Page 1 of 1
Cristian Calude, proba pe echipe, R.III, P.III
Posted: Tue Nov 18, 2008 4:21 pm
by Laurian Filip
Sa se determine numerele de forma \( \overline{xy} \), pentru care exista cifrele distincte a,b,c, astfel incat sa fie verificata egalitatea: \( \sqrt{\overline{ba}}+\sqrt{ c}+\sqrt{a}=\sqrt{\overline{xy} \)
Posted: Thu Jun 25, 2009 9:47 pm
by Mateescu Constantin
\( \overline{xy},\ a,\ b,\ c\in\mathbb{N}\ \Longrightarrow \sqrt{\overline{ba}},\ \sqrt{a},\ \sqrt{c}\in\mathbb{N} \Longrightarrow \overline{ba},\ a,\ c \) sunt patrate perfecte.
\( \overline{ba}\in\{16,\ 25,\ 36,\ 49,\ 64,\ 81\}\ \Longrightarrow a\in\{1,4,5,6,9\} \)
Dar \( a,c\in\{0,1,4,9\} \), deci \( a\in\{1,4,9\} \)
\( a=1\ \Longrightarrow \overline{ba}=81,\ c\in\{0,4,9}\ \Longrightarrow \overline{xy}\in\{10,12,13\} \)
\( a=4\ \Longrightarrow \overline{ba}=64,\ c\in\{0,1,9}\ \Longrightarrow \overline{xy}\in\{10,11,13\} \)
\( a=9\ \Longrightarrow \overline{ba}=49,\ c\in\{0,1}\ \Longrightarrow \overline{xy}\in\{10,11\} \)
In concluzie \( \overline{xy}\in\{10,11,12,13\} \).
Posted: Fri Jun 26, 2009 7:38 am
by mihai++
Cred ca ati uitat un radical caci nicio solutie pe care ati dat-o nu verifica.
Posted: Fri Jun 26, 2009 11:21 am
by Mateescu Constantin
E scris gresit in enunt, pt ca membrul drept al ecuatiei este \( \overline{xy} \)