       M v
0
  m m<M  M
 u  v Mv
0
=Mu+mv,
1
2
Mv
2
0
=
1
2
Mu
2
+
1
2
mv
2
.
 u  v =
2M
M +m
v
0
, u =
M−m
M +m
v
0
.
 0<m<M  v
0
<v < 2v
0
  n  m
1
,m
2
,...,m
n
 m < m
1
< m
2
< ... < m
n
< M  M v
0
  m
n
 v
n
= 2Mv
0
/(m
n
+M)  m
n−1
 v
n−1
= 2m
n
v
n
/(m
n−1
+m
n
)  m
n−2
  m
1
  m v v =
m
1
m
2
···m
n
M
(m+m
1
)(m
1
+m
2
)···(m
n−1
+m
n
)(m
n
+M)
·2
n+1
v
0
.
Obzornik mat. ﬁz. 55 (2008) 5
Marko Razpet
 m
1
< m
2
< ... < m
n
 v  m < m
1
< m
2
< ... < m
n
< M  q =
n+1
p
M/m a b a < b x
1
 x
2
 x
n
  u:R
n
+
→R
+
 u(x
1
,x
2
,...,x
n
) =
x
1
x
2
···x
n
(a+x
1
)(x
1
+x
2
)···(x
n−1
+x
n
)(x
n
+b)
, ℄  n = 1 u(x
1
) =
x
1
(a+x
1
)(x
1
+b)
.
     n a
1
,a
2
,...,a
n
 A(a
1
,a
2
,...,a
n
) = (a
1
+a
2
+···+a
n
)/n
 G(a
1
,a
2
,...,a
n
) =
n
√
a
1
a
2
···a
n
 a
1
,a
2
,...,a
n
 Obzornik mat. ﬁz. 55 (2008) 5
Huygensova naloga
 G(a
1
,a
2
,...,a
n
)≤A(a
1
,a
2
,...,a
n
), a
1
=a
2
=... =a
n
 n u
1
,u
2
,...,u
n
 (1+u
1
)(1+u
2
)···(1+u
n
)≥
 
1+G(u
1
,u
2
,...,u
n
)

n
, u
1
=u
2
=... =u
n
 (1+u
1
)(1+u
2
)···(1+u
n
) = 1+
n
X
k=1
s
k
(u
1
,u
2
,...,u
n
).
 s
k
 k = 1,2,...,n u
1
,u
2
,...,u
n
 s
1
(u
1
,u
2
,...,u
n
) = u
1
+u
2
+···+u
n
,
s
2
(u
1
,u
2
,...,u
n
) = u
1
u
2
+u
1
u
3
+···+u
n−1
u
n
,
 s
n
(u
1
,u
2
,...,u
n
) = u
1
u
2
···u
n
.
 s
1
 n =
 
n
1

 s
2
  
n
2

 s
k
 k s
k
 c(n,k) =
 
n
k

 a
i
=a
i
(s
k
) s
k
=c(n,k)A(a
1
,a
2
,...,a
c(n,k)
).
 u
r
 s
k
 c(n− 1,k− 1) n− 1 u
1
,u
2
,...,u
n
 u
r
 k−
1 a
i
=
a
i
(s
k
) s
k
 G(a
1
,a
2
,...,a
c(n,k)
) = (a
1
a
2
···a
c(n,k)
)
1
c(n,k)
= (u
1
u
2
···u
n
)
c(n−1,k−1)
c(n,k)
.
161–167
Marko Razpet
 s
k
 s
k
(u
1
,u
2
,...,u
n
)≥c(n,k)(u
1
u
2
···u
n
)
m(n,k)
 k = 1,2,...,n m(n,k) =c(n−1,k−1)/c(n,k) n≥ 1 k≥ 1 m(n,k) =
 
n−1
k−1

 
n
k
 =
(n−1)!
(k−1)!(n−k)!
·
k!(n−k)!
n!
=
k
n
,
 0! = 1 c(0,0) = 1 s
k
(u
1
,u
2
,...,u
n
)≥c(n,k)(u
1
u
2
···u
n
)
k/n
=

n
k

(G(u
1
,u
2
,...,u
n
))
k
.
  a
1
=
a
2
= ... = a
c(n,k)
 s
k
 k ≥ 1 s
1
 u
1
=u
2
=... =u
n
 s
k
 (1+u
1
)(1+u
2
)···(1+u
n
)≥ 1+
n
X
k=1

