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Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2021 Mediterranean Mathematics Olympiad
2021 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
4
1
Hide problems
\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}+\sqrt{x_5} <= 20
Let
x
1
,
x
2
,
x
3
,
x
4
,
x
5
x_1,x_2,x_3,x_4,x_5
x
1
,
x
2
,
x
3
,
x
4
,
x
5
ve non-negative real numbers, so that
x
1
≤
4
x_1\le4
x
1
≤
4
and
x
1
+
x
2
≤
13
x_1+x_2\le13
x
1
+
x
2
≤
13
and
x
1
+
x
2
+
x
3
≤
29
x_1+x_2+x_3\le29
x
1
+
x
2
+
x
3
≤
29
and
x
1
+
x
2
+
x
3
+
x
4
≤
54
x_1+x_2+x_3+x_4\le54
x
1
+
x
2
+
x
3
+
x
4
≤
54
and
x
1
+
x
2
+
x
3
+
x
4
+
x
5
≤
90
x_1+x_2+x_3+x_4+x_5\le90
x
1
+
x
2
+
x
3
+
x
4
+
x
5
≤
90
. Prove that
x
1
+
x
2
+
x
3
+
x
4
+
x
5
≤
20
\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}+\sqrt{x_5}\le20
x
1
+
x
2
+
x
3
+
x
4
+
x
5
≤
20
.
3
1
Hide problems
incenter of DXY is independent of choice of points E,F , equilateral
Let
A
B
C
ABC
A
BC
be an equiangular triangle with circumcircle
ω
\omega
ω
. Let point
F
∈
A
B
F\in AB
F
∈
A
B
and point
E
∈
A
C
E\in AC
E
∈
A
C
so that
∠
A
B
E
+
∠
A
C
F
=
6
0
∘
\angle ABE+\angle ACF=60^{\circ}
∠
A
BE
+
∠
A
CF
=
6
0
∘
. The circumcircle of triangle
A
F
E
AFE
A
FE
intersects the circle
ω
\omega
ω
in the point
D
D
D
. The halflines
D
E
DE
D
E
and
D
F
DF
D
F
intersect the line through
B
B
B
and
C
C
C
in the points
X
X
X
and
Y
Y
Y
. Prove that the incenter of the triangle
D
X
Y
DXY
D
X
Y
is independent of the choice of
E
E
E
and
F
F
F
.(The angles in the problem statement are not directed. It is assumed that
E
E
E
and
F
F
F
are chosen in such a way that the halflines
D
E
DE
D
E
and
D
F
DF
D
F
indeed intersect the line through
B
B
B
and
C
C
C
.)
2
1
Hide problems
diophantine p_1 p_2...p_8 (x_1/p_1 + ... +x_8/p_8)=N
For every sequence
p
1
<
p
2
<
⋯
<
p
8
p_1<p_2<\cdots<p_8
p
1
<
p
2
<
⋯
<
p
8
of eight prime numbers, determine the largest integer
N
N
N
for which the following equation has no solution in positive integers
x
1
,
…
,
x
8
x_1,\ldots,x_8
x
1
,
…
,
x
8
:
p
1
p
2
⋯
p
8
(
x
1
p
1
+
x
2
p
2
+
⋯
+
x
8
p
8
)
=
N
p_1\, p_2\, \cdots\, p_8 \left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right) ~~=~~ N
p
1
p
2
⋯
p
8
(
p
1
x
1
+
p
2
x
2
+
⋯
+
p
8
x
8
)
=
N
Proposed by Gerhard Woeginger, Austria
1
1
Hide problems
min M, exists integer Polynomial P, P(1)=aM , P(2)=bM, P(4)=cM
Determine the smallest positive integer
M
M
M
with the following property: For every choice of integers
a
,
b
,
c
a,b,c
a
,
b
,
c
, there exists a polynomial
P
(
x
)
P(x)
P
(
x
)
with integer coefficients so that
P
(
1
)
=
a
M
P(1)=aM
P
(
1
)
=
a
M
and
P
(
2
)
=
b
M
P(2)=bM
P
(
2
)
=
b
M
and
P
(
4
)
=
c
M
P(4)=cM
P
(
4
)
=
c
M
. Proposed by Gerhard Woeginger, Austria