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National and Regional Contests
Bulgaria Contests
Bulgaria JBMO Team Selection Test
2014 Bulgaria JBMO TST
2014 Bulgaria JBMO TST
Part of
Bulgaria JBMO Team Selection Test
Subcontests
(8)
5
1
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Geometry in a 45-60-75 triangle
From the foot
D
D
D
of the height
C
D
CD
C
D
in the triangle
A
B
C
,
ABC,
A
BC
,
perpendiculars to
B
C
BC
BC
and
A
C
AC
A
C
are drawn, which they intersect at points
M
M
M
and
N
.
N.
N
.
Let
∠
C
A
B
=
6
0
∘
,
∠
C
B
A
=
4
5
∘
,
\angle CAB = 60^{\circ} , \angle CBA = 45^{\circ} ,
∠
C
A
B
=
6
0
∘
,
∠
CB
A
=
4
5
∘
,
and
H
H
H
be the orthocentre of
M
N
C
.
MNC.
MNC
.
If
O
O
O
is the midpoint of
C
D
,
CD,
C
D
,
find
∠
C
O
H
.
\angle COH.
∠
CO
H
.
1
1
Hide problems
Geometry in a square
Points
M
M
M
and
N
N
N
lie on the sides
B
C
BC
BC
and
C
D
CD
C
D
of the square
A
B
C
D
,
ABCD,
A
BC
D
,
respectively, and
∠
M
A
N
=
4
5
∘
\angle MAN = 45^{\circ}
∠
M
A
N
=
4
5
∘
. The circle through
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
intersects
A
M
AM
A
M
and
A
N
AN
A
N
again at
P
P
P
and
Q
Q
Q
, respectively. Prove that
M
N
∣
∣
P
Q
.
MN || PQ.
MN
∣∣
PQ
.
8
1
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Perfect square for all k
Find the smallest positive integer
n
,
n,
n
,
such that
3
k
+
n
k
+
(
3
n
)
k
+
201
4
k
3^k+n^k+ (3n)^k+ 2014^k
3
k
+
n
k
+
(
3
n
)
k
+
201
4
k
is a perfect square for all natural numbers
k
,
k,
k
,
but not a perfect cube, for all natural numbers
k
.
k.
k
.
7
1
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Broken line in 9x1 rectangle
A
9
×
1
9\times 1
9
×
1
rectangle is divided into unit squares. A broken line, from the lower left to the upper right corner, goes through all
20
20
20
vertices of the unit squares and consists of
19
19
19
line segments. How many such lines are there?
4
1
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Chessboard tromino tiling
Removing a unit square from a
2
×
2
2\times 2
2
×
2
square we get a piece called L-tromino. From the fourth line of a
7
×
7
7 \times 7
7
×
7
cheesboard some unit squares have been removed. The resulting chessboard is cut in L-trominos. Determine the number and location of the removed squares.
3
1
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Perfect square
Determine the last four digits of a perfect square of a natural number, knowing that the last three of them are the same.
2
1
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Maximum of a+b+c
Find the maximum possible value of
a
+
b
+
c
,
a + b + c ,
a
+
b
+
c
,
if
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers such that
a
2
+
b
2
+
c
2
=
a
3
+
b
3
+
c
3
.
a^2 + b^2 + c^2 = a^3 + b^3 + c^3 .
a
2
+
b
2
+
c
2
=
a
3
+
b
3
+
c
3
.
6
1
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Find a+b
If
a
,
b
a,b
a
,
b
are real numbers such that
a
3
+
12
a
2
+
49
a
+
69
=
0
a^3 +12a^2 + 49a + 69 = 0
a
3
+
12
a
2
+
49
a
+
69
=
0
and
b
3
−
9
b
2
+
28
b
−
31
=
0
,
b^3 - 9b^2 + 28b - 31 = 0,
b
3
−
9
b
2
+
28
b
−
31
=
0
,
find
a
+
b
.
a + b .
a
+
b
.