Prove that in two any triangles and the following inequality holds: ( +

Prove that in two any triangles and the following inequality holds: ( +

SOCIETATEA DE ȘTIINȚE MATEMATICE DIN ROMÂNIA
ROMANIAN MATHEMATICAL SOCIETY
Filiala Mehedinți - Mehedinți Branch
www.ssmrmh.ro
Prove that in two any triangles
( + )
and
+( + )
the following inequality holds:
+( + )
≥ √
Proposed by Nguyen Viet Hung – Hanoi – Vietnam
Solution by Kevin Soto Palacios – Huarmey – Peru
1. Para todos los ℝ : , , , , , , se cumple la siguiente desigualdad:
Demonstración:
( + ) +(
+ ) +(
+ ) ≥
(
+
)(
+
+
+
)
+
)
Aplicando: Cauchy – Schwarz:
= ( + ) +(
+ ) +(
(
+
+
)+ (
≥ (
+
+
)(
=
+
+
+
≥
+ ) =(
+
) (
)+
(
(
+
+
+
+ )( + + ) − (
+
+
+
)+ (
+
)( + +
+
)( + + )
La igualdad se alcanza cuando:
+
+
) −(
)−(
+
+
+
+
)≥
)
= =
2. Siendo , , ∧ ′ , ′ , ′ los lados de los triángulos
y
la siguiente desigualdad:
+
+
≥ √ ∧ ′ ′+ ′ ′+ ′ ′ ≥ √
′
′
, se cumple
′
Demostración:
+
=(
+
⇒
)(
+
)+(
≥ √ (
+
)(
)+(
)(
)
≥ √ ... (Válido en un triángulo
)
)≥
SOCIETATEA DE ȘTIINȚE MATEMATICE DIN ROMÂNIA
ROMANIAN MATHEMATICAL SOCIETY
Filiala Mehedinți - Mehedinți Branch
www.ssmrmh.ro
Realizamos los siguientes cambios:
= ,
+
+
+
= ,
= , =
+
=
′ ′
=
′
+
+
, =
+
′ ′
+
′
, =
′
≥ √
′ ′
≥ √
′
Aplicando las desigualdad (1) ∧ (2) se tiene lo siguiente:
( + ) ′+( + )
′
+( + )
≥
√
′
(
≥
√
+
′
=
)(
+
′
+
+
)≥