MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia JBMO TST
2016 Junior Balkan Team Selection Test
2016 Junior Balkan Team Selection Test
Part of
Serbia JBMO TST
Subcontests
(4)
3
1
Hide problems
Serbia Junior TST 2016
In two neigbouring cells(dimensions
1
×
1
1\times 1
1
×
1
) of square table
10
×
10
10\times 10
10
×
10
there is hidden treasure. John needs to guess these cells. In one
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<span class='latex-italic'>move</span>
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he can choose some cell of the table and can get information whether there is treasure in it or not. Determine minimal number of
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<span class='latex-italic'>move</span>
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's, with properly strategy, that always allows John to find cells in which is treasure hidden.
4
1
Hide problems
Serbia Junior TST 2016
Let
a
,
b
,
c
∈
R
+
a,b,c\in \mathbb{R}^+
a
,
b
,
c
∈
R
+
, prove that:
2
a
3
a
+
b
+
2
b
3
b
+
c
+
2
c
3
c
+
a
≤
3
(
a
+
b
+
c
)
\frac{2a}{\sqrt{3a+b}}+\frac{2b}{\sqrt{3b+c}}+\frac{2c}{\sqrt{3c+a}}\leq \sqrt{3(a+b+c)}
3
a
+
b
2
a
+
3
b
+
c
2
b
+
3
c
+
a
2
c
≤
3
(
a
+
b
+
c
)
1
1
Hide problems
Serbian Junior TST
Let rightangled
△
A
B
C
\triangle ABC
△
A
BC
be given with right angle at vertex
C
C
C
. Let
D
D
D
be foot of altitude from
C
C
C
and let
k
k
k
be circle that touches
B
D
BD
B
D
at
E
E
E
,
C
D
CD
C
D
at
F
F
F
and circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
G
G
G
.
a
.
)
a.)
a
.
)
Prove that points
A
A
A
,
F
F
F
and
G
G
G
are collinear.
b
.
)
b.)
b
.
)
Express radius of circle
k
k
k
in terms of sides of
△
A
B
C
\triangle ABC
△
A
BC
.
2
1
Hide problems
Minimal number of divisors
Find minimal number of divisors that can number
∣
201
6
m
−
3
6
n
∣
|2016^m-36^n|
∣201
6
m
−
3
6
n
∣
have,where
m
m
m
and
n
n
n
are natural numbers.