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Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
1999 Mediterranean Mathematics Olympiad
1999 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
1
1
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Infinite set of points on circle with rational distances
Do there exist a circle and an infinite set of points on it such that the distance between any two of the points is rational?
2
1
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Plane figure with area > n can cover n + 1 lattice points
A plane figure of area
A
>
n
A > n
A
>
n
is given, where
n
n
n
is a positive integer. Prove that this figure can be placed onto a Cartesian plane so that it covers at least
n
+
1
n+1
n
+
1
points with integer coordinates.
3
1
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Strange Inequality
Let
a
,
b
,
c
≠
0
a,b,c\not= 0
a
,
b
,
c
=
0
and
x
,
y
,
z
∈
R
+
x,y,z\in\mathbb{R}^+
x
,
y
,
z
∈
R
+
such that
x
+
y
+
z
=
3
x+y+z=3
x
+
y
+
z
=
3
. Prove that
3
2
1
a
2
+
1
b
2
+
1
c
2
≥
x
1
+
a
2
+
y
1
+
b
2
+
z
1
+
c
2
\frac{3}{2}\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}\geq\frac{x}{1+a^2}+\frac{y}{1+b^2}+\frac{z}{1+c^2}
2
3
a
2
1
+
b
2
1
+
c
2
1
≥
1
+
a
2
x
+
1
+
b
2
y
+
1
+
c
2
z
[color=#FF0000]Mod: before the edit, it was
3
2
(
1
a
2
+
1
b
2
+
1
c
2
)
≥
x
1
+
a
2
+
y
1
+
b
2
+
z
1
+
c
2
\frac{3}{2}\left (\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right )\geq\frac{x}{1+a^2}+\frac{y}{1+b^2}+\frac{z}{1+c^2}
2
3
(
a
2
1
+
b
2
1
+
c
2
1
)
≥
1
+
a
2
x
+
1
+
b
2
y
+
1
+
c
2
z
4
1
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Brutal Algebra expression
In triangle
△
A
B
C
\triangle ABC
△
A
BC
we have
B
C
=
a
,
C
A
=
b
,
A
B
=
c
BC=a,CA=b,AB=c
BC
=
a
,
C
A
=
b
,
A
B
=
c
and
∠
B
=
4
∠
A
\angle B=4\angle A
∠
B
=
4∠
A
Show that
a
b
2
c
3
=
(
b
2
−
a
2
−
a
c
)
(
(
a
2
−
b
2
)
2
−
a
2
c
2
)
ab^2c^3=(b^2-a^2-ac)((a^2-b^2)^2-a^2c^2)
a
b
2
c
3
=
(
b
2
−
a
2
−
a
c
)
((
a
2
−
b
2
)
2
−
a
2
c
2
)