MathDB
Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2017 Mediterranean Mathematics Olympiad
2017 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
Problem 4
1
Hide problems
Inequality [Mediterranean Mathematics Olympiad 2017, P4]
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
and
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers with
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove that
(
x
2
+
y
2
+
z
2
)
(
a
3
x
2
+
2
y
2
+
b
3
y
2
+
2
z
2
+
c
3
z
2
+
2
x
2
)
≥
1
9
.
\left(x^2+y^2+z^2\right) \left( \frac{a^3}{x^2+2y^2} + \frac{b^3}{y^2+2z^2} + \frac{c^3}{z^2+2x^2} \right) \ge\frac19.
(
x
2
+
y
2
+
z
2
)
(
x
2
+
2
y
2
a
3
+
y
2
+
2
z
2
b
3
+
z
2
+
2
x
2
c
3
)
≥
9
1
.
Problem 3
1
Hide problems
Balearic sets
A set
S
S
S
of integers is Balearic, if there are two (not necessarily distinct) elements
s
,
s
′
∈
S
s,s'\in S
s
,
s
′
∈
S
whose sum
s
+
s
′
s+s'
s
+
s
′
is a power of two; otherwise it is called a non-Balearic set. Find an integer
n
n
n
such that
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n\}
{
1
,
2
,
…
,
n
}
contains a 99-element non-Balearic set, whereas all the 100-element subsets are Balearic.
Problem 2
1
Hide problems
Least value of n [Mediterranean Mathematics Olympiad 2017, P2]
Determine the smallest integer
n
n
n
for which there exist integers
x
1
,
…
,
x
n
x_1,\ldots,x_n
x
1
,
…
,
x
n
and positive integers
a
1
,
…
,
a
n
a_1,\ldots,a_n
a
1
,
…
,
a
n
so that \begin{align*} x_1+\cdots+x_n &=0,\\ a_1x_1+\cdots+a_nx_n&>0, \text{ and }\\ a_1^2x_1+\cdots+a_n^2x_n &<0. \end{align*}
Problem 1
1
Hide problems
Equilateral triangle and constant function
Let
A
B
C
ABC
A
BC
be an equilateral triangle, and let
P
P
P
be some point in its circumcircle. Determine all positive integers
n
n
n
, for which the value of the sum
S
n
(
P
)
=
∣
P
A
∣
n
+
∣
P
B
∣
n
+
∣
P
C
∣
n
S_n (P) = |PA|^n + |PB|^n + |PC|^n
S
n
(
P
)
=
∣
P
A
∣
n
+
∣
PB
∣
n
+
∣
PC
∣
n
is independent of the choice of point
P
P
P
.