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Contests
National and Regional Contests
Bulgaria Contests
Bulgaria JBMO Team Selection Test
2018 Bulgaria JBMO TST
2018 Bulgaria JBMO TST
Part of
Bulgaria JBMO Team Selection Test
Subcontests
(4)
1
2
Hide problems
minimum of f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b)
For real numbers
a
a
a
and
b
b
b
, define
f
(
a
,
b
)
=
a
2
+
b
2
+
26
a
+
86
b
+
2018
.
f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}.
f
(
a
,
b
)
=
a
2
+
b
2
+
26
a
+
86
b
+
2018
.
Find the smallest possible value of the expression
f
(
a
,
b
)
+
f
(
a
,
−
b
)
+
f
(
−
a
,
b
)
+
f
(
−
a
,
−
b
)
.
f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b).
f
(
a
,
b
)
+
f
(
a
,
−
b
)
+
f
(
−
a
,
b
)
+
f
(
−
a
,
−
b
)
.
Find the length of diagonal AC
In the quadrilateral
A
B
C
D
ABCD
A
BC
D
, we have
∡
B
A
D
=
10
0
∘
\measuredangle BAD = 100^{\circ}
∡
B
A
D
=
10
0
∘
,
∡
B
C
D
=
13
0
∘
\measuredangle BCD = 130^{\circ}
∡
BC
D
=
13
0
∘
, and
A
B
=
A
D
=
1
AB=AD=1
A
B
=
A
D
=
1
centimeter. Find the length of diagonal
A
C
AC
A
C
.
4
2
Hide problems
cells of any 2x3 rectangles have different colors
Each cell of an infinite table (infinite in all directions) is colored with one of
n
n
n
given colors. All six cells of any
2
×
3
2\times 3
2
×
3
(or
3
×
2
3 \times 2
3
×
2
) rectangle have different colors. Find the smallest possible value of
n
n
n
.
672 numbers, what is the minimum of the largest one?
The real numbers
a
1
≤
a
2
≤
⋯
≤
a
672
a_1 \leq a_2 \leq \cdots \leq a_{672}
a
1
≤
a
2
≤
⋯
≤
a
672
are given such that
a
1
+
a
2
+
⋯
+
a
672
=
2018.
a_1 + a_2 + \cdots + a_{672} = 2018.
a
1
+
a
2
+
⋯
+
a
672
=
2018.
For any
n
≤
672
n \leq 672
n
≤
672
, there are
n
n
n
of these numbers with an integer sum. What is the smallest possible value of
a
672
a_{672}
a
672
?
2
2
Hide problems
(a^2+b^2)/(2a^5b^5) + (81a^2b^2)/4 + 9ab \geq 81
For all positive reals
a
a
a
and
b
b
b
, show that
a
2
+
b
2
2
a
5
b
5
+
81
a
2
b
2
4
+
9
a
b
>
18.
\frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab > 18.
2
a
5
b
5
a
2
+
b
2
+
4
81
a
2
b
2
+
9
ab
>
18.
CD=BE [Bulgaria JBMO TST 2018]
Let
A
B
C
ABC
A
BC
be a triangle and
A
A
1
AA_1
A
A
1
be the angle bisector of
A
A
A
(
A
1
∈
B
C
A_1 \in BC
A
1
∈
BC
). The point
P
P
P
is on the segment
A
A
1
AA_1
A
A
1
and
M
M
M
is the midpoint of the side
B
C
BC
BC
. The point
Q
Q
Q
is on the line connecting
P
P
P
and
M
M
M
such that
M
M
M
is the midpoint of
P
Q
PQ
PQ
. Define
D
D
D
and
E
E
E
as the intersections of
B
Q
BQ
BQ
,
A
C
AC
A
C
, and
C
Q
CQ
CQ
,
A
B
AB
A
B
. Prove that
C
D
=
B
E
CD=BE
C
D
=
BE
.
3
2
Hide problems
n^6 + 5n^3 + 4n + 116 is the product of two or more consecutive numbers
Find all positive integers
n
n
n
such that the number
n
6
+
5
n
3
+
4
n
+
116
n^6 + 5n^3 + 4n + 116
n
6
+
5
n
3
+
4
n
+
116
is the product of two or more consecutive numbers.
Weird inequality from Bulgaria JBMO TST 2017
Prove for all positive real numbers
m
,
n
,
p
,
q
m,n,p,q
m
,
n
,
p
,
q
that
m
t
+
n
+
p
+
q
+
n
t
+
p
+
q
+
m
+
p
t
+
q
+
m
+
n
+
q
t
+
m
+
n
+
p
≥
4
5
,
\frac{m}{t+n+p+q} + \frac{n}{t+p+q+m} + \frac{p}{t+q+m+n} + \frac{q}{t+m+n+p} \geq \frac{4}{5},
t
+
n
+
p
+
q
m
+
t
+
p
+
q
+
m
n
+
t
+
q
+
m
+
n
p
+
t
+
m
+
n
+
p
q
≥
5
4
,
where
t
=
m
+
n
+
p
+
q
2
.
t=\frac{m+n+p+q}{2}.
t
=
2
m
+
n
+
p
+
q
.