n
k

 
G(u
1
,u
2
,...,u
n
)

k
 (1+u
1
)(1+u
2
)···(1+u
n
)≥
 
1+G(u
1
,u
2
,...,u
n
)

n
.
 G(1+u
1
,1+u
2
,...,1+u
n
)≥ 1+G(u
1
,u
2
,...,u
n
).
 u
1
=u
2
=... =u
n
  k

n
k

=n

n−1
k−1

,
 m(n,k) ℄  n k  
n
k

 k k
 
n
k

 n k− 1 n− 1  
n−1
k−1

 n
 
n−1
k−1

 Obzornik mat. ﬁz. 55 (2008) 5
Huygensova naloga
 a
1
 a
2
 a
n
 b
1
 b
2
 b
n
 u
1
=a
1
/b
1
 u
2
=a
2
/b
2
 u
n
=a
n
/b
n
 
1+
a
1
b
1

1+
a
2
b
2

···

1+
a
n
b
n

≥

1+G

a
1
b
1
,
a
2
b
2
,...,
a
n
b
n

n
,
 G(a
1
+b
1
,a
2
+b
2
,...,a
n
+b
n
)≥G(a
1
,a
2
,...,a
n
)+G(b
1
,b
2
,...,b
n
). [a
1
,a
2
,...,a
n
] [b
1
,b
2
,...,b
n
]  ℄ ℄ a
1
 a
2
 a
n
 b
1
 b
2
 b
n
 c
1
 c
2
 c
n
 G(a
1
+b
1
+c
1
,a
2
+b
2
+c
2
,...,a
n
+b
n
+c
n
)≥
≥G(a
1
,a
2
,...,a
n
)+G(b
1
,b
2
,...,b
n
)+G(c
1
,c
2
,...,c
n
). [a
1
,a
2
,...,a
n
] [b
1
 b
2
,...,b
n
] [c
1
,c
2
,...,c
n
]  ℄ x y z (x+y)(y +z)(z +x)≥ 8xyz.
 n = 3 a
1
=x, a
2
=y, a
3
=z, b
1
=y, b
2
=z, b
3
=x.
 (x+y)(y +z)(z +x) = (G(x+y,y +z,z +x))
3
≥
≥ (G(x,y,z)+G(y,z,x))
3
= (2
3
√
xyz)
3
= 8xyz,
 x/y = y/z = z/x = λ x =λy y =λz z =λx xyz =λ
3
xyz λ = 1 x =y =z 161–167
Marko Razpet
   v(x
1
,x
2
,...,x
n
) =
(a+x
1
)(x
1
+x
2
)···(x
n−1
+x
n
)(x
n
+b)
x
1
x
2
···x
n
b
,  u b  v  u v(x
1
,x
2
,...,x
n
) =

1+
a
x
1

1+
x
1
x
2

···

1+
x
n−1
x
n


1+
x
n
b

.
 n+1 u
1
=
a
x
1
, u
2
=
x
1
x
2
, ... , u
n
=
x
n−1
x
n
, u
n+1
=
x
n
b
.
 G(u
1
,u
2
,...,u
n
,u
n+1
) =
n+1
r
a
b
,
 v(x
1
,x
2
,...,x
n
)≥

1+
n+1
r
a
b

n+1
=

n+1
√
a+
n+1
√
b

n+1
b
.
 bu(x
1
,x
2
,...,x
n
)v(x
1
,x
2
,...,x
n
) = 1 u(x
1
,x
2
,...,x
n
)≤
1

n+1
√
a+
n+1
√
b

n+1
.
 x
1
,x
2
,...,x
n
  u u
1
=u
2
=... =u
n
=u
n+1
  [x
1
,x
2
,,...,x
n
,b], [a,x
1
,...,x
n−1
,x
n
],
 q x
1
a
=
x
2
x
1
=... =
x
n
x
n−1
=
b
x
n
=q.
 Obzornik mat. ﬁz. 55 (2008) 5
Huygensova naloga
  x
1
=aq, x
2
=x
1
q =aq
2
, ... , x
n
=x
n−1
q =aq
n
, b =aq
n+1
.
 0<a<b q > 1 a<x
1
<x
2
<...<x
n
<b.
 a,x
1
,x
2
,...,x
n
,b  q =
n+1
p
b/a  u (aq,aq
2
,...,aq
n
) R
n
+
 1


n+1
√
a+
n+1
√
b

n+1
   ℄   ℄ ℄  ℄ ℄ 161–167