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Problems
Contests
National and Regional Contests
Greece Contests
Greece JBMO TST
1999 Greece JBMO TST
1999 Greece JBMO TST
Part of
Greece JBMO TST
Subcontests
(5)
4
1
Hide problems
3n is cube, 4n is perfect 4th power, 5n perfect 5th power, 6n perfect 6th power
Examine whether exists
n
∈
N
∗
n \in N^*
n
∈
N
∗
, such that: (a)
3
n
3n
3
n
is perfect cube,
4
n
4n
4
n
is perfect fourth power and
5
n
5n
5
n
perfect fifth power (b)
3
n
3n
3
n
is perfect cube,
4
n
4n
4
n
is perfect fourth power,
5
n
5n
5
n
perfect fifth power and
6
n
6n
6
n
perfect sixth power
2
1
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(a+b+c)/2-ab/(a+b)-bc/(b+c)-ca/(c+a)\ge 0, a(1+b)+b(1+c)+c(1+a)\ge 6\sqrt{abc}
For
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
, prove that (i)
a
+
b
+
c
2
−
a
b
a
+
b
−
b
c
b
+
c
−
c
a
c
+
a
≥
0
\frac{a+b+c}{2}-\frac{ab}{a+b}-\frac{bc}{b+c}-\frac{ca}{c+a}\ge 0
2
a
+
b
+
c
−
a
+
b
ab
−
b
+
c
b
c
−
c
+
a
c
a
≥
0
(ii)
a
(
1
+
b
)
+
b
(
1
+
c
)
+
c
(
1
+
a
)
≥
6
a
b
c
a(1+b)+b(1+c)+c(1+a)\ge 6\sqrt{abc}
a
(
1
+
b
)
+
b
(
1
+
c
)
+
c
(
1
+
a
)
≥
6
ab
c
1
1
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100 points in a circle, green or yellow, every 5 color changes
A circle is divided in
100
100
100
equal parts and the points of this division are colored green or yellow, such that when between two points of division
A
,
B
A,B
A
,
B
there are exactly
4
4
4
division points and the point
A
A
A
is green, then the point
B
B
B
shall be yellow. Which points are more, the green or the yellow ones?
3
1
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\overline{abc}+\overline{acb}=\overline{199x} , find digits
Find digits
a
,
b
,
c
,
x
a,b,c,x
a
,
b
,
c
,
x
(
a
>
0
a>0
a
>
0
) such that
a
b
c
‾
+
a
c
b
‾
=
199
x
‾
\overline{abc}+\overline{acb}=\overline{199x}
ab
c
+
a
c
b
=
199
x
5
1
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area of locus of symmetric points of a triangle
Φ
\Phi
Φ
is the union of all triangles that are symmetric of the triangle
A
B
C
ABC
A
BC
wrt a point
O
O
O
, as point
O
O
O
moves along the triangle's sides. If the area of the triangle is
E
E
E
, find the area of
Φ
\Phi
Φ
.