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Romania Contests
JBMO TST - Romania
2017 Junior Balkan Team Selection Tests - Romania
2017 Junior Balkan Team Selection Tests - Romania
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JBMO TST - Romania
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Real Number satisfying inequality
Given
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
real numbers, prove that there exists a real number
y
y
y
, such that,
{
y
−
x
1
}
+
{
y
−
x
2
}
+
.
.
.
+
{
y
−
x
n
}
≤
n
−
1
2
\{y-x_1\}+\{y-x_2\}+...+\{y-x_n\} \leq \frac{n-1}{2}
{
y
−
x
1
}
+
{
y
−
x
2
}
+
...
+
{
y
−
x
n
}
≤
2
n
−
1
equation x + y = z has a monochrome solution with x \ne y
Determine the smallest positive integer
n
n
n
such that, for any coloring of the elements of the set
{
2
,
3
,
.
.
.
,
n
}
\{2,3,...,n\}
{
2
,
3
,
...
,
n
}
with two colors, the equation
x
+
y
=
z
x + y = z
x
+
y
=
z
has a monochrome solution with
x
≠
y
x \ne y
x
=
y
.(We say that the equation
x
+
y
=
z
x + y = z
x
+
y
=
z
has a monochrome solution if there exist
a
,
b
,
c
a, b, c
a
,
b
,
c
distinct, having the same color, such that
a
+
b
=
c
a + b = c
a
+
b
=
c
.)
difference between odd pos. divisors and even pos. divisors inequality
Let
n
n
n
be a positive integer. For each of the numbers
1
,
2
,
.
.
,
n
1, 2,.., n
1
,
2
,
..
,
n
we compute the difference between the number of its odd positive divisors and its even positive divisors. Prove that the sum of these differences is at least
0
0
0
and at most
n
n
n
.
1
4
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