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Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
1987 Romania Team Selection Test
1987 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(11)
11
1
Hide problems
Real polynomial and system of equations
Let
P
(
X
,
Y
)
=
X
2
+
2
a
X
Y
+
Y
2
P(X,Y)=X^2+2aXY+Y^2
P
(
X
,
Y
)
=
X
2
+
2
a
X
Y
+
Y
2
be a real polynomial where
∣
a
∣
≥
1
|a|\geq 1
∣
a
∣
≥
1
. For a given positive integer
n
n
n
,
n
≥
2
n\geq 2
n
≥
2
consider the system of equations:
P
(
x
1
,
x
2
)
=
P
(
x
2
,
x
3
)
=
…
=
P
(
x
n
−
1
,
x
n
)
=
P
(
x
n
,
x
1
)
=
0.
P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 .
P
(
x
1
,
x
2
)
=
P
(
x
2
,
x
3
)
=
…
=
P
(
x
n
−
1
,
x
n
)
=
P
(
x
n
,
x
1
)
=
0.
We call two solutions
(
x
1
,
x
2
,
…
,
x
n
)
(x_1,x_2,\ldots,x_n)
(
x
1
,
x
2
,
…
,
x
n
)
and
(
y
1
,
y
2
,
…
,
y
n
)
(y_1,y_2,\ldots,y_n)
(
y
1
,
y
2
,
…
,
y
n
)
of the system to be equivalent if there exists a real number
λ
≠
0
\lambda \neq 0
λ
=
0
,
x
1
=
λ
y
1
x_1=\lambda y_1
x
1
=
λ
y
1
,
…
\ldots
…
,
x
n
=
λ
y
n
x_n= \lambda y_n
x
n
=
λ
y
n
. How many nonequivalent solutions does the system have? Mircea Becheanu
9
1
Hide problems
\sum \sum ij \cos ( \alpha_i - \alpha_j ) \geq 0
Prove that for all real numbers
α
1
,
α
2
,
…
,
α
n
\alpha_1,\alpha_2,\ldots,\alpha_n
α
1
,
α
2
,
…
,
α
n
we have
∑
i
=
1
n
∑
j
=
1
n
i
j
cos
(
α
i
−
α
j
)
≥
0.
\sum_{i=1}^n \sum_{j=1}^n ij \cos {(\alpha_i - \alpha_j )} \geq 0.
i
=
1
∑
n
j
=
1
∑
n
ij
cos
(
α
i
−
α
j
)
≥
0.
Octavian Stanasila
8
1
Hide problems
solid geometry - ABCD a square and AE and CF perpendiculars
Let
A
B
C
D
ABCD
A
BC
D
be a square and
a
a
a
be the length of his edges. The segments
A
E
AE
A
E
and
C
F
CF
CF
are perpendicular on the square's plane in the same half-space and they have the length
A
E
=
a
AE=a
A
E
=
a
,
C
F
=
b
CF=b
CF
=
b
where
a
<
b
<
a
3
a<b<a\sqrt 3
a
<
b
<
a
3
. If
K
K
K
denoted the set of the interior points of the square
A
B
C
D
ABCD
A
BC
D
determine
min
M
∈
K
(
max
(
E
M
,
F
M
)
)
\min_{M\in K} \left( \max ( EM, FM ) \right)
min
M
∈
K
(
max
(
EM
,
FM
)
)
and
max
M
∈
K
(
min
(
E
M
,
F
M
)
)
\max_{M\in K} \left( \min (EM,FM) \right)
max
M
∈
K
(
min
(
EM
,
FM
)
)
. Octavian Stanasila
6
1
Hide problems
covering of the plane with regular hexagons
The plane is covered with network of regular congruent disjoint hexagons. Prove that there cannot exist a square which has its four vertices in the vertices of the hexagons. Gabriel Nagy
5
1
Hide problems
Least number n for which there exists permutations
Let
A
A
A
be the set
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n\}
{
1
,
2
,
…
,
n
}
,
n
≥
2
n\geq 2
n
≥
2
. Find the least number
n
n
n
for which there exist permutations
α
\alpha
α
,
β
\beta
β
,
γ
\gamma
γ
,
δ
\delta
δ
of the set
A
A
A
with the property:
∑
i
=
1
n
α
(
i
)
β
(
i
)
=
19
10
∑
i
=
1
n
γ
(
i
)
δ
(
i
)
.
