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Contests
International Contests
Romanian Masters of Mathematics Collection
2023 Romanian Master of Mathematics
2023 Romanian Master of Mathematics
Part of
Romanian Masters of Mathematics Collection
Subcontests
(6)
6
1
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2023 RMM, Problem 6
Let
r
,
g
,
b
r,g,b
r
,
g
,
b
be non negative integers and
Γ
\Gamma
Γ
be a connected graph with
r
+
g
+
b
+
1
r+g+b+1
r
+
g
+
b
+
1
vertices. Its edges are colored in red green and blue. It turned out that
Γ
\Gamma
Γ
containsA spanning tree with exactly
r
r
r
red edges. A spanning tree with exactly
g
g
g
green edges. A spanning tree with exactly
b
b
b
blue edges.Prove that
Γ
\Gamma
Γ
contains a spanning tree with exactly
r
r
r
red edges,
g
g
g
green edges and
b
b
b
blue edges.
5
1
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2023 RMM, Problem 5
Let
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
be non constant polynomials with real coefficients, such that
P
(
Q
(
x
)
)
=
R
(
S
(
x
)
)
P(Q(x))=R(S(x))
P
(
Q
(
x
))
=
R
(
S
(
x
))
and the degree of
P
P
P
is multiple of the degree of
R
.
R.
R
.
Prove that there exists a polynomial
T
T
T
with real coefficients such that
P
(
x
)
=
R
(
T
(
x
)
)
\displaystyle P(x)=R(T(x))
P
(
x
)
=
R
(
T
(
x
))
4
1
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2023 RMM, Problem 4
An acute triangle
A
B
C
ABC
A
BC
is given and
H
H
H
and
O
O
O
be its orthocenter and circumcenter respectively. Let
K
K
K
be the midpoint of
A
H
AH
A
H
and
ℓ
\ell
ℓ
be a line through
O
.
O.
O
.
Let
P
P
P
and
Q
Q
Q
be the projections of
B
B
B
and
C
C
C
on
ℓ
.
\ell.
ℓ
.
Prove that
K
P
+
K
Q
≥
B
C
KP+KQ\ge BC
K
P
+
K
Q
≥
BC
3
1
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Extremely Weird Algebra
Let
n
≥
2
n\geq 2
n
≥
2
be an integer and let
f
f
f
be a
4
n
4n
4
n
-variable polynomial with real coefficients. Assume that, for any
2
n
2n
2
n
points
(
x
1
,
y
1
)
,
…
,
(
x
2
n
,
y
2
n
)
(x_1,y_1),\dots,(x_{2n},y_{2n})
(
x
1
,
y
1
)
,
…
,
(
x
2
n
,
y
2
n
)
in the Cartesian plane,
f
(
x
1
,
y
1
,
…
,
x
2
n
,
y
2
n
)
=
0
f(x_1,y_1,\dots,x_{2n},y_{2n})=0
f
(
x
1
,
y
1
,
…
,
x
2
n
,
y
2
n
)
=
0
if and only if the points form the vertices of a regular
2
n
2n
2
n
-gon in some order, or are all equal. Determine the smallest possible degree of
f
f
f
.(Note, for example, that the degree of the polynomial
g
(
x
,
y
)
=
4
x
3
y
4
+
y
x
+
x
−
2
g(x,y)=4x^3y^4+yx+x-2
g
(
x
,
y
)
=
4
x
3
y
4
+
y
x
+
x
−
2
is
7
7
7
because
7
=
3
+
4
7=3+4
7
=
3
+
4
.)Ankan Bhattacharya
2
1
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Beautiful Triangles
Fix an integer
n
≥
3
n \geq 3
n
≥
3
. Let
S
\mathcal{S}
S
be a set of
n
n
n
points in the plane, no three of which are collinear. Given different points
A
,
B
,
C
A,B,C
A
,
B
,
C
in
S
\mathcal{S}
S
, the triangle
A
B
C
ABC
A
BC
is nice for
A
B
AB
A
B
if
[
A
B
C
]
≤
[
A
B
X
]
[ABC] \leq [ABX]
[
A
BC
]
≤
[
A
BX
]
for all
X
X
X
in
S
\mathcal{S}
S
different from
A
A
A
and
B
B
B
. (Note that for a segment
A
B
AB
A
B
there could be several nice triangles). A triangle is beautiful if its vertices are all in
S
\mathcal{S}
S
and is nice for at least two of its sides.Prove that there are at least
1
2
(
n
−
1
)
\frac{1}{2}(n-1)
2
1
(
n
−
1
)
beautiful triangles.
1
1
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Funny Diophantine
Determine all prime numbers
p
p
p
and all positive integers
x
x
x
and
y
y
y
satisfying
x
3
+
y
3
=
p
(
x
y
+
p
)
.
x^3+y^3=p(xy+p).
x
3
+
y
3
=
p
(
x
y
+
p
)
.