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Problems
Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
1999 Romania Team Selection Test
1999 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(13)
17
1
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A polyhedron in space has 3 edges which form a triangle?
A polyhedron
P
P
P
is given in space. Find whether there exist three edges in
P
P
P
which can be the sides of a triangle. Justify your answer! Barbu Berceanu
15
1
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Native and foreign scientists exchange messages
The participants to an international conference are native and foreign scientist. Each native scientist sends a message to a foreign scientist and each foreign scientist sends a message to a native scientist. There are native scientists who did not receive a message. Prove that there exists a set
S
S
S
of native scientists such that the outer
S
S
S
scientists are exactly those who received messages from those foreign scientists who did not receive messages from scientists belonging to
S
S
S
. Radu Niculescu
13
1
Hide problems
n points on a circle, find the max nr. of acute triangles
Let
n
≥
3
n\geq 3
n
≥
3
and
A
1
,
A
2
,
…
,
A
n
A_1,A_2,\ldots,A_n
A
1
,
A
2
,
…
,
A
n
be points on a circle. Find the largest number of acute triangles that can be considered with vertices in these points.G. Eckstein
11
1
Hide problems
Show that x^n+ax+p is irreducible if p>|a|+1, p prime
Let
a
,
n
a,n
a
,
n
be integer numbers,
p
p
p
a prime number such that
p
>
∣
a
∣
+
1
p>|a|+1
p
>
∣
a
∣
+
1
. Prove that the polynomial
f
(
x
)
=
x
n
+
a
x
+
p
f(x)=x^n+ax+p
f
(
x
)
=
x
n
+
a
x
+
p
cannot be represented as a product of two integer polynomials. Laurentiu Panaitopol
8
1
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unbounded sequence of positive integers by Manolescu
Let
a
a
a
be a positive real number and
{
x
n
}
n
≥
1
\{x_n\}_{n\geq 1}
{
x
n
}
n
≥
1
a sequence of real numbers such that
x
1
=
a
x_1=a
x
1
=
a
and
x
n
+
1
≥
(
n
+
2
)
x
n
−
∑
k
=
1
n
−
1
k
x
k
,
∀
n
≥
1.
x_{n+1} \geq (n+2)x_n - \sum^{n-1}_{k=1}kx_k, \ \forall \ n\geq 1.
x
n
+
1
≥
(
n
+
2
)
x
n
−
k
=
1
∑
n
−
1
k
x
k
,
∀
n
≥
1.
Prove that there exists a positive integer
n
n
n
such that
x
n
>
1999
!
x_n > 1999!
x
n
>
1999
!
. Ciprian Manolescu
7
1
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Arithmetic and geometric progressions with integers
Prove that for any integer
n
n
n
,
n
≥
3
n\geq 3
n
≥
3
, there exist
n
n
n
positive integers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
in arithmetic progression, and
n
n
n
positive integers in geometric progression
b
1
,
b
2
,
…
,
b
n
b_1,b_2,\ldots,b_n
b
1
,
b
2
,
…
,
b
n
such that
b
1
<
a
1
<
b
2
<
a
2
<
⋯
<
b
n
<
a
n
.
b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n .
b
1
<
a
1
<
b
2
<
a
2
<
⋯
<
b
n
<
a
n
.
Give an example of two such progressions having at least five terms.Mihai Baluna
5
1
Hide problems
an inequality with distinct positive integers by Panaitopol
Let
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
be distinct positive integers. Prove that
x
1
2
+
x
2
2
+
⋯
+
x
n
2
≥
2
n
+
1
3
(
x
1
+
x
2
+
⋯
+
x
n
)
.
x_1^2+x_2^2 + \cdots + x_n^2 \geq \frac {2n+1}3 ( x_1+x_2+\cdots + x_n).
x
1
2
+
x
2
2
+
⋯
+
x
n
2
≥
3
2
n
+
1
(
x
1
+
x
2
+
⋯
+
x
n
)
.
Laurentiu Panaitopol
3
1
Hide problems
Sum of two consecutive perfect squares
Prove that for any positive integer
n
n
n
, the number
S
n
=
(
2
n
+
1
0
)
⋅
2
2
n
+
(
2
n
+
1
2
)
⋅
2
2
n
−
2
⋅
3
+
⋯
+
(
2
n
+
1
2
n
)
⋅
3
n
S_n = {2n+1\choose 0}\cdot 2^{2n}+{2n+1\choose 2}\cdot 2^{2n-2}\cdot 3 +\cdots + {2n+1 \choose 2n}\cdot 3^n
S
n
=
(
0
2
n
+
1
)
⋅
2
2
n
+
(
2
2
n
+
1
)
⋅
2
2
n
−
2
⋅
3
+
⋯
+
(
2
n
2
n
+
1
)
⋅
3
n
is the sum of two consecutive perfect squares. Dorin Andrica
2
1
Hide problems
KLM is equilateral iff BAC = 60 degrees.
