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Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
1978 Romania Team Selection Test
1978 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(9)
9
1
Hide problems
sequences for IMO preparation
A sequence
(
x
n
)
n
≥
0
\left( x_n\right)_{n\ge 0}
(
x
n
)
n
≥
0
of real numbers satisfies
x
0
>
1
=
x
n
+
1
(
x
n
−
⌊
x
n
⌋
)
,
x_0>1=x_{n+1}\left( x_n-\left\lfloor x_n\right\rfloor\right) ,
x
0
>
1
=
x
n
+
1
(
x
n
−
⌊
x
n
⌋
)
,
for each
n
≥
1.
n\ge 1.
n
≥
1.
Prove that if
(
x
n
)
n
≥
0
\left( x_n\right)_{n\ge 0}
(
x
n
)
n
≥
0
is periodic, then
x
0
x_0
x
0
is a root of a quadratic equation. Study the converse.
8
1
Hide problems
fog=hog (similar functions)
For any set
A
A
A
we say that two functions
f
,
g
:
A
⟶
A
f,g:A\longrightarrow A
f
,
g
:
A
⟶
A
are similar, if there exists a bijection
h
:
A
⟶
A
h:A\longrightarrow A
h
:
A
⟶
A
such that
f
∘
h
=
h
∘
g
.
f\circ h=h\circ g.
f
∘
h
=
h
∘
g
.
a) If
A
A
A
has three elements, construct a finite, arbitrary number functions, having as domain and codomain
A
,
A,
A
,
that are two by two similar, and every other function with the same domain and codomain as the ones determined is similar to, at least, one of them. b) For
A
=
R
,
A=\mathbb{R} ,
A
=
R
,
show that the functions
sin
\sin
sin
and
−
sin
-\sin
−
sin
are similar.
7
2
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polynomials L of deg(L)=3,4
Let
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
be polynomials of degree
3
3
3
with real coefficients such that
P
(
x
)
≤
Q
(
x
)
≤
R
(
x
)
,
P(x)\le Q(x)\le R(x) ,
P
(
x
)
≤
Q
(
x
)
≤
R
(
x
)
,
for every real
x
.
x.
x
.
Suppose
P
−
R
P-R
P
−
R
admits a root. Show that
Q
=
k
P
+
(
1
−
k
)
R
,
Q=kP+(1-k)R,
Q
=
k
P
+
(
1
−
k
)
R
,
for some real number
k
∈
[
0
,
1
]
.
k\in [0,1] .
k
∈
[
0
,
1
]
.
What happens if
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
are of degree
4
,
4,
4
,
under the same circumstances?
Barycenter at TST
a) Prove that for any natural number
n
≥
1
,
n\ge 1,
n
≥
1
,
there is a set
M
\mathcal{M}
M
of
n
n
n
points from the Cartesian plane such that the barycenter of every subset of
M
\mathcal{M}
M
has integral coordinates (both coordinates are integer numbers).b) Show that if a set
N
\mathcal{N}
N
formed by an infinite number of points from the Cartesian plane is given such that no three of them are collinear, then there exists a finite subset of
N
,
\mathcal{N} ,
N
,
the barycenter of which has non-integral coordinates.
6
2
Hide problems
projection of polyhedrons
Show that there is no polyhedron whose projection on the plane is a nondegenerate triangle.
infimum = 0
a) Prove that
0
=
inf
{
∣
x
2
+
y
3
+
y
5
∣
∣
x
,
y
,
z
∈
Z
,
x
2
+
y
2
+
z
2
>
0
}
0=\inf\{ |x\sqrt 2+y\sqrt 3+y\sqrt 5|\big| x,y,z\in\mathbb{Z} ,x^2+y^2+z^2>0 \}
0
=
in
f
{
∣
x
2
+
y
3
+
y
5
∣
x
,
y
,
z
∈
Z
,
x
2
+
y
2
+
z
2
>
0
}
b) Prove that there exist three positive rational numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that the expression
E
(
x
,
y
,
z
)
:
=
x
a
+
y
b
+
z
c
E(x,y,z):=xa+yb+zc
E
(
x
,
y
,
z
)
:=
x
a
+
y
b
+
zc
vanishes for infinitely many integer triples
(
x
,
y
,
z
)
,
(x,y,z),
(
x
,
y
,
z
)
,
but it doesn´t get arbitrarily close to
0.
0.
0.
5
2
Hide problems
1978 geometry.
Prove that there is no square with its four vertices on four concentric circles whose radii form an arithmetic progression.
A locus contained in an equilateral triangle
Find locus of points
M
M
M
inside an equilateral triangle
A
B
C
ABC
A
BC
such that
∠
M
B
C
+
∠
M
C
A
+
∠
M
A
B
=
π
/
2.
\angle MBC+\angle MCA +\angle MAB={\pi}/{2}.
∠
MBC
+
∠
MC
A
+
∠
M
A
B
=
π
/
2
.
4
4
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3
4
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2
4
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1
4
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