MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
IMOC Shortlist
2023-IMOC
A6
A6
Part of
2023-IMOC
Problems
(1)
Make c1 large and c2 small
Source: 2023 IMOC A6
9/9/2023
We define
f
(
x
,
y
,
z
)
=
∣
x
y
∣
x
2
+
y
2
+
∣
y
z
∣
y
2
+
z
2
+
∣
z
x
∣
z
2
+
x
2
.
f(x,y,z)=|xy|\sqrt{x^2+y^2}+|yz|\sqrt{y^2+z^2}+|zx|\sqrt{z^2+x^2}.
f
(
x
,
y
,
z
)
=
∣
x
y
∣
x
2
+
y
2
+
∣
yz
∣
y
2
+
z
2
+
∣
z
x
∣
z
2
+
x
2
.
Find the best constants
c
1
,
c
2
∈
R
c_1,c_2\in\mathbb{R}
c
1
,
c
2
∈
R
such that
c
1
(
x
2
+
y
2
+
z
2
)
3
/
2
≤
f
(
x
,
y
,
z
)
≤
c
1
(
x
2
+
y
2
+
z
2
)
3
/
2
c_1(x^2+y^2+z^2)^{3/2}\leq f(x,y,z)\leq c_1(x^2+y^2+z^2)^{3/2}
c
1
(
x
2
+
y
2
+
z
2
)
3/2
≤
f
(
x
,
y
,
z
)
≤
c
1
(
x
2
+
y
2
+
z
2
)
3/2
hold for all reals
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfying
x
+
y
+
z
=
0
x+y+z=0
x
+
y
+
z
=
0
.Proposed by Untro368.
IMOC
algebra
inequalities