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Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
2001 Romania Team Selection Test
2001 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(4)
4
2
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Finding pairwise friends from each of the 3 schools
Three schools have
200
200
200
students each. Every student has at least one friend in each school (if the student
a
a
a
is a friend of the student
b
b
b
then
b
b
b
is a friend of
a
a
a
). It is known that there exists a set
E
E
E
of
300
300
300
students (among the
600
600
600
) such that for any school
S
S
S
and any two students
x
,
y
∈
E
x,y\in E
x
,
y
∈
E
but not in
S
S
S
, the number of friends in
S
S
S
of
x
x
x
and
y
y
y
are different. Show that one can find a student in each school such that they are friends with each other.
Defining the sequence from neighbours of P
Consider a convex polyhedron
P
P
P
with vertices
V
1
,
…
,
V
p
V_1,\ldots ,V_p
V
1
,
…
,
V
p
. The distinct vertices
V
i
V_i
V
i
and
V
j
V_j
V
j
are called neighbours if they belong to the same face of the polyhedron. To each vertex
V
k
V_k
V
k
we assign a number
v
k
(
0
)
v_k(0)
v
k
(
0
)
, and construct inductively the sequence
v
k
(
n
)
(
n
≥
0
)
v_k(n)\ (n\ge 0)
v
k
(
n
)
(
n
≥
0
)
as follows:
v
k
(
n
+
1
)
v_k(n+1)
v
k
(
n
+
1
)
is the average of the
v
j
(
n
)
v_j(n)
v
j
(
n
)
for all neighbours
V
j
V_j
V
j
of
V
k
V_k
V
k
. If all numbers
v
k
(
n
)
v_k(n)
v
k
(
n
)
are integers, prove that there exists the positive integer
N
N
N
such that all
v
k
(
n
)
v_k(n)
v
k
(
n
)
are equal for
n
≥
N
n\ge N
n
≥
N
.
3
3
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Inequality for sides of a triangle a,b,c
The sides of a triangle have lengths
a
,
b
,
c
a,b,c
a
,
b
,
c
. Prove that: \begin{align*}(-a+b+c)(a-b+c)\, +\, & (a-b+c)(a+b-c)+(a+b-c)(-a+b+c)\\ &\le\sqrt{abc}(\sqrt{a}+\sqrt{b}+\sqrt{c})\end{align*}
Two rays among n form acute angle
Find the least
n
∈
N
n\in N
n
∈
N
such that among any
n
n
n
rays in space sharing a common origin there exist two which form an acute angle.
Tangents to the circumcircle of ABC
The tangents at
A
A
A
and
B
B
B
to the circumcircle of the acute triangle
A
B
C
ABC
A
BC
intersect the tangent at
C
C
C
at the points
D
D
D
and
E
E
E
, respectively. The line
A
E
AE
A
E
intersects
B
C
BC
BC
at
P
P
P
and the line
B
D
BD
B
D
intersects
A
C
AC
A
C
at
R
R
R
. Let
Q
Q
Q
and
S
S
S
be the midpoints of the segments
A
P
AP
A
P
and
B
R
BR
BR
respectively. Prove that
∠
A
B
Q
=
∠
B
A
S
\angle ABQ=\angle BAS
∠
A
BQ
=
∠
B
A
S
.
2
3
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Product of injective/surjective functions
a) Let
f
,
g
:
Z
→
Z
f,g:\mathbb{Z}\rightarrow\mathbb{Z}
f
,
g
:
Z
→
Z
be one to one maps. Show that the function
h
:
Z
→
Z
h:\mathbb{Z}\rightarrow\mathbb{Z}
h
:
Z
→
Z
defined by
h
(
x
)
=
f
(
x
)
g
(
x
)
h(x)=f(x)g(x)
h
(
x
)
=
f
(
x
)
g
(
x
)
, for all
x
∈
Z
x\in\mathbb{Z}
x
∈
Z
, cannot be a surjective function.b) Let
f
:
Z
→
Z
f:\mathbb{Z}\rightarrow\mathbb{Z}
f
:
Z
→
Z
be a surjective function. Show that there exist surjective functions
g
,
h
:
Z
→
Z
g,h:\mathbb{Z}\rightarrow\mathbb{Z}
g
,
h
:
Z
→
Z
such that
f
(
x
)
=
g
(
x
)
h
(
x
)
f(x)=g(x)h(x)
f
(
x
)
=
g
(
x
)
h
(
x
)
, for all
x
∈
Z
x\in\mathbb{Z}
x
∈
Z
.
Tangents AA',BB',CC',DD' form p with axis of symmetry
The vertices
A
,
B
,
C
A,B,C
A
,
B
,
C
and
D
D
D
of a square lie outside a circle centred at
M
M
M
. Let
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
AA',BB',CC',DD'
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
be tangents to the circle. Assume that the segments
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
AA',BB',CC',DD'
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
are the consecutive sides of a quadrilateral
p
p
p
in which a circle is inscribed. Prove that
p
p
p
has an axis of symmetry.
No function satisfies inequality
Prove that there is no function
f
:
(
0
,
∞
)
→
(
0
,
∞
)
f:(0,\infty )\rightarrow (0,\infty)
f
:
(
0
,
∞
)
→
(
0
,
∞
)
such that
f
(
x
+
y
)
≥
f
(
x
)
+
y
f
(
f
(
x
)
)
f(x+y)\ge f(x)+yf(f(x))
f
(
x
+
y
)
≥
f
(
x
)
+
y
f
(
f
(
x
))
for every
x
,
y
∈
(
0
,
∞
)
x,y\in (0,\infty )
x
,
y
∈
(
0
,
∞
)
.
1
4
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