MathDB
Problems
Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
1996 Romania Team Selection Test
1996 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(15)
16
1
Hide problems
System with 9 coefficients
Let
n
≥
3
n\geq 3
n
≥
3
be an integer and let
S
⊂
{
1
,
2
,
…
,
n
3
}
\mathcal{S} \subset \{1,2,\ldots, n^3\}
S
⊂
{
1
,
2
,
…
,
n
3
}
be a set with
3
n
2
3n^2
3
n
2
elements. Prove that there exist nine distinct numbers
a
1
,
a
2
,
…
,
a
9
∈
S
a_1,a_2,\ldots,a_9 \in \mathcal{S}
a
1
,
a
2
,
…
,
a
9
∈
S
such that the following system has a solution in nonzero integers: \begin{eqnarray*} a_1x + a_2y +a_3 z &=& 0 \\ a_4x + a_5 y + a_6 z &=& 0 \\ a_7x + a_8y + a_9z &=& 0. \end{eqnarray*} Marius Cavachi
14
1
Hide problems
Quadrilateral condition in inequality terms
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be real numbers. Prove that the following conditions are equivalent: (i)
x
,
y
,
z
x,y,z
x
,
y
,
z
are positive numbers and
1
x
+
1
y
+
1
z
≤
1
\dfrac 1x + \dfrac 1y + \dfrac 1z \leq 1
x
1
+
y
1
+
z
1
≤
1
; (ii)
a
2
x
+
b
2
y
+
c
2
z
>
d
2
a^2x+b^2y+c^2z>d^2
a
2
x
+
b
2
y
+
c
2
z
>
d
2
holds for every quadrilateral with sides
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
.
13
1
Hide problems
square root begs for a well known inequality
Let
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
be positive real numbers and
x
n
+
1
=
x
1
+
x
2
+
⋯
+
x
n
x_{n+1} = x_1 + x_2 + \cdots + x_n
x
n
+
1
=
x
1
+
x
2
+
⋯
+
x
n
. Prove that
∑
k
=
1
n
x
k
(
x
n
+
1
−
x
k
)
≤
∑
k
=
1
n
x
n
+
1
(
x
n
+
1
−
x
k
)
.
\sum_{k=1}^n \sqrt { x_k (x_{n+1} - x_k)} \leq \sqrt { \sum_{k=1}^n x_{n+1}(x_{n+1}-x_k)}.
k
=
1
∑
n
x
k
(
x
n
+
1
−
x
k
)
≤
k
=
1
∑
n
x
n
+
1
(
x
n
+
1
−
x
k
)
.
Mircea Becheanu
12
1
Hide problems
M is a set of n points with should lie on a circle
Let
n
≥
3
n\geq 3
n
≥
3
be an integer and let
p
≥
2
n
−
3
p\geq 2n-3
p
≥
2
n
−
3
be a prime number. For a set
M
M
M
of
n
n
n
points in the plane, no 3 collinear, let
f
:
M
→
{
0
,
1
,
…
,
p
−
1
}
f: M\to \{0,1,\ldots, p-1\}
f
:
M
→
{
0
,
1
,
…
,
p
−
1
}
be a function such that (i) exactly one point of
M
M
M
maps to 0, (ii) if a circle
C
\mathcal{C}
C
passes through 3 distinct points of
A
,
B
,
C
∈
M
A,B,C\in M
A
,
B
,
C
∈
M
then
∑
P
∈
M
∩
C
f
(
P
)
≡
0
(
m
o
d
p
)
\sum_{P\in M\cap \mathcal{C}} f(P) \equiv 0 \pmod p
∑
P
∈
M
∩
C
f
(
P
)
≡
0
(
mod
p
)
. Prove that all the points in
M
M
M
lie on a circle.
11
1
Hide problems
\alpha^{3pq} -\alpha \equiv 0 \pmod {3pq}
Find all primes
p
,
q
p,q
p
,
q
such that
α
3
p
q
−
α
≡
0
(
m
o
d
3
p
q
)
\alpha^{3pq} -\alpha \equiv 0 \pmod {3pq}
α
3
pq
−
α
≡
0
(
mod
3
pq
)
for all integers
α
\alpha
α
.
