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National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2016 Junior Balkan Team Selection Tests - Moldova
2016 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(8)
8
1
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Nicu plays the next game on the computer
Nicu plays the Next game on the computer. Initially the number
S
S
S
in the computer has the value
S
=
0
S = 0
S
=
0
. At each step Nicu chooses a certain number
a
a
a
(
0
<
a
<
1
0 <a <1
0
<
a
<
1
) and enters it in computer. The computer arbitrarily either adds this number
a
a
a
to the number
S
S
S
or it subtracts from
S
S
S
and displays on the screen the new result for
S
S
S
. After that Nicu does Next step. It is known that among any
100
100
100
consecutive operations the computer the at least once apply the assembly. Give an arbitrary number
M
>
0
M> 0
M
>
0
. Show that there is a strategy for Nicu that will always allow him after a finite number of steps to get a result
S
>
M
S> M
S
>
M
.[hide=original wording]Nicu joacă la calculator următorul joc. Iniţial numărul S din calculator are valoarea S = 0. La fiecare pas Nicu alege un număr oarecare a (0 < a < 1) şi îl introduce în calculator. Calculatorul, în mod arbitrar, sau adună acest număr a la numărul S sau îl scade din S şi afişează pe ecran rezultatul nou pentru S. După aceasta Nicu face următorul pas. Se ştie că printre oricare 100 de operaţii consecutive calculatorul cel puţin o dată aplică adunarea. Fie dat un număr arbitrar M > 0. Să se arate că există o strategie pentru Nicu care oricând îi va permite lui după un număr finit de paşi să obţină un rezulat S > M.
6
1
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5^x=y^4+4y+1
Determine all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of natural numbers satisfying the equation
5
x
=
y
4
+
4
y
+
1
5^x=y^4+4y+1
5
x
=
y
4
+
4
y
+
1
.
5
1
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a+b =? if a^3-a^2+a-5=0 and b^3-2b^2+2b+4=0
Real numbers
a
a
a
and
b
b
b
satisfy the system of equations
{
a
3
−
a
2
+
a
−
5
=
0
b
3
−
2
b
2
+
2
b
+
4
=
0
\begin{cases} a^3-a^2+a-5=0 \\ b^3-2b^2+2b+4=0 \end{cases}
{
a
3
−
a
2
+
a
−
5
=
0
b
3
−
2
b
2
+
2
b
+
4
=
0
Find the numerical value of the sum
a
+
b
a+ b
a
+
b
.
7
1
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DF= CG wanted, perpendiculars inside a square
Let
A
B
C
D
ABCD
A
BC
D
ba a square and let point
E
E
E
be the midpoint of side
A
D
AD
A
D
. Points
G
G
G
and
F
F
F
are located on the segment
(
B
E
)
(BE)
(
BE
)
such that the lines
A
G
AG
A
G
and
C
F
CF
CF
are perpendicular on the line
B
E
BE
BE
. Prove that
D
F
=
C
G
DF= CG
D
F
=
CG
.
4
1
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Moldova Junior TST for JBMO 2016 - Problem 4
Find all solutions for (x,y) , both integers such that:
x
y
=
3
(
x
2
+
y
2
−
1
)
xy=3(\sqrt{x^2+y^2}-1)
x
y
=
3
(
x
2
+
y
2
−
1
)
3
1
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Moldova Junior TST for JBMO 2016 - Problem 3
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
∡
C
=
∡
B
=
36
\measuredangle C=\measuredangle B=36
∡
C
=
∡
B
=
36
. The point
M
M
M
is in interior of
A
B
C
ABC
A
BC
such that
∡
M
B
C
=
2
4
∘
,
∡
B
C
M
=
3
0
∘
\measuredangle MBC=24^{\circ} , \measuredangle BCM=30^{\circ}
∡
MBC
=
2
4
∘
,
∡
BCM
=
3
0
∘
N
=
A
M
∩
B
C
.
N = AM \cap BC.
N
=
A
M
∩
BC
.
. Find
∡
M
C
B
\measuredangle MCB
∡
MCB
.
2
1
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Moldova Junior TST for JBMO 2016 - Problem 2
This is a really easy one for Junior level :p
a
2
+
b
2
+
c
2
+
a
b
+
b
c
+
a
c
=
6
a^2+b^2+c^2+ab+bc+ac=6
a
2
+
b
2
+
c
2
+
ab
+
b
c
+
a
c
=
6
a,b,c>0 Find max{a+b+c}
1
1
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Moldova Junior TST for JBMO 2016 - Problem 1
a
3
b
3
\frac{a^3}{b^3}
b
3
a
3
+
a
3
+
1
b
3
+
1
\frac{a^3+1}{b^3+1}
b
3
+
1
a
3
+
1
+...+
a
3
+
2015
b
3
+
2015
\frac{a^3+2015}{b^3+2015}
b
3
+
2015
a
3
+
2015
=2016 b - positive integer, b can't be 0 a - real Find
a
3
b
3
\frac{a^3}{b^3}
b
3
a
3
*
a
3
+
1
b
3
+
1
\frac{a^3+1}{b^3+1}
b
3
+
1
a
3
+
1
*...*
a
3
+
2015
b
3
+
2015
\frac{a^3+2015}{b^3+2015}
b
3
+
2015
a
3
+
2015