Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
 Already signed up? New to KöMaL?

# Problem P. 5095. (January 2019)

P. 5095. Two resistors of resistance values $\displaystyle R_{1}$ and $\displaystyle R_{2}$ are connected in series and their equivalent resistance is $\displaystyle R_{1}+R_{2}$. Two other resistors of resistance $\displaystyle R$ are connected in the circuit, one of them and $\displaystyle R_{1}$ are connected in parallel, whilst the other one and $\displaystyle R_{2}$ are connected in series. Is there a value of $\displaystyle R$ such that the equivalent resistance of the whole circuit remains $\displaystyle R_{1}+R_{2}$?

(4 pont)

Deadline expired on February 11, 2019.

Sorry, the solution is available only in Hungarian. Google translation

$\displaystyle \left(\frac{1}{R_1}+\frac{1}{R}\right)^{-1}+R_2+R=R_1+R_2,$

vagyis

$\displaystyle \frac{R_1R}{R_1+R}+R=R_1.$

Bevezetve az $\displaystyle x=R/R_1$ jelölést, a fenti összefüggés az

$\displaystyle x^2+x-1=0$

másodfokú egyenletre vezet, aminek pozitív gyöke:

$\displaystyle x=\frac{\sqrt{5}-1}{2}\approx 0{,}62.$

A keresett ellenállás nagysága tehát $\displaystyle R_2$-től függetlenül $\displaystyle R\approx 0{,}62\,R_1.$

### Statistics:

 91 students sent a solution. 4 points: 76 students. 3 points: 8 students. 2 points: 4 students. 1 point: 1 student. 0 point: 2 students.

Problems in Physics of KöMaL, January 2019