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Problems
Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2013 JBMO TST - Macedonia
2013 JBMO TST - Macedonia
Part of
JBMO TST - Macedonia
Subcontests
(5)
5
1
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equation
Let
p
,
r
p, r
p
,
r
be prime numbers, and
q
q
q
natural. Solve the equation
(
p
+
q
+
r
)
2
=
2
p
2
+
2
q
2
+
r
2
(p+q+r)^2=2p^2+2q^2+r^2
(
p
+
q
+
r
)
2
=
2
p
2
+
2
q
2
+
r
2
.
4
1
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m points in hexagon
A regular hexagon with side length
1
1
1
is given. There are
m
m
m
points in its interior such that no
3
3
3
are collinear. The hexagon is divided into triangles (triangulated), such that every point of the
m
m
m
given and every vertex of the hexagon is a vertex of such a triangle. The triangles don't have common interior points. Prove that there exists a triangle with area not greater than
3
3
4
(
m
+
2
)
\frac{3 \sqrt{3}}{4(m+2)}
4
(
m
+
2
)
3
3
.
3
1
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ineq
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
and
a
b
c
=
1
abc=1
ab
c
=
1
. Prove that
1
2
(
a
+
b
+
c
)
+
1
1
+
a
+
1
1
+
b
+
1
1
+
c
≥
3
\frac{1}{2}\ (\sqrt{a}\ +\sqrt{b}\ + \sqrt{c}\ ) +\frac{1}{1+a}\ + \frac{1}{1+b}\ + \frac{1}{1+c}\ge\ 3
2
1
(
a
+
b
+
c
)
+
1
+
a
1
+
1
+
b
1
+
1
+
c
1
≥
3
. ( The official problem is with
a
b
c
=
1
abc = 1
ab
c
=
1
but it can be proved without using it. )
2
1
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Areas
A triangle
A
B
C
ABC
A
BC
is given, and a segment
P
Q
=
t
PQ=t
PQ
=
t
on
B
C
BC
BC
such that
P
P
P
is between
B
B
B
and
Q
Q
Q
and
Q
Q
Q
is between
P
P
P
and
C
C
C
. Let
P
P
1
∣
∣
A
B
PP_1 || AB
P
P
1
∣∣
A
B
,
P
1
P_1
P
1
is on
A
C
AC
A
C
, and
P
P
2
∣
∣
A
C
PP_2 || AC
P
P
2
∣∣
A
C
,
P
2
P_2
P
2
is on
A
B
AB
A
B
. Points
Q
1
Q_1
Q
1
and
Q
2
Q_2
Q
2
аrе defined similar. Prove that the sum of the areas of
P
Q
Q
1
P
1
PQQ_1P_1
PQ
Q
1
P
1
and
P
Q
Q
2
P
2
PQQ_2P_2
PQ
Q
2
P
2
does not depend from the position of
P
Q
PQ
PQ
on
B
C
BC
BC
.
1
1
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Rational number
Let
x
x
x
be a real number such that
x
3
x^3
x
3
and
x
2
+
x
x^2+x
x
2
+
x
are rational numbers. Prove that
x
x
x
is rational.