MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1988 Swedish Mathematical Competition
3
3
Part of
1988 Swedish Mathematical Competition
Problems
(1)
find n such that x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 <= 0 if sum x_i=0
Source: 1988 Swedish Mathematical Competition p3
3/28/2021
Show that if
x
1
+
x
2
+
x
3
=
0
x_1+x_2+x_3 = 0
x
1
+
x
2
+
x
3
=
0
for real numbers
x
1
,
x
2
,
x
3
x_1,x_2,x_3
x
1
,
x
2
,
x
3
, then
x
1
x
2
+
x
2
x
3
+
x
3
x
1
≤
0
x_1x_2+x_2x_3+x_3x_1\le 0
x
1
x
2
+
x
2
x
3
+
x
3
x
1
≤
0
.Find all
n
≥
4
n \ge 4
n
≥
4
for which
x
1
+
x
2
+
.
.
.
+
x
n
=
0
x_1+x_2+...+x_n = 0
x
1
+
x
2
+
...
+
x
n
=
0
implies
x
1
x
2
+
x
2
x
3
+
.
.
.
+
x
n
−
1
x
n
+
x
n
x
1
≤
0
x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 \le 0
x
1
x
2
+
x
2
x
3
+
...
+
x
n
−
1
x
n
+
x
n
x
1
≤
0
.
inequalities
Sum
algebra