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Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2020 Mediterranean Mathematics Olympiad
2020 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
4
1
Hide problems
collinear wanted, PQ= PR, 3 circles related
Let
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
be three points on a circle
k
1
k_1
k
1
with
∣
P
Q
∣
=
∣
P
R
∣
|PQ|=|PR|
∣
PQ
∣
=
∣
PR
∣
and
∣
P
Q
∣
>
∣
Q
R
∣
|PQ|>|QR|
∣
PQ
∣
>
∣
QR
∣
. Let
k
2
k_2
k
2
be the circle with center in
P
P
P
that goes through
Q
Q
Q
and
R
R
R
. The circle with center
Q
Q
Q
through
R
R
R
intersects
k
1
k_1
k
1
in another point
X
≠
R
X\ne R
X
=
R
and intersects
k
2
k_2
k
2
in another point
Y
≠
R
Y\ne R
Y
=
R
. The two points
X
X
X
and
R
R
R
lie on different sides of the line through
P
Q
PQ
PQ
. Show that the three points
P
P
P
,
X
X
X
,
Y
Y
Y
lie on a common line.
3
1
Hide problems
sum \frac{ab}{\sqrt[4]{3c^2+16}} \le 4/3 \sqrt[4]{12} if a+b+c=4, a,b,c>0
Prove that all postive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
with
a
+
b
+
c
=
4
a+b+c=4
a
+
b
+
c
=
4
satisfy the inequality
a
b
3
c
2
+
16
4
+
b
c
3
a
2
+
16
4
+
c
a
3
b
2
+
16
4
≤
4
3
12
4
\frac{ab}{\sqrt[4]{3c^2+16}}+ \frac{bc}{\sqrt[4]{3a^2+16}}+ \frac{ca}{\sqrt[4]{3b^2+16}} \le\frac43 \sqrt[4]{12}
4
3
c
2
+
16
ab
+
4
3
a
2
+
16
b
c
+
4
3
b
2
+
16
c
a
≤
3
4
4
12
2
1
Hide problems
at least n^2 integers written in the form x+yz with x,y,z\in S
Let
S
S
S
be a set of
n
≥
2
n\ge2
n
≥
2
positive integers. Prove that there exist at least
n
2
n^2
n
2
integers that can be written in the form
x
+
y
z
x+yz
x
+
yz
with
x
,
y
,
z
∈
S
x,y,z\in S
x
,
y
,
z
∈
S
.Proposed by Gerhard Woeginger, Austria
1
1
Hide problems
gcd(m,n)=d and gcd(m,4n+1)=1
Determine all integers
m
≥
2
m\ge2
m
≥
2
for which there exists an integer
n
≥
1
n\ge1
n
≥
1
with
gcd
(
m
,
n
)
=
d
\gcd(m,n)=d
g
cd
(
m
,
n
)
=
d
and
gcd
(
m
,
4
n
+
1
)
=
1
\gcd(m,4n+1)=1
g
cd
(
m
,
4
n
+
1
)
=
1
.Proposed by Gerhard Woeginger, Austria