Let
A
B
C
ABC
A
BC
be an acute triangle. The interior angle bisectors of
∠
A
B
C
\angle ABC
∠
A
BC
and
∠
A
C
B
\angle ACB
∠
A
CB
meet the opposite sides in
L
L
L
and
M
M
M
respectively. Prove that there is a point
K
K
K
in the interior of the side
B
C
BC
BC
such that the triangle
K
L
M
KLM
K
L
M
is equilateral if and only if
∠
B
A
C
=
6
0
∘
\angle BAC = 60^\circ
∠
B
A
C
=
6
0
∘
.
16
1
Hide problems
Fun sets [3-element subsets of n-element set; [sqrt(2n)]]
Let
X
X
X
be a set with
n
n
n
elements, and let
A
1
A_{1}
A
1
,
A
2
A_{2}
A
2
, ...,
A
m
A_{m}
A
m
be subsets of
X
X
X
such that: 1)
∣
A
i
∣
=
3
|A_{i}|=3
∣
A
i
∣
=
3
for every
i
∈
{
1
,
2
,
.
.
.
,
m
}
i\in\left\{1,2,...,m\right\}
i
∈
{
1
,
2
,
...
,
m
}
; 2)
∣
A
i
∩
A
j
∣
≤
1
|A_{i}\cap A_{j}|\leq 1
∣
A
i
∩
A
j
∣
≤
1
for all
i
,
j
∈
{
1
,
2
,
.
.
.
,
m
}
i,j\in\left\{1,2,...,m\right\}
i
,
j
∈
{
1
,
2
,
...
,
m
}
such that
i
≠
j
i \neq j
i
=
j
. Prove that there exists a subset
A
A
A
of
X
X
X
such that
A
A
A
has at least
[
2
n
]
\left[\sqrt{2n}\right]
[
2
n
]
elements, and for every
i
∈
{
1
,
2
,
.
.
.
,
m
}
i\in\left\{1,2,...,m\right\}
i
∈
{
1
,
2
,
...
,
m
}
, the set
A
A
A
does not contain
A
i
A_{i}
A
i
. Alternative formulation. Let
X
X
X
be a finite set with
n
n
n
elements and
A
1
,
A
2
,
…
,
A
m
A_{1},A_{2},\ldots, A_{m}
A
1
,
A
2
,
…
,
A
m
be three-elements subsets of
X
X
X
, such that
∣
A
i
∩
A
j
∣
≤
1
|A_{i}\cap A_{j}|\leq 1
∣
A
i
∩
A
j
∣
≤
1
, for every
i
≠
j
i\neq j
i
=
j
. Prove that there exists
A
⊆
X
A\subseteq X
A
⊆
X
with
∣
A
∣
≥
⌊
2
n
⌋
|A|\geq \lfloor \sqrt{2n}\rfloor
∣
A
∣
≥
⌊
2
n
⌋
, such that none of
A
i
A_{i}
A
i
's is a subset of
A
A
A
.
9
1
Hide problems
ugly geometry
Let
O
,
A
,
B
,
C
O,A,B,C
O
,
A
,
B
,
C
be variable points in the plane such that
O
A
=
4
OA=4
O
A
=
4
,
O
B
=
2
3
OB=2\sqrt3
OB
=
2
3
and
O
C
=
22
OC=\sqrt {22}
OC
=
22
. Find the maximum value of the area
A
B
C
ABC
A
BC
. Mihai Baluna
4
1
Hide problems
Product(x_i)=1
Show that for all positive real numbers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
with product 1, the following inequality holds
1
n
−
1
+
x
1
+
1
n
−
1
+
x
2
+
⋯
+
1
n
−
1
+
x
n
≤
1.
\frac 1{n-1+x_1 } +\frac 1{n-1+x_2} + \cdots + \frac 1{n-1+x_n} \leq 1.
n
−
1
+
x
1
1
+
n
−
1
+
x
2
1
+
⋯
+
n
−
1
+
x
n
1
≤
1.
1
1
Hide problems
sum of digits divisible by...
a) Prove that it is possible to choose one number out of any 39 consecutive positive integers, having the sum of its digits divisible by 11; b) Find the first 38 consecutive positive integers none of which have the sum of its digits divisible by 11.