MathDB
Problems
Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
1991 Romania Team Selection Test
1991 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(10)
9
1
Hide problems
MNPQR , ABCDE similar pentagons, regular?
The diagonals of a pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
determine another pentagon
M
N
P
Q
R
MNPQR
MNPQR
. If
M
N
P
Q
R
MNPQR
MNPQR
and
A
B
C
D
E
ABCDE
A
BC
D
E
are similar, must
A
B
C
D
E
ABCDE
A
BC
D
E
be regular?
7
1
Hide problems
x_1^2x_2^2...x_n^2+x_2^2x_3^2...x_{n+1}^2+...+x_{2n}^2x_1^2...x_{n-1}^2<1/n^{2n}
Let
x
1
,
x
2
,
.
.
.
,
x
2
n
x_1,x_2,...,x_{2n}
x
1
,
x
2
,
...
,
x
2
n
be positive real numbers with the sum
1
1
1
. Prove that
x
1
2
x
2
2
.
.
.
x
n
2
+
x
2
2
x
3
2
.
.
.
x
n
+
1
2
+
.
.
.
+
x
2
n
2
x
1
2
.
.
.
x
n
−
1
2
<
1
n
2
n
x_1^2x_2^2...x_n^2+x_2^2x_3^2...x_{n+1}^2+...+x_{2n}^2x_1^2...x_{n-1}^2 <\frac{1}{n^{2n}}
x
1
2
x
2
2
...
x
n
2
+
x
2
2
x
3
2
...
x
n
+
1
2
+
...
+
x
2
n
2
x
1
2
...
x
n
−
1
2
<
n
2
n
1
6
1
Hide problems
here is a regular n-gon whose all vertices lie on the considered arcs
Let
n
≥
3
n \ge 3
n
≥
3
be an integer. A finite number of disjoint arcs with the total sum of length
1
−
1
n
1 -\frac{1}{n}
1
−
n
1
are given on a circle of perimeter
1
1
1
. Prove that there is a regular
n
n
n
-gon whose all vertices lie on the considered arcs
2
2
Hide problems
concurrent sets of planes
Let
A
1
A
2
A
3
A
4
A_1A_2A_3A_4
A
1
A
2
A
3
A
4
be a tetrahedron. For any permutation
(
i
,
j
,
k
,
h
)
(i, j,k,h)
(
i
,
j
,
k
,
h
)
of
1
,
2
,
3
,
4
1,2,3,4
1
,
2
,
3
,
4
denote: -
P
i
P_i
P
i
– the orthogonal projection of
A
i
A_i
A
i
on
A
j
A
k
A
h
A_jA_kA_h
A
j
A
k
A
h
; -
B
i
j
B_{ij}
B
ij
– the midpoint of the edge
A
i
A
j
A_iAj
A
i
A
j
, -
C
i
j
C_{ij}
C
ij
– the midpoint of segment
P
i
P
j
P_iP_j
P
i
P
j
-
β
i
j
\beta_{ij}
β
ij
– the plane
B
i
j
P
h
P
k
B_{ij}P_hP_k
B
ij
P
h
P
k
-
δ
i
j
\delta_{ij}
δ
ij
– the plane
B
i
j
P
i
P
j
B_{ij}P_iP_j
B
ij
P
i
P
j
-
α
i
j
\alpha_{ij}
α
ij
– the plane through
C
i
j
C_{ij}
C
ij
orthogonal to
A
k
A
h
A_kA_h
A
k
A
h
-
γ
i
j
\gamma_{ij}
γ
ij
– the plane through
C
i
j
C_{ij}
C
ij
orthogonal to
A
i
A
j
A_iA_j
A
i
A
j
. Prove that if the points
P
i
P_i
P
i
are not in a plane, then the following sets of planes are concurrent: (a)
α
i
j
\alpha_{ij}
α
ij
, (b)
β
i
j
\beta_{ij}
β
ij
, (c)
γ
i
j
\gamma_{ij}
γ
ij
, (d)
δ
i
j
\delta_{ij}
δ
ij
.
a_{n+2 }= a_{n+1} +a_n +k, least k for which a1991 and 1991 not coprime
The sequence (
a
n
a_n
a
n
) is defined by
a
1
=
a
2
=
1
a_1 = a_2 = 1
a
1
=
a
2
=
1
and
a
n
+
2
=
a
n
+
1
+
a
n
+
k
a_{n+2 }= a_{n+1} +a_n +k
a
n
+
2
=
a
n
+
1
+
a
n
+
k
, where
k
k
k
is a positive integer. Find the least
k
k
k
for which
a
1991
a_{1991}
a
1991
and
1991
1991
1991
are not coprime.
