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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2008 Moldova Team Selection Test
2008 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(4)
4
3
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Probably Known Property of homogeneous Polynomials in R[X,Y]
A non-zero polynomial
S
∈
R
[
X
,
Y
]
S\in\mathbb{R}[X,Y]
S
∈
R
[
X
,
Y
]
is called homogeneous of degree
d
d
d
if there is a positive integer
d
d
d
so that S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y) for any
λ
∈
R
\lambda\in\mathbb{R}
λ
∈
R
. Let
P
,
Q
∈
R
[
X
,
Y
]
P,Q\in\mathbb{R}[X,Y]
P
,
Q
∈
R
[
X
,
Y
]
so that
Q
Q
Q
is homogeneous and
P
P
P
divides
Q
Q
Q
(that is,
P
∣
Q
P|Q
P
∣
Q
). Prove that
P
P
P
is homogeneous too.
Nice: find number of even permutations with no fixed points
Find the number of even permutations of
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n\}
{
1
,
2
,
…
,
n
}
with no fixed points.
Good sets and coloring of all positive integers.
A non-empty set
S
S
S
of positive integers is said to be good if there is a coloring with
2008
2008
2008
colors of all positive integers so that no number in
S
S
S
is the sum of two different positive integers (not necessarily in
S
S
S
) of the same color. Find the largest value
t
t
t
can take so that the set S\equal{}\{a\plus{}1,a\plus{}2,a\plus{}3,\ldots,a\plus{}t\} is good, for any positive integer
a
a
a
. [hide="P.S."]I have the feeling that I've seen this problem before, so if I'm right, maybe someone can post some links...
3
3
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Centroids of triangles inscribed/circumscribed to 2 circles.
Let
Γ
(
I
,
r
)
\Gamma(I,r)
Γ
(
I
,
r
)
and
Γ
(
O
,
R
)
\Gamma(O,R)
Γ
(
O
,
R
)
denote the incircle and circumcircle, respectively, of a triangle
A
B
C
ABC
A
BC
. Consider all the triangels
A
i
B
i
C
i
A_iB_iC_i
A
i
B
i
C
i
which are simultaneously inscribed in
Γ
(
O
,
R
)
\Gamma(O,R)
Γ
(
O
,
R
)
and circumscribed to
Γ
(
I
,
r
)
\Gamma(I,r)
Γ
(
I
,
r
)
. Prove that the centroids of these triangles are concyclic.
Prove that some circles are concurrent.
Let
ω
\omega
ω
be the circumcircle of
A
B
C
ABC
A
BC
and let
D
D
D
be a fixed point on
B
C
BC
BC
,
D
≠
B
D\neq B
D
=
B
,
D
≠
C
D\neq C
D
=
C
. Let
X
X
X
be a variable point on
(
B
C
)
(BC)
(
BC
)
,
X
≠
D
X\neq D
X
=
D
. Let
Y
Y
Y
be the second intersection point of
A
X
AX
A
X
and
ω
\omega
ω
. Prove that the circumcircle of
X
Y
D
XYD
X
Y
D
passes through a fixed point.
Some circles and a constant segment.
In triangle
A
B
C
ABC
A
BC
the bisector of
∠
A
C
B
\angle ACB
∠
A
CB
intersects
A
B
AB
A
B
at
D
D
D
. Consider an arbitrary circle
O
O
O
passing through
C
C
C
and
D
D
D
, so that it is not tangent to
B
C
BC
BC
or
C
A
CA
C
A
. Let O\cap BC \equal{} \{M\} and O\cap CA \equal{} \{N\}. a) Prove that there is a circle
S
S
S
so that
D
M
DM
D
M
and
D
N
DN
D
N
are tangent to
S
S
S
in
M
M
M
and
N
N
N
, respectively. b) Circle
S
S
S
intersects lines
B
C
BC
BC
and
C
A
CA
C
A
in
P
P
P
and
Q
Q
Q
respectively. Prove that the lengths of
M
P
MP
MP
and
N
Q
NQ
NQ
do not depend on the choice of circle
O
O
O
.
2
3
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Minimum Value of a Non-symmetric Expression
Let
a
1
,
…
,
a
n
a_1,\ldots,a_n
a
1
,
…
,
a
n
be positive reals so that a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le\frac n2. Find the minimal value of \sqrt{a_1^2\plus{}\frac1{a_2^2}}\plus{}\sqrt{a_2^2\plus{}\frac1{a_3^2}}\plus{}\ldots\plus{}\sqrt{a_n^2\plus{}\frac1{a_1^2}}.
Again Warm-Up Prob for a TST-Partition of a set into triples
We say the set
{
1
,
2
,
…
,
3
k
}
\{1,2,\ldots,3k\}
{
1
,
2
,
…
,
3
k
}
has property
D
D
D
if it can be partitioned into disjoint triples so that in each of them a number equals the sum of the other two. (a) Prove that
{
1
,
2
,
…
,
3324
}
\{1,2,\ldots,3324\}
{
1
,
2
,
…
,
3324
}
has property
D
D
D
. (b) Prove that
{
1
,
2
,
…
,
3309
}
\{1,2,\ldots,3309\}
{
1
,
2
,
…
,
3309
}
hasn't property
D
D
D
.
Prove that a binomial coefficient is not divisible by p.
Let
p
p
p
be a prime number and
k
,
n
k,n
k
,
n
positive integers so that \gcd(p,n)\equal{}1. Prove that
(
n
⋅
p
k
p
k
)
\binom{n\cdot p^k}{p^k}
(
p
k
n
⋅
p
k
)
and
p
p
p
are coprime.
1
3
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Warm-up Diophantine Equation.
Let
p
p
p
be a prime number. Solve in
N
0
×
N
0
\mathbb{N}_0\times\mathbb{N}_0
N
0
×
N
0
the equation x^3\plus{}y^3\minus{}3xy\equal{}p\minus{}1.
Solve a system in real numbers.
Find all solutions
(
x
,
y
)
∈
R
×
R
(x,y)\in \mathbb{R}\times\mathbb R
(
x
,
y
)
∈
R
×
R
of the following system: \begin{cases}x^3 \plus{} 3xy^2 \equal{} 49, \\ x^2 \plus{} 8xy \plus{} y^2 \equal{} 8y \plus{} 17x.\end{cases}
Find a solution for a^2+b^2+c^2+d^2+e^2=abcde in integers>0.
Determine a subset
A
⊂
N
∗
A\subset \mathbb{N}^*
A
⊂
N
∗
having
5
5
5
different elements, so that the sum of the squares of its elements equals their product. Do not simply post the subset, show how you found it.