Determining current in a circuit
If you write the nodal equation for node V1,

$$(V_1 - 100)/5 + V_1/5 + (V_1 - 5i)/10 = 0$$

and relation for i using ohms law in R1,

$$(100 - V_1)/5 = i$$

On solving you get V1 = 50V and i = 10A

\$ \implies V_2 = 5i = 50V \$

Voltage across R3 = \$V_1 - V_2 = 0\$. So obviously the current thru it is zero too, by ohms law.  You can cross check your calculated answers using ohms law in all resistors.

It is not always necessary that a finite value of current flows through each resistor in a circuit. Current flows thru a resistor only if a potential difference exists between its ends.