MathDB
Problems
Contests
National and Regional Contests
France Contests
France JBMO TST
2020 JBMO TST of France
3
3
Part of
2020 JBMO TST of France
Problems
(1)
Algebra functions
Source: First JBMO TST of France 2020, Problem 3
3/4/2020
Let n be a nonzero natural number. We say about a function f ∶ R ⟶ R that is n-positive if, for any real numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
with the property that
x
1
+
x
2
+
.
.
.
+
x
n
=
0
x_1+x_2+...+x_n = 0
x
1
+
x
2
+
...
+
x
n
=
0
, the inequality
f
(
x
1
)
+
f
(
x
2
)
+
.
.
.
+
f
(
x
n
)
=
>
0
f(x_1)+f(x_2)+...+f(x_n)=>0
f
(
x
1
)
+
f
(
x
2
)
+
...
+
f
(
x
n
)
=>
0
is true a) Is it true that any 2020-positive function is also 1010-positive? b) Is it true that any 1010-positive function is 2020-positive?
function
algebra