This is called a steady state heat transfer problem.
In this problem, you are given some boundary conditions for the wire. Specifically; you are told that the copper end is at 100°C, the aluminum end is 10°C and (if I'm reading the question properly) the aluminum in the middle is also 10°C.
The thing you seem to be unsure about is, "shouldn't the copper end go below 100°C if the other end is cooling it?"
The answer is yes; if the situation was just a copper wire and an aluminum wire with those temperatures left on their own, an equilibrium would be reached and the end of the copper wire would drop below 100°C (and the heat transfer (Q) between them would be 0). Determining the equilibrium temperatures is not what this question is looking for.
Instead; this question is a steady state, where the temperatures given are fixed at those locations. You can then solve for the heat transfer at those given temperatures.
There are two ways to interpret what this math represents.
The first is that you are taking a snapshot of how this situation started, and you are determining the heat transfer at that instant. That is, you are determining the heat transfer (Q) at the specific time when one end is 100°C and the aluminum is 10°C. Obviously this will only really apply for an instant at the start, and then the ends would begin to move towards equilibrium temperatures.
The other interpretation is that there is a heat source and heat sink capable of keeping the temperatures constant. That is to say that the hot end is being supplied a constant heat (Q) that it is losing to the aluminum rod, and that the aluminum rod constantly has that heat being taken out, so that all temperatures are maintained.
The second interpretation is how I like to consider steady state. Another example would be if you had a CPU processor running at 65 W for example, you know that it is always giving out 65 W of heat, so it can maintain a steady temperature above room temperature and keep a constant temperature gradient.
In this case it doesn't matter how or why the ends are held at those temperatures, we just accept that they are for the purpose of the exercise of determining heat transfer between the two.