IMO Shortlist

To find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying the given functional equation, we start with the condition:
$$f(0)\neq 0,$$
and the functional equation for all $x, y \in \mathbb{R}$:
$$f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}.$$

### Step 1: Test Simple Functions
Let's start by testing the functions given in the reference answer: $f(x) = -1$ and $f(x) = x - 1$.

#### Case 1: $f(x) = -1$
Substitute $f(x) = -1$ into the equation:
$$f(x+y)^2 = f(-1)^2 = 1.$$
$$2f(x)f(y) = 2(-1)(-1) = 2.$$
$$\max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\} = \max \{-1, -2\} = -1.$$
Thus, the right side becomes:
$$2 - 1 = 1.$$
This matches the left side, confirming $f(x) = -1$ is a solution.

#### Case 2: $f(x) = x - 1$
Substituting $f(x) = x - 1$:
$$f(x+y) = (x+y) - 1.$$
Hence, the left side is:
$$((x+y)-1)^2 = (x+y)^2 - 2(x+y) + 1.$$

Now the right side:
$$2f(x)f(y) = 2(x-1)(y-1) = 2(xy - x - y + 1).$$
$$f(x^2+y^2) = (x^2+y^2) - 1.$$
$$f(x^2)+f(y^2) = (x^2-1)+(y^2-1) = x^2+y^2-2.$$
Thus:
$$\max \left\{ (x^2+y^2)-1, (x^2+y^2)-2 \right\} = (x^2+y^2)-1.$$

Equating both sides:
$$(x+y)^2 - 2(x+y) + 1 = 2(xy - x - y + 1) + (x^2+y^2) - 1.$$

Simplifying right side:
$$2xy - 2x - 2y + 2 + x^2 + y^2 - 1 = x^2 + y^2 + 2xy - 2x - 2y + 1.$$

This simplifies to:
$$(x+y)^2 - 2(x+y) + 1,$$
confirming $f(x) = x - 1$ is another valid solution.

### Conclusion
After verifying the two functions, we conclude that the functions satisfying the original functional equation are:
$$f(x) = -1 \quad \text{and} \quad f(x) = x - 1.$$

Thus, the solutions are:
$$\boxed{f(x) = -1 \text{ and } f(x) = x - 1}.$$