MathDB
Problems
Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2015 FYROM JBMO Team Selection Test
2015 FYROM JBMO Team Selection Test
Part of
JBMO TST - Macedonia
Subcontests
(5)
5
1
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Junior Combinatorics
A
A
A
and
B
B
B
are two identical convex polygons, each with an area of
2015
2015
2015
. The polygon
A
A
A
is divided into polygons
A
1
,
A
2
,
.
.
.
,
A
2015
A_1, A_2,...,A_{2015}
A
1
,
A
2
,
...
,
A
2015
, while
B
B
B
is divided into polygons
B
1
,
B
2
,
.
.
.
,
B
2015
B_1, B_2,...,B_{2015}
B
1
,
B
2
,
...
,
B
2015
. Each of these smaller polygons has a positive area. Furthermore,
A
1
,
A
2
,
.
.
.
,
A
2015
A_1, A_2,...,A_{2015}
A
1
,
A
2
,
...
,
A
2015
and
B
1
,
B
2
,
.
.
.
,
B
2015
B_1, B_2,...,B_{2015}
B
1
,
B
2
,
...
,
B
2015
are colored in
2015
2015
2015
distinct colors, such that
A
i
A_i
A
i
and
A
j
A_j
A
j
are differently colored for every distinct
i
i
i
and
j
j
j
and
B
i
B_i
B
i
and
B
j
B_j
B
j
are also differently colored for every distinct
i
i
i
and
j
j
j
. After
A
A
A
and
B
B
B
overlap, we calculate the sum of the areas with the same colors. Prove that we can color the polygons such that this sum is at least
1
1
1
.
4
1
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Geometry with iff
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute angled triangle and let
k
k
k
be its circumscribed circle. A point
O
O
O
is given in the interior of the triangle, such that
C
E
=
C
F
CE=CF
CE
=
CF
, where
E
E
E
and
F
F
F
are on
k
k
k
and
E
E
E
lies on
A
O
AO
A
O
while
F
F
F
lies on
B
O
BO
BO
. Prove that
O
O
O
is on the angle bisector of
∠
A
C
B
\angle ACB
∠
A
CB
if and only if
A
C
=
B
C
AC=BC
A
C
=
BC
.
3
1
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Inequality
Let
a
,
b
a, b
a
,
b
and
c
c
c
be positive real numbers. Prove that
∏
c
y
c
(
16
a
2
+
8
b
+
17
)
≥
2
12
∏
c
y
c
(
a
+
1
)
\prod_{cyc}(16a^2+8b+17)\geq2^{12}\prod_{cyc}(a+1)
∏
cyc
(
16
a
2
+
8
b
+
17
)
≥
2
12
∏
cyc
(
a
+
1
)
.
2
1
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Easy Geometry
A circle
k
k
k
with center
O
O
O
and radius
r
r
r
and a line
p
p
p
which has no common points with
k
k
k
, are given. Let
E
E
E
be the foot of the perpendicular from
O
O
O
to
p
p
p
. Let
M
M
M
be an arbitrary point on
p
p
p
, distinct from
E
E
E
. The tangents from the point
M
M
M
to the circle
k
k
k
are
M
A
MA
M
A
and
M
B
MB
MB
. If
H
H
H
is the intersection of
A
B
AB
A
B
and
O
E
OE
OE
, then prove that
O
H
=
r
2
O
E
OH=\frac{r^2}{OE}
O
H
=
OE
r
2
.
1
1
Hide problems
Diophantine
Solve the equation
x
2
+
y
4
+
1
=
6
z
x^2+y^4+1=6^z
x
2
+
y
4
+
1
=
6
z
in the set of integers.