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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2016 Moldova Team Selection Test
2016 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(10)
4
1
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the decimal representation of $p^k$ contains $n$ consecutive equal digits.
Show that for every prime number
p
p
p
and every positive integer
n
≥
2
n\geq2
n
≥
2
there exists a positive integer
k
k
k
such that the decimal representation of
p
k
p^k
p
k
contains
n
n
n
consecutive equal digits.
3
1
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Find the smallest value of $\frac{AO_1+BO_1}{AB},$
Let
A
B
C
ABC
A
BC
be a triangle with
∠
C
=
90
\angle C=90
∠
C
=
90
. The tangent points of the inscribed circle with the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
are
M
,
N
M, N
M
,
N
and
P
.
P.
P
.
Points
M
1
,
N
1
,
P
1
M_1, N_1, P_1
M
1
,
N
1
,
P
1
are symmetric to points
M
,
N
,
P
M, N, P
M
,
N
,
P
with respect to midpoints of sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
.
AB.
A
B
.
Find the smallest value of
A
O
1
+
B
O
1
A
B
,
\frac{AO_1+BO_1}{AB},
A
B
A
O
1
+
B
O
1
,
where
O
1
O_1
O
1
is the circumcenter of triangle
M
1
N
1
P
1
.
M_1N_1P_1.
M
1
N
1
P
1
.
11
1
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yet another geometry
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral. Circle with diameter
A
B
AB
A
B
intersects
C
A
CA
C
A
,
C
B
CB
CB
,
D
A
DA
D
A
, and
D
B
DB
D
B
in
E
E
E
,
F
F
F
,
G
G
G
, and
H
H
H
, respectively (all different from
A
A
A
and
B
B
B
). The lines
E
F
EF
EF
and
G
H
GH
G
H
intersect in
I
I
I
. Prove that the bisector of
∠
G
I
F
\angle GIF
∠
G
I
F
and the line
C
D
CD
C
D
are perpendicular.
10
1
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Ez Pascal and Polars
Let
A
1
A
2
⋯
A
14
A_{1}A_{2} \cdots A_{14}
A
1
A
2
⋯
A
14
be a regular
14
−
14-
14
−
gon. Prove that
A
1
A
3
∩
A
5
A
11
∩
A
6
A
9
≠
∅
A_{1}A_{3}\cap A_{5}A_{11}\cap A_{6}A_{9}\ne \emptyset
A
1
A
3
∩
A
5
A
11
∩
A
6
A
9
=
∅
.
9
1
Hide problems
you wouldn't believe this is from TST if I didn't tell you
Let
α
∈
(
0
,
π
2
)
\alpha \in \left( 0, \dfrac{\pi}{2}\right)
α
∈
(
0
,
2
π
)
.Find the minimum value of the expression
P
=
(
1
+
cos
α
)
(
1
+
1
sin
α
)
+
(
1
+
sin
α
)
(
1
+
1
cos
α
)
.
P = (1+\cos\alpha)\left(1+\frac{1}{\sin \alpha} \right)+(1+\sin \alpha)\left(1+\frac{1}{\cos \alpha} \right) .
P
=
(
1
+
cos
α
)
(
1
+
sin
α
1
)
+
(
1
+
sin
α
)
(
1
+
cos
α
1
)
.
7
1
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geo... again >:(
Let
Ω
\Omega
Ω
and
O
O
O
be the circumcircle of acute triangle
A
B
C
ABC
A
BC
and its center, respectively.
M
≠
O
M\ne O
M
=
O
is an arbitrary point in the interior of
A
B
C
ABC
A
BC
such that
A
M
AM
A
M
,
B
M
BM
BM
, and
C
M
CM
CM
intersect
Ω
\Omega
Ω
at
A
1
A_{1}
A
1
,
B
1
B_{1}
B
1
, and
C
1
C_{1}
C
1
, respectiuvely. Let
A
2
A_{2}
A
2
,
B
2
B_{2}
B
2
, and
C
2
C_{2}
C
2
be the circumcenters of
M
B
C
MBC
MBC
,
M
C
A
MCA
MC
A
, and
M
A
B
MAB
M
A
B
, respectively. It is to be proven that
A
1
A
2
A_{1}A_{2}
A
1
A
2
,
B
1
B
2
B_{1}B_{2}
B
1
B
2
,
C
1
C
2
C_{1}C{2}
C
1
C
2
concur.
6
1
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only for 200iq kids
Let
n
∈
Z
>
0
n\in \mathbb{Z}_{> 0}
n
∈
Z
>
0
. The set
S
S
S
contains all positive integers written in decimal form that simultaneously satisfy the following conditions: [*] each element of
S
S
S
has exactly
n
n
n
digits; [*] each element of
S
S
S
is divisible by
3
3
3
; [*] each element of
S
S
S
has all its digits from the set
{
3
,
5
,
7
,
9
}
\{3,5,7,9\}
{
3
,
5
,
7
,
9
}
Find
∣
S
∣
\mid S\mid
∣
S
∣
5
1
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Moldova 2016 tst B4
The sequence of polynomials
(
P
n
(
X
)
)
n
∈
Z
>
0
\left( P_{n}(X)\right)_{n\in Z_{>0}}
(
P
n
(
X
)
)
n
∈
Z
>
0
is defined as follows:
P
1
(
X
)
=
2
X
P_{1}(X)=2X
P
1
(
X
)
=
2
X
P
2
(
X
)
=
2
(
X
2
+
1
)
P_{2}(X)=2(X^2+1)
P
2
(
X
)
=
2
(
X
2
+
1
)
P
n
+
2
(
X
)
=
2
X
⋅
P
n
+
1
(
X
)
−
(
X
2
−
1
)
P
n
(
X
)
P_{n+2}(X)=2X\cdot P_{n+1}(X)-(X^2-1)P_{n}(X)
P
n
+
2
(
X
)
=
2
X
⋅
P
n
+
1
(
X
)
−
(
X
2
−
1
)
P
n
(
X
)
, for all positive integers
n
n
n
. Find all
n
n
n
for which
X
2
+
1
∣
P
n
(
X
)
X^2+1\mid P_{n}(X)
X
2
+
1
∣
P
n
(
X
)
8
1
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n balls with different rays
Let us have
n
n
n
(
n
>
3
n>3
n
>
3
) balls with different rays. On each ball it is written an integer number. Determine the greatest natural number
d
d
d
such that for any numbers written on the balls, we can always find at least 4 different ways to choose some balls with the sum of the numbers written on them divisible by
d
d
d
.
1
1
Hide problems
Moldova TST 2016,B1
If
x
1
,
x
2
,
.
.
.
,
x
n
>
0
x_1,x_2,...,x_n>0
x
1
,
x
2
,
...
,
x
n
>
0
and
x
1
2
+
x
2
2
+
.
.
.
+
x
n
2
=
1
n
x_1^2+x_2^2+...+x_n^2=\dfrac{1}{n}
x
1
2
+
x
2
2
+
...
+
x
n
2
=
n
1
,prove that
∑
x
i
+
∑
1
x
i
⋅
x
i
+
1
≥
n
3
+
1.
\sum x_i+\sum \dfrac{1}{x_i \cdot x_{i+1}} \ge n^3+1.
∑
x
i
+
∑
x
i
⋅
x
i
+
1
1
≥
n
3
+
1.