Can we write algorithms without conditional statements?
Here is a Turing complete "programming" language. It has just two constants, called $K$ and $S$, and one operation called "application", written $x \cdot y$. The two constants satisfy the rules
$$(K \cdot x) \cdot y = x$$
and
$$((S \cdot x) \cdot y) \cdot z = (x \cdot z) \cdot (y \cdot z).$$
This is known as a combinatory algebra and is well known to be Turing complete. There is an actual programming language based on $S$ and $K$, called unlambda. I think we can agree that there is no builtin conditional. Of course, since this language is Turing complete, it can also simulate conditional statements.