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Poland - Second Round
1956 Poland - Second Round
5
\sqrt{A} + \sqrt{B} + \sqrt{C} < 4 \sqrt{3}
\sqrt{A} + \sqrt{B} + \sqrt{C} < 4 \sqrt{3}
Source: Polish MO second round 1956 p5
August 29, 2024
trigonometry
Problem Statement
Prove that the numbers
A
A
A
,
B
B
B
,
C
C
C
defined by the formulas
A
=
t
g
β
t
g
γ
+
5
,
B
=
t
g
γ
t
g
α
+
5
,
C
=
t
g
α
t
g
β
+
5
,
A = tg \beta tg \gamma + 5,\\ B = tg \gamma tg \alpha + 5,\\ C = tg \alpha tg \beta + 5,
A
=
t
g
βt
g
γ
+
5
,
B
=
t
g
γ
t
gα
+
5
,
C
=
t
gα
t
g
β
+
5
,
where
α
>
0
\alpha>0
α
>
0
,
β
>
0
\beta > 0
β
>
0
,
γ
>
0
\gamma > 0
γ
>
0
and
α
+
β
+
γ
=
9
0
∘
\alpha + \beta + \gamma = 90^\circ
α
+
β
+
γ
=
9
0
∘
, satisfy the inequality
A
+
B
+
C
<
4
3
.
\sqrt{A} + \sqrt{B} + \sqrt{C} < 4 \sqrt{3}.
A
+
B
+
C
<
4
3
.
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