\sum_{i=1}^n \alpha(i) \beta (i) = \dfrac {19}{10} \sum^n_{i=1} \gamma(i)\delta(i) .
i
=
1
∑
n
α
(
i
)
β
(
i
)
=
10
19
i
=
1
∑
n
γ
(
i
)
δ
(
i
)
.
Marcel Chirita
3
1
Hide problems
x+y cannot be divisible by x-y
Let
A
A
A
be the set
A
=
{
1
,
2
,
…
,
n
}
A = \{ 1,2, \ldots, n\}
A
=
{
1
,
2
,
…
,
n
}
. Determine the maximum number of elements of a subset
B
⊂
A
B\subset A
B
⊂
A
such that for all elements
x
,
y
x,y
x
,
y
from
B
B
B
,
x
+
y
x+y
x
+
y
cannot be divisible by
x
−
y
x-y
x
−
y
. Mircea Lascu, Dorel Mihet
2
1
Hide problems
Find all positive integers A which can be represented ...
Find all positive integers
A
A
A
which can be represented in the form:
A
=
(
m
−
1
n
)
(
n
−
1
p
)
(
p
−
1
m
)
A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right)
A
=
(
m
−
n
1
)
(
n
−
p
1
)
(
p
−
m
1
)
where
m
≥
n
≥
p
≥
1
m\geq n\geq p \geq 1
m
≥
n
≥
p
≥
1
are integer numbers. Ioan Bogdan
1
1
Hide problems
3x3 matrix - find max {det A}
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be distinct real numbers such that
a
+
b
+
c
>
0
a+b+c>0
a
+
b
+
c
>
0
. Let
M
M
M
be the set of
3
×
3
3\times 3
3
×
3
matrices with the property that each line and each column contain all given numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
. Find
{
max
{
det
A
∣
A
∈
M
}
\{\max \{ \det A \mid A \in M \}
{
max
{
det
A
∣
A
∈
M
}
and the number of matrices which realise the maximum value. Mircea Becheanu
10
1
Hide problems
divizibility
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be integer numbers such that
(
a
+
b
+
c
)
∣
(
a
2
+
b
2
+
c
2
)
(a+b+c) \mid (a^{2}+b^{2}+c^{2})
(
a
+
b
+
c
)
∣
(
a
2
+
b
2
+
c
2
)
. Show that there exist infinitely many positive integers
n
n
n
such that
(
a
+
b
+
c
)
∣
(
a
n
+
b
n
+
c
n
)
(a+b+c) \mid (a^{n}+b^{n}+c^{n})
(
a
+
b
+
c
)
∣
(
a
n
+
b
n
+
c
n
)
. Laurentiu Panaitopol
4
1
Hide problems
ineq lp
Let P(X) \equal{} a_{n}X^{n} \plus{} a_{n \minus{} 1}X^{n \minus{} 1} \plus{} \ldots \plus{} a_{1}X \plus{} a_{0} be a real polynomial of degree
n
n
n
. Suppose
n
n
n
is an even number and: a)
a
0
>
0
a_{0} > 0
a
0
>
0
,
a
n
>
0
a_{n} > 0
a
n
>
0
; b) a_{1}^{2} \plus{} a_{2}^{2} \plus{} \ldots \plus{} a_{n \minus{} 1}^{2}\leq\frac {4\min(a_{0}^{2} , a_{n}^{2})}{n \minus{} 1}. Prove that
P
(
x
)
≥
0
P(x)\geq 0
P
(
x
)
≥
0
for all real values
x
x
x
. Laurentiu Panaitopol
7
1
Hide problems
n|(3^n - 2^n)
Determine all positive integers
n
n
n
such that
n
n
n
divides
3
n
−
2
n
3^n - 2^n
3
n
−
2
n
.