MathDB
Problems
Contests
National and Regional Contests
France Contests
France Team Selection Test
2007 France Team Selection Test
2
2
Part of
2007 France Team Selection Test
Problems
(2)
France TST 2007
Source: Problem 5
5/16/2007
Find all functions
f
:
Z
→
Z
f: \mathbb{Z}\rightarrow\mathbb{Z}
f
:
Z
→
Z
such that for all
x
,
y
∈
Z
x,y \in \mathbb{Z}
x
,
y
∈
Z
:
f
(
x
−
y
+
f
(
y
)
)
=
f
(
x
)
+
f
(
y
)
.
f(x-y+f(y))=f(x)+f(y).
f
(
x
−
y
+
f
(
y
))
=
f
(
x
)
+
f
(
y
)
.
function
induction
algebra
functional equation
algebra proposed
France TST 2007
Source: Problem 2
5/16/2007
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive reals such taht
a
+
b
+
c
+
d
=
1
a+b+c+d=1
a
+
b
+
c
+
d
=
1
. Prove that:
6
(
a
3
+
b
3
+
c
3
+
d
3
)
≥
a
2
+
b
2
+
c
2
+
d
2
+
1
8
.
6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.
6
(
a
3
+
b
3
+
c
3
+
d
3
)
≥
a
2
+
b
2
+
c
2
+
d
2
+
8
1
.
inequalities
search
symmetry
inequalities proposed