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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia JBMO TST
2009 Junior Balkan Team Selection Test
2009 Junior Balkan Team Selection Test
Part of
Serbia JBMO TST
Subcontests
(4)
4
2
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Divisibility by 11 with digits 5 and 8
In the decimal expression of a
2009
2009
2009
-digit natural number there are only the digits
5
5
5
and
8
8
8
. Prove that we can get a
2008
2008
2008
-digit number divisible by
11
11
11
if we remove just one digit from the number.
Strange condition, easy inequality
For positive real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
the inequality \frac1{x^2\plus{}1}\plus{}\frac1{y^2\plus{}1}\plus{}\frac1{z^2\plus{}1}\equal{}\frac12 holds. Prove the inequality \frac1{x^3\plus{}2}\plus{}\frac1{y^3\plus{}2}\plus{}\frac1{z^3\plus{}2}<\frac13.
3
2
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Minimum number of fields with figures
On each field of the board
n
×
n
n\times n
n
×
n
there is one figure, where
n
≥
2
n\ge 2
n
≥
2
. In one move we move every figure on one of its diagonally adjacent fields. After one move on one field there can be more than one figure. Find the least number of fields on which there can be all figures after some number of moves.
Proving a kite
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral, such that \angle CBD\equal{}2\cdot\angle ADB, \angle ABD\equal{}2\cdot\angle CDB and AB\equal{}CB. Prove that quadrilateral
A
B
C
D
ABCD
A
BC
D
is a kite.
2
2
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Ratio in isosceles right triangle
In isosceles right triangle
A
B
C
ABC
A
BC
a circle is inscribed. Let
C
D
CD
C
D
be the hypotenuse height (
D
∈
A
B
D\in AB
D
∈
A
B
), and let
P
P
P
be the intersection of inscribed circle and height
C
D
CD
C
D
. In which ratio does the circle divide segment
A
P
AP
A
P
?
Choosing numbers with condition
From the set
{
1
,
2
,
3
,
…
,
2009
}
\{1,2,3,\ldots,2009\}
{
1
,
2
,
3
,
…
,
2009
}
we choose
1005
1005
1005
numbers, such that sum of any
2
2
2
numbers isn't neither
2009
2009
2009
nor
2010
2010
2010
. Find all ways on we can choose these
1005
1005
1005
numbers.
1
2
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Expressing number as a sum of 2 squares
Given are natural numbers
a
,
b
a,b
a
,
b
and
n
n
n
such that a^2\plus{}2nb^2 is a complete square. Prove that the number a^2\plus{}nb^2 can be written as a sum of squares of
2
2
2
natural numbers.
Easy divisibility
Find all two digit numbers
A
B
‾
\overline{AB}
A
B
such that
A
B
‾
\overline{AB}
A
B
divides
A
0
B
‾
\overline{A0B}
A
0
B
.