Question

e. Determinati numerele naturale abc, cu proprietatea că

a+1/b+3 =b+2/c+4=c+4/a


Problema din poza 252

Answer 1

$\displaystyle\\\boxed{\it \frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n}=\frac{\sum_{i=1}^{n} a_i}{\sum_{i=1}^{n}b_i}}.\\------------------------\\\frac{a+1}{b+3}=\frac{b+2}{c+4}=\frac{c+4}{a}=\frac{a+b+c+3+4}{a+b+c+3+4}=1 \implies\\\\ \frac{a+1}{b+3}=1 \implies a+1=b+3 \implies a=b+2.~(1)\\\\\frac{b+2}{c+4}=1 \implies b+2=c+4 \implies b=c+2.~(2)\\\\\frac{c+4}{a}=1 \implies a=c+4.~(3).$

$\displaystyle\\\boxed{daca~c=0 \implies a=4,~b=2}.\\\\\boxed{daca~c=1 \implies a=5,~b=3}.\\\\\boxed{daca~c=2 \implies a=6,~b=4}.\\\\\boxed{daca~c=3 \implies a=7,~b=5}.\\\\\boxed{daca~c=4 \implies a=8,~b=6}.\\\\\boxed{daca~c=5 \implies a=9,~b=7}.\\\\\boxed{\overline{abc}\in\left\{ 420,531,642,753,864,975 \right\}}.\\------------------------\\\boxed{daca~c>5 \implies a=c+4 \geq 10,~dar~a=cifra}.$