4
2
Hide problems
(a_m,a_n) = a_{(m,n)}, a_n = \prod_{d|n} b_d
A sequence
(
a
n
)
(a_n)
(
a
n
)
of positive integers satisfies
(
a
m
,
a
n
)
=
a
(
m
,
n
)
(a_m,a_n) = a_{(m,n)}
(
a
m
,
a
n
)
=
a
(
m
,
n
)
for all
m
,
n
m,n
m
,
n
. Prove that there is a unique sequence
(
b
n
)
(b_n)
(
b
n
)
of positive integers such that
a
n
=
∏
d
∣
n
b
d
a_n = \prod_{d|n} b_d
a
n
=
∏
d
∣
n
b
d
f : S \to (0,1) with $f(S_1 \cup S_2) = f(S_1)+ f(S_2), from set of polygonals
Let
S
S
S
be the set of all polygonal areas in a plane. Prove that there is a function
f
:
S
→
(
0
,
1
)
f : S \to (0,1)
f
:
S
→
(
0
,
1
)
which satisfies
f
(
S
1
∪
S
2
)
=
f
(
S
1
)
+
f
(
S
2
)
f(S_1 \cup S_2) = f(S_1)+ f(S_2)
f
(
S
1
∪
S
2
)
=
f
(
S
1
)
+
f
(
S
2
)
for any
S
1
,
S
2
∈
S
S_1,S_2 \in S
S
1
,
S
2
∈
S
which have common points only on their borders
8
1
Hide problems
Composition of function
Let
n
,
a
,
b
n, a, b
n
,
a
,
b
be integers with
n
≥
2
n \geq 2
n
≥
2
and
a
∉
{
0
,
1
}
a \notin \{0, 1\}
a
∈
/
{
0
,
1
}
and let
u
(
x
)
=
a
x
+
b
u(x) = ax + b
u
(
x
)
=
a
x
+
b
be the function defined on integers. Show that there are infinitely many functions
f
:
Z
→
Z
f : \mathbb{Z} \rightarrow \mathbb{Z}
f
:
Z
→
Z
such that
f
n
(
x
)
=
f
(
f
(
⋯
f
⏟
n
(
x
)
⋯
)
)
=
u
(
x
)
f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)
f
n
(
x
)
=
n
f
(
f
(
⋯
f
(
x
)
⋯
))
=
u
(
x
)
for all
x
x
x
. If
a
=
1
a = 1
a
=
1
, show that there is a
b
b
b
for which there is no
f
f
f
with
f
n
(
x
)
≡
u
(
x
)
f_n(x) \equiv u(x)
f
n
(
x
)
≡
u
(
x
)
.
5
1
Hide problems
Excribed circles and orthocenters
In a triangle
A
1
A
2
A
3
A_1A_2A_3
A
1
A
2
A
3
, the excribed circles corresponding to sides
A
2
A
3
A_2A_3
A
2
A
3
,
A
3
A
1
A_3A_1
A
3
A
1
,
A
1
A
2
A_1A_2
A
1
A
2
touch these sides at
T
1
T_1
T
1
,
T
2
T_2
T
2
,
T
3
T_3
T
3
, respectively. If
H
1
H_1
H
1
,
H
2
H_2
H
2
,
H
3
H_3
H
3
are the orthocenters of triangles
A
1
T
2
T
3
A_1T_2T_3
A
1
T
2
T
3
,
A
2
T
3
T
1
A_2T_3T_1
A
2
T
3
T
1
,
A
3
T
1
T
2
A_3T_1T_2
A
3
T
1
T
2
, respectively, prove that lines
H
1
T
1
H_1T_1
H
1
T
1
,
H
2
T
2
H_2T_2
H
2
T
2
,
H
3
T
3
H_3T_3
H
3
T
3
are concurrent.
3
2
Hide problems
min (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}, coloring inequality
Let
C
C
C
be a coloring of all edges and diagonals of a convex
n
n
n
−gon in red and blue (in Romanian, rosu and albastru). Denote by
q
r
(
C
)
q_r(C)
q
r
(
C
)
(resp.
q
a
(
C
)
q_a(C)
q
a
(
C
)
) the number of quadrilaterals having all its edges and diagonals red (resp. blue). Prove:
m
i
n
C
(
q
r
(
C
)
+
q
a
(
C
)
)
≤
1
32
(
n
4
)
\underset{C}{min} (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}
C
min
(
q
r
(
C
)
+
q
a
(
C
))
≤
32
1
(
4
n
)
A nice identity
Prove the following identity for every
n
∈
N
n\in N
n
∈
N
: \sum_{j\plus{}h\equal{}n,j\geq h}\frac{(\minus{}1)^h2^{j\minus{}h}\binom{j}{h}}{j}\equal{}\frac{2}{n}
1
2
Hide problems
(a+1)/b+b/a
Suppose that
a
,
b
a,b
a
,
b
are positive integers for which A\equal{}\frac{a\plus{}1}{b}\plus{}\frac{b}{a} is an integer.Prove that A\equal{}3.
area > 4
Let
M
=
{
A
1
,
A
2
,
…
,
A
5
}
M=\{A_{1},A_{2},\ldots,A_{5}\}
M
=
{
A
1
,
A
2
,
…
,
A
5
}
be a set of five points in the plane such that the area of each triangle
A
i
A
j
A
k
A_{i}A_{j}A_{k}
A
i
A
j
A
k
, is greater than 3. Prove that there exists a triangle with vertices in
M
M
M
and having the area greater than 4. Laurentiu Panaitopol
10
1
Hide problems
Is it possible to solve this by generating functions?
Let
a
1
<
a
2
<
⋯
<
a
n
a_1<a_2<\cdots<a_n
a
1
<
a
2
<
⋯
<
a
n
be positive integers. Some colouring of
Z
\mathbb{Z}
Z
is periodic with period
t
t
t
such that for each
x
∈
Z
x\in \mathbb{Z}
x
∈
Z
exactly one of
x
+
a
1
,
x
+
a
2
,
…
,
x
+
a
n
x+a_1,x+a_2,\dots,x+a_n
x
+
a
1
,
x
+
a
2
,
…
,
x
+
a
n
is coloured. Prove that
n
∣
t
n\mid t
n
∣
t
. Andrei Radulescu-Banu