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Switzerland Contests
Switzerland - Final Round
2016 Switzerland - Final Round
2
2
Part of
2016 Switzerland - Final Round
Problems
(1)
sum (ab+ 1)/(a^2 + ca + 1) >= 3/2 for triangle sidelengths
Source: Switzerland - 2016 Swiss MO Final Round p2
12/30/2022
Let
a
,
b
a, b
a
,
b
and
c
c
c
be the sides of a triangle, that is:
a
+
b
>
c
a + b > c
a
+
b
>
c
,
b
+
c
>
a
b + c > a
b
+
c
>
a
and
c
+
a
>
b
c + a > b
c
+
a
>
b
. Show that:
a
b
+
1
a
2
+
c
a
+
1
+
b
c
+
1
b
2
+
a
b
+
1
+
c
a
+
1
c
2
+
b
c
+
1
>
3
2
\frac{ab+ 1}{a^2 + ca + 1} +\frac{bc + 1}{b^2 + ab + 1} +\frac{ca + 1}{c^2 + bc + 1} > \frac32
a
2
+
c
a
+
1
ab
+
1
+
b
2
+
ab
+
1
b
c
+
1
+
c
2
+
b
c
+
1
c
a
+
1
>
2
3
algebra
inequalities
Geometric Inequalities