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Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2003 Moldova Team Selection Test
2003 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(4)
4
3
Hide problems
Infinitely many solutions (a,b,c)
Prove that the equation \frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c}\plus{}\frac{1}{abc} \equal{} \frac {12}{a \plus{} b \plus{} c} has infinitely many solutions
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
in natural numbers.
Minimum number of coins on a chesstable
On the fields of a chesstable of dimensions
n
×
n
n\times n
n
×
n
, where
n
≥
4
n\geq 4
n
≥
4
is a natural number, are being put coins. We shall consider a diagonal of table each diagonal formed by at least
2
2
2
fields. What is the minimum number of coins put on the table, s.t. on each column, row and diagonal there is at least one coin? Explain your answer.
Balanced table containing the numbers 1,2,...,n^2
A square-table of dimensions
n
×
n
n\times n
n
×
n
, where
n
∈
N
∗
n\in N^*
n
∈
N
∗
, is filled arbitrarly with the numbers
1
,
2
,
.
.
.
,
n
2
1,2,...,n^2
1
,
2
,
...
,
n
2
such that every number appears on the table exactly one time. From each row of the table is chosen the least number and then denote by
x
x
x
the biggest number from the numbers chosen. From each column of the table is chosen the least number and then denote by
y
y
y
the biggest number from the numbers chosen. The table is called balanced iff x \equal{} y. How many balanced tables we can obtain?
3
3
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Prove that FI and BC are perpendicular
The sides
[
A
B
]
[AB]
[
A
B
]
and
[
A
C
]
[AC]
[
A
C
]
of the triangle
A
B
C
ABC
A
BC
are tangent to the incircle with center
I
I
I
of the
△
A
B
C
\triangle ABC
△
A
BC
at the points
M
M
M
and
N
N
N
, respectively. The internal bisectors of the
△
A
B
C
\triangle ABC
△
A
BC
drawn form
B
B
B
and
C
C
C
intersect the line
M
N
MN
MN
at the points
P
P
P
and
Q
Q
Q
, respectively. Suppose that
F
F
F
is the intersection point of the lines
C
P
CP
CP
and
B
Q
BQ
BQ
. Prove that
F
I
⊥
B
C
FI\perp BC
F
I
⊥
BC
.
Four points are concyclic
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral inscribed in a circle of center
O
O
O
. Let M and N be the midpoints of diagonals
A
C
AC
A
C
and
B
D
BD
B
D
, respectively and let
P
P
P
be the intersection point of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
of the given quadrilateral .It is known that the points
O
,
M
,
N
p
O,M,Np
O
,
M
,
Np
are distinct. Prove that the points
O
,
N
,
A
,
C
O,N,A,C
O
,
N
,
A
,
C
are concyclic if and only if the points
O
,
M
,
B
,
D
O,M,B,D
O
,
M
,
B
,
D
are concyclic. Proposer: Dorian Croitoru
Locus of a point satisfying a given relation
Consider a point
M
M
M
found in the same plane with the triangle
A
B
C
ABC
A
BC
, but not found on any of the lines
A
B
,
B
C
AB,BC
A
B
,
BC
and
C
A
CA
C
A
. Denote by
S
1
,
S
2
S_1,S_2
S
1
,
S
2
and
S
3
S_3
S
3
the areas of the triangles
A
M
B
,
B
M
C
AMB,BMC
A
MB
,
BMC
and
C
M
A
CMA
CM
A
, respectively. Find the locus of
M
M
M
satisfying the relation: (MA^2\plus{}MB^2\plus{}MC^2)^2\equal{}16(S_1^2\plus{}S_2^2\plus{}S_3^2)
1
3
Hide problems
Polynomial with positive real roots
Let
n
>
0
n>0
n
>
0
be a natural number. Determine all the polynomials of degree
2
n
2n
2
n
with real coefficients in the form P(X)\equal{}X^{2n}\plus{}(2n\minus{}10)X^{2n\minus{}1}\plus{}a_2X^{2n\minus{}2}\plus{}...\plus{}a_{2n\minus{}2}X^2\plus{}(2n\minus{}10)X\plus{}1, if it is known that all the roots of them are positive reals. Proposer: Baltag Valeriu
Sides of triangle divided into 2002 congruent segments
Each side of an arbitrarly triangle is divided into
2002
2002
2002
congruent segments. After that, each vertex is joined with all "division" points on the opposite side. Prove that the number of the regions formed, in which the triangle is divided, is divisible by
6
6
6
. Proposer: Dorian Croitoru
Quadratic permutations
Let
n
∈
N
∗
n\in N^*
n
∈
N
∗
. A permutation
(
a
1
,
a
2
,
.
.
.
,
a
n
)
(a_1,a_2,...,a_n)
(
a
1
,
a
2
,
...
,
a
n
)
of the numbers
(
1
,
2
,
.
.
.
,
n
)
(1,2,...,n)
(
1
,
2
,
...
,
n
)
is called quadratic iff at least one of the numbers a_1,a_1\plus{}a_2,...,a_1\plus{}a_2\plus{}a\plus{}...\plus{}a_n is a perfect square. Find the greatest natural number
n
≤
2003
n\leq 2003
n
≤
2003
, such that every permutation of
(
1
,
2
,
.
.
.
,
n
)
(1,2,...,n)
(
1
,
2
,
...
,
n
)
is quadratic.
2
3
Hide problems
Three variables cyclic inequality
The positive reals
x
,
y
x,y
x
,
y
and
z
z
z
are satisfying the relation x \plus{} y \plus{} z\geq 1. Prove that: \frac {x\sqrt {x}}{y \plus{} z} \plus{} \frac {y\sqrt {y}}{z \plus{} x} \plus{} \frac {z\sqrt {z}}{x \plus{} y}\geq \frac {\sqrt {3}}{2} Proposer: Baltag Valeriu
Geometric inequality involving lenghts of bisectors
Consider the triangle
A
B
C
ABC
A
BC
with side-lenghts equal to
a
,
b
,
c
a,b,c
a
,
b
,
c
. Let p\equal{}\frac{a\plus{}b\plus{}c}{2},
R
R
R
-the radius of circumcircle of the triangle
A
B
C
ABC
A
BC
,
r
r
r
-the radius of the incircle of the triangle
A
B
C
ABC
A
BC
and let
l
a
,
l
b
,
l
c
l_a,l_b,l_c
l
a
,
l
b
,
l
c
be the lenghts of bisectors drawn from
A
,
B
A,B
A
,
B
and
C
C
C
, respectively, in the triangle
A
B
C
ABC
A
BC
. Prove that: l_al_b\plus{}l_bl_c\plus{}l_cl_a\leq p\sqrt{3r^2\plus{}12Rr} Proposer: Baltag Valeriu
Minimum and maximum value of a sum
Let
a
1
,
a
2
,
.
.
.
,
a
2003
≥
0
a_1,a_2,...,a_{2003}\geq 0
a
1
,
a
2
,
...
,
a
2003
≥
0
, such that a_1\plus{}a_2\plus{}...\plus{}a_{2003}\equal{}2 and a_1a_2\plus{}a_2a_3\plus{}...\plus{}a_{2003}a_1\equal{}1. Determine the minimum and maximum value of a_1^2\plus{}a_2^2\plus{}...\plus{}a_{2003}^2.