10
1
Hide problems
vertices of a rectangle between colored latticial points
Let
n
n
n
and
r
r
r
be positive integers and
A
A
A
be a set of lattice points in the plane such that any open disc of radius
r
r
r
contains a point of
A
A
A
. Show that for any coloring of the points of
A
A
A
in
n
n
n
colors there exists four points of the same color which are the vertices of a rectangle.
9
1
Hide problems
Find the minimal value of F(x_1,x_2,\ldots,x_n)
Let
n
≥
3
n\geq 3
n
≥
3
be an integer and let
x
1
,
x
2
,
…
,
x
n
−
1
x_1,x_2,\ldots,x_{n-1}
x
1
,
x
2
,
…
,
x
n
−
1
be nonnegative integers such that \begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n \\ x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*} Find the minimal value of
F
(
x
1
,
x
2
,
…
,
x
n
)
=
∑
k
=
1
n
−
1
k
(
2
n
−
k
)
x
k
F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k
F
(
x
1
,
x
2
,
…
,
x
n
)
=
∑
k
=
1
n
−
1
k
(
2
n
−
k
)
x
k
.
5
1
Hide problems
Three circles internally and externally tangent
Let
A
A
A
and
B
B
B
be points on a circle
C
\mathcal{C}
C
with center
O
O
O
such that
∠
A
O
B
=
π
2
\angle AOB = \dfrac {\pi}2
∠
A
OB
=
2
π
. Circles
C
1
\mathcal{C}_1
C
1
and
C
2
\mathcal{C}_2
C
2
are internally tangent to
C
\mathcal{C}
C
at
A
A
A
and
B
B
B
respectively and are also externally tangent to one another. The circle
C
3
\mathcal{C}_3
C
3
lies in the interior of
∠
A
O
B
\angle AOB
∠
A
OB
and it is tangent externally to
C
1
\mathcal{C}_1
C
1
,
C
2
\mathcal{C}_2
C
2
at
P
P
P
and
R
R
R
and internally tangent to
C
\mathcal{C}
C
at
S
S
S
. Evaluate the value of
∠
P
S
R
\angle PSR
∠
PSR
.
4
1
Hide problems
ABCD cyclic quadrilateral and 16 incenters
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral and let
M
M
M
be the set of incenters and excenters of the triangles
B
C
D
BCD
BC
D
,
C
D
A
CDA
C
D
A
,
D
A
B
DAB
D
A
B
,
A
B
C
ABC
A
BC
(so 16 points in total). Prove that there exist two sets
K
\mathcal{K}
K
and
L
\mathcal{L}
L
of four parallel lines each, such that every line in
K
∪
L
\mathcal{K} \cup \mathcal{L}
K
∪
L
contains exactly four points of
M
M
M
.
2
1
Hide problems
Find greatest positive integer for which there exist ...
Find the greatest positive integer
n
n
n
for which there exist
n
n
n
nonnegative integers
x
1
,
x
2
,
…
,
x
n
x_1, x_2,\ldots , x_n
x
1
,
x
2
,
…
,
x
n
, not all zero, such that for any
ε
1
,
ε
2
,
…
,
ε
n
\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n
ε
1
,
ε
2
,
…
,
ε
n
from the set
{
−
1
,
0
,
1
}
\{-1, 0, 1\}
{
−
1
,
0
,
1
}
, not all zero,
ε
1
x
1
+
ε
2
x
2
+
⋯
+
ε
n
x
n
\varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n
ε
1
x
1
+
ε
2
x
2
+
⋯
+
ε
n
x
n
is not divisible by
n
3
n^3
n
3
.
15
1
Hide problems
Function on a set of n concentric circles
Let
S
S
S
be a set of
n
n
n
concentric circles in the plane. Prove that if a function
f
:
S
→
S
f: S\to S
f
:
S
→
S
satisfies the property
d
(
f
(
A
)
,
f
(
B
)
)
≥
d
(
A
,
B
)
d( f(A),f(B)) \geq d(A,B)
d
(
f
(
A
)
,
f
(
B
))
≥
d
(
A
,
B
)
for all
A
,
B
∈
S
A,B \in S
A
,
B
∈
S
, then
d
(
f
(
A
)
,
f
(
B
)
)
=
d
(
A
,
B
)
d(f(A),f(B)) = d(A,B)
d
(
f
(
A
)
,
f
(
B
))
=
d
(
A
,
B
)
, where
d
d
d
is the euclidean distance function.
7
1
Hide problems
Summation function
Let
a
∈
R
a\in \mathbb{R}
a
∈
R
and
f
1
(
x
)
,
f
2
(
x
)
,
…
,
f
n
(
x
)
:
R
→
R
f_1(x),f_2(x),\ldots,f_n(x): \mathbb{R} \rightarrow \mathbb{R}
f
1
(
x
)
,
f
2
(
x
)
,
…
,
f
n
(
x
)
:
R
→
R
are the additive functions such that for every
x
∈
R
x\in \mathbb{R}
x
∈
R
we have
f
1
(
x
)
f
2
(
x
)
⋯
f
n
(
x
)
=
a
x
n
f_1(x)f_2(x) \cdots f_n(x) =ax^n
f
1
(
x
)
f
2
(
x
)
⋯
f
n
(
x
)
=
a
x
n
. Show that there exists
b
∈
R
b\in \mathbb {R}
b
∈
R
and
i
∈
{
1
,
2
,
…
,
n
}
i\in {\{1,2,\ldots,n}\}
i
∈
{
1
,
2
,
…
,
n
}
such that for every
x
∈
R
x\in \mathbb{R}
x
∈
R
we have
f
i
(
x
)
=
b
x
f_i(x)=bx
f
i
(
x
)
=
b
x
.
1
1
Hide problems
n-gon function
Let
f
:
R
2
→
R
f: \mathbb{R}^2 \rightarrow \mathbb{R}
f
:
R
2
→
R
be a function such that for every regular
n
n
n
-gon
A
1
A
2
…
A
n
A_1A_2 \ldots A_n
A
1
A
2
…
A
n
we have
f
(
A
1
)
+
f
(
A
2
)
+
⋯
+
f
(
A
n
)
=
0
f(A_1)+f(A_2)+\cdots +f(A_n)=0
f
(
A
1
)
+
f
(
A
2
)
+
⋯
+
f
(
A
n
)
=
0
. Prove that
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
for all reals
x
x
x
.
8
1
Hide problems
very beautiful problem
Let
p
1
,
p
2
,
…
,
p
k
p_1,p_2,\ldots,p_k
p
1
,
p
2
,
…
,
p
k
be the distinct prime divisors of
n
n
n
and let
a
n
=
1
p
1
+
1
p
2
+
⋯
+
1
p
k
a_n=\frac {1}{p_1}+\frac {1}{p_2}+\cdots+\frac {1}{p_k}
a
n
=
p
1
1
+
p
2
1
+
⋯
+
p
k
1
for
n
≥
2
n\geq 2
n
≥
2
. Show that for every positive integer
N
≥
2
N\geq 2
N
≥
2
the following inequality holds:
∑
k
=
2
N
a
2
a
3
⋯
a
k
<
1
\sum_{k=2}^{N} a_2a_3 \cdots a_k <1
∑
k
=
2
N
a
2
a
3
⋯
a
k
<
1
Laurentiu Panaitopol
3
1
Hide problems
not easy ,i think
Let
x
,
y
∈
R
x,y\in \mathbb{R}
x
,
y
∈
R
. Show that if the set
A
x
,
y
=
{
cos
(
n
π
x
)
+
cos
(
n
π
y
)
∣
n
∈
N
}
A_{x,y}=\{ \cos {(n\pi x)}+\cos {(n\pi y)} \mid n\in \mathbb{N}\}
A
x
,
y
=
{
cos
(
nπ
x
)
+
cos
(
nπ
y
)
∣
n
∈
N
}
is finite then
x
,
y
∈
Q
x,y \in \mathbb{Q}
x
,
y
∈
Q
. Vasile Pop