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National and Regional Contests
Netherlands Contests
Dutch IMO TST
2024 Dutch IMO TST
3
A strangely asymmetric inequality with square roots
A strangely asymmetric inequality with square roots
Source: Dutch TST 2024, 2.3
June 28, 2024
inequalities
inequalities proposed
algebra proposed
algebra
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers such that
0
≤
a
≤
b
≤
c
0 \le a \le b \le c
0
≤
a
≤
b
≤
c
and
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Show that
a
b
b
−
a
+
b
c
c
−
b
+
a
c
c
−
a
<
1
4
.
ab\sqrt{b-a}+bc\sqrt{c-b}+ac\sqrt{c-a}<\frac{1}{4}.
ab
b
−
a
+
b
c
c
−
b
+
a
c
c
−
a
<
4
1
.
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