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International Contests
Mediterranean Mathematics Olympiad
2003 Mediterranean Mathematics Olympiad
2003 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
3
1
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Maybe old inequality with a+b+c=3
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be non-negative numbers with
a
+
b
+
c
=
3
a+b+c = 3
a
+
b
+
c
=
3
. Prove the inequality
a
b
2
+
1
+
b
c
2
+
1
+
c
a
2
+
1
≥
3
2
.
\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac 32.
b
2
+
1
a
+
c
2
+
1
b
+
a
2
+
1
c
≥
2
3
.
2
1
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Prove that CAP = 2 CPA
In a triangle
A
B
C
ABC
A
BC
with
B
C
=
C
A
+
1
2
A
B
BC = CA + \frac 12 AB
BC
=
C
A
+
2
1
A
B
, point
P
P
P
is given on side
A
B
AB
A
B
such that
B
P
:
P
A
=
1
:
3
BP : PA = 1 : 3
BP
:
P
A
=
1
:
3
. Prove that
∠
C
A
P
=
2
∠
C
P
A
.
\angle CAP = 2 \angle CPA.
∠
C
A
P
=
2∠
CP
A
.
1
1
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Prove that the equation has no rational solutions
Prove that the equation
x
2
+
y
2
+
z
2
=
x
+
y
+
z
+
1
x^2 + y^2 + z^2 = x + y + z + 1
x
2
+
y
2
+
z
2
=
x
+
y
+
z
+
1
has no rational solutions.
4
1
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very nice problem
Consider a system of infinitely many spheres made of metal, with centers at points
(
a
,
b
,
c
)
∈
Z
3
(a, b, c) \in \mathbb Z^3
(
a
,
b
,
c
)
∈
Z
3
. We say that the system is stable if the temperature of each sphere equals the average temperature of the six closest spheres. Assuming that all spheres in a stable system have temperatures between
0
∘
C
0^\circ C
0
∘
C
and
1
∘
C
1^\circ C
1
∘
C
, prove that all the spheres have the same temperature.