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National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2015 Switzerland - Final Round
10
10
Part of
2015 Switzerland - Final Round
Problems
(1)
sum (n + 2)\sqrt{a^2 + b^2} >= n(a + b + c + d)
Source: Switzerland - 2015 Swiss MO Final Round p10
1/14/2023
Find the largest natural number
n
n
n
such that for all real numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
the following holds:
(
n
+
2
)
a
2
+
b
2
+
(
n
+
1
)
a
2
+
c
2
+
(
n
+
1
)
a
2
+
d
2
≥
n
(
a
+
b
+
c
+
d
)
(n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d)
(
n
+
2
)
a
2
+
b
2
+
(
n
+
1
)
a
2
+
c
2
+
(
n
+
1
)
a
2
+
d
2
≥
n
(
a
+
b
+
c
+
d
)
algebra
inequalities