## 1. Abridged Problem Statement

Two circular disks move on a plane. For each disk, we know its initial center position, velocity, radius, and mass. The disks move in straight lines with constant velocity unless they collide. During a collision, only the velocity components along the line connecting the centers may change, and total momentum and kinetic energy are conserved.

Given a time `t`, determine the positions and velocities of both disks after `t` seconds. Initially and at time `t`, the disks do not intersect or touch.

Input contains the data for disk `A`, disk `B`, and time `t`.

Output the final position and velocity of each disk, with at least 3 digits after the decimal point.

---

## 2. Detailed Editorial

We have two moving circles/disks. Since there are no walls or other objects, each disk moves uniformly until the possible collision between the two disks.

The main tasks are:

1. Determine whether the disks collide before time `t`.
2. If they do not collide, simply move them forward for time `t`.
3. If they collide, move them to the collision time, update velocities using elastic collision rules, then move them for the remaining time.

---

### Step 1: Relative Motion

Let:

```text
pos_a, vel_a = position and velocity of disk A
pos_b, vel_b = position and velocity of disk B
```

Instead of considering both disks moving, we can observe disk `A` relative to disk `B`.

Define:

```text
p = pos_a - pos_b
v = vel_a - vel_b
```

Here:

- `p` is the relative position of A with respect to B.
- `v` is the relative velocity of A with respect to B.

The disks touch when the distance between their centers equals the sum of their radii:

```text
|p + v*s| = r_a + r_b
```

where `s` is the time elapsed from the start.

Let:

```text
R = r_a + r_b
```

Then collision happens when:

```text
|p + v*s|^2 = R^2
```

Expanding:

```text
(v · v) s^2 + 2(p · v) s + (p · p - R^2) = 0
```

This is a quadratic equation:

```text
a s^2 + b s + c = 0
```

where:

```text
a = v · v
b = 2(p · v)
c = p · p - R^2
```

Since the disks do not initially touch or overlap, `c > 0`.

---

### Step 2: Find Collision Time

Compute the discriminant:

```text
D = b^2 - 4ac
```

If:

```text
D < 0
```

there is no collision.

Otherwise, the possible collision times are:

```text
s1 = (-b - sqrt(D)) / (2a)
s2 = (-b + sqrt(D)) / (2a)
```

The first possible contact time is `s1`.

If:

```text
0 < s1 <= t
```

then the disks collide during the interval.

Otherwise, there is no collision before time `t`.

Because there are only two disks and no external forces or walls, after an elastic collision they move apart, so at most one collision needs to be processed.

---

### Step 3: If There Is No Collision

If there is no collision, final positions are simply:

```text
pos_a = pos_a + vel_a * t
pos_b = pos_b + vel_b * t
```

Velocities remain unchanged.

---

### Step 4: If There Is a Collision

Suppose the collision happens at time `hit`.

First move both disks to the collision point:

```text
pos_a = pos_a + vel_a * hit
pos_b = pos_b + vel_b * hit
```

At the moment of collision, define the unit normal vector from disk B to disk A:

```text
n = unit(pos_a - pos_b)
```

Only the velocity components along `n` are affected.

Let the relative velocity along the normal be:

```text
rel = (vel_a - vel_b) · n
```

For a perfectly elastic collision of two masses, the velocity updates are:

```text
vel_a = vel_a - n * (2 * m_b / (m_a + m_b) * rel)
vel_b = vel_b + n * (2 * m_a / (m_a + m_b) * rel)
```

These formulas conserve both total momentum and kinetic energy.

Then move the disks for the remaining time:

```text
rest = t - hit

pos_a = pos_a + vel_a * rest
pos_b = pos_b + vel_b * rest
```

---

### Complexity

The solution uses only a constant number of arithmetic operations.

```text
Time complexity:  O(1)
Memory complexity: O(1)
```

---

## 3. Provided C++ Solution with Detailed Comments

```cpp
// Include the standard C++ library header containing most commonly used STL tools.
#include <bits/stdc++.h>

// Use the standard namespace to avoid writing std:: everywhere.
using namespace std;

// Overload output operator for pairs.
// This allows writing cout << pair_object.
template<typename T1, typename T2>
ostream& operator<<(ostream& out, const pair<T1, T2>& x) {
    // Print first and second element separated by a space.
    return out << x.first << ' ' << x.second;
}

// Overload input operator for pairs.
// This allows writing cin >> pair_object.
template<typename T1, typename T2>
istream& operator>>(istream& in, pair<T1, T2>& x) {
    // Read first and second element.
    return in >> x.first >> x.second;
}

// Overload input operator for vectors.
// Reads all elements of an already-sized vector.
template<typename T>
istream& operator>>(istream& in, vector<T>& a) {
    // Iterate through all elements by reference.
    for(auto& x: a) {
        // Read each element.
        in >> x;
    }

    // Return the input stream.
    return in;
};

// Overload output operator for vectors.
template<typename T>
ostream& operator<<(ostream& out, const vector<T>& a) {
    // Iterate through all elements.
    for(auto x: a) {
        // Print each element followed by a space.
        out << x << ' ';
    }

    // Return the output stream.
    return out;
};

// Coordinate type used for geometry calculations.
using coord_t = double;

// Structure representing a 2D point or vector.
struct Point {
    // Small epsilon used for floating-point comparisons.
    static constexpr coord_t eps = 1e-9;

    // Value of pi.
    static inline const coord_t PI = acos((coord_t)-1.0);

    // Coordinates of the point/vector.
    coord_t x, y;

    // Constructor with default coordinates equal to zero.
    Point(coord_t x = 0, coord_t y = 0) : x(x), y(y) {}

    // Vector addition.
    Point operator+(const Point& p) const { return Point(x + p.x, y + p.y); }

    // Vector subtraction.
    Point operator-(const Point& p) const { return Point(x - p.x, y - p.y); }

    // Multiplication of a vector by a scalar.
    Point operator*(coord_t c) const { return Point(x * c, y * c); }

    // Division of a vector by a scalar.
    Point operator/(coord_t c) const { return Point(x / c, y / c); }

    // Dot product of two vectors.
    coord_t operator*(const Point& p) const { return x * p.x + y * p.y; }

    // Cross product of two vectors.
    coord_t operator^(const Point& p) const { return x * p.y - y * p.x; }

    // Exact equality comparison.
    bool operator==(const Point& p) const { return x == p.x && y == p.y; }

    // Exact inequality comparison.
    bool operator!=(const Point& p) const { return x != p.x || y != p.y; }

    // Lexicographical comparison.
    bool operator<(const Point& p) const {
        return x != p.x ? x < p.x : y < p.y;
    }

    // Reverse lexicographical comparison.
    bool operator>(const Point& p) const {
        return x != p.x ? x > p.x : y > p.y;
    }

    // Less-than-or-equal lexicographical comparison.
    bool operator<=(const Point& p) const {
        return x != p.x ? x < p.x : y <= p.y;
    }

    // Greater-than-or-equal lexicographical comparison.
    bool operator>=(const Point& p) const {
        return x != p.x ? x > p.x : y >= p.y;
    }

    // Squared length of the vector.
    coord_t norm2() const { return x * x + y * y; }

    // Length of the vector.
    coord_t norm() const { return sqrt(norm2()); }

    // Polar angle of the vector.
    coord_t angle() const { return atan2(y, x); }

    // Rotate the vector by angle a radians.
    Point rotate(coord_t a) const {
        return Point(x * cos(a) - y * sin(a), x * sin(a) + y * cos(a));
    }

    // Return a perpendicular vector rotated 90 degrees counterclockwise.
    Point perp() const { return Point(-y, x); }

    // Return the unit vector in the same direction.
    Point unit() const { return *this / norm(); }

    // Return a unit normal vector.
    Point normal() const { return perp().unit(); }

    // Projection of vector p onto this vector.
    Point project(const Point& p) const {
        return *this * (*this * p) / norm2();
    }

    // Reflection of vector p across the line in the direction of this vector.
    Point reflect(const Point& p) const {
        return *this * 2 * (*this * p) / norm2() - p;
    }

    // Output a point as "x y".
    friend ostream& operator<<(ostream& os, const Point& p) {
        return os << p.x << ' ' << p.y;
    }

    // Read a point as "x y".
    friend istream& operator>>(istream& is, Point& p) {
        return is >> p.x >> p.y;
    }

    // Compute orientation of three points.
    // Returns 1 for counterclockwise, -1 for clockwise, and 0 for collinear.
    friend int ccw(const Point& a, const Point& b, const Point& c) {
        // Cross product of AB and AC.
        coord_t v = (b - a) ^ (c - a);

        // If cross product is close to zero, points are collinear.
        if(-eps <= v && v <= eps) {
            return 0;
        } else if(v > 0) {
            return 1;
        } else {
            return -1;
        }
    }

    // Check whether point p lies on segment AB.
    friend bool point_on_segment(
        const Point& a, const Point& b, const Point& p
    ) {
        return ccw(a, b, p) == 0 &&
               p.x >= min(a.x, b.x) - eps &&
               p.x <= max(a.x, b.x) + eps &&
               p.y >= min(a.y, b.y) - eps &&
               p.y <= max(a.y, b.y) + eps;
    }

    // Check whether point p is inside or on triangle ABC.
    friend bool point_in_triangle(
        const Point& a, const Point& b, const Point& c, const Point& p
    ) {
        int d1 = ccw(a, b, p);
        int d2 = ccw(b, c, p);
        int d3 = ccw(c, a, p);

        return (d1 >= 0 && d2 >= 0 && d3 >= 0) ||
               (d1 <= 0 && d2 <= 0 && d3 <= 0);
    }

    // Find intersection point of two infinite lines.
    friend Point line_line_intersection(
        const Point& a1, const Point& b1, const Point& a2, const Point& b2
