
is strictly increasing.(Poland) Andrzej Komisarski and Marcin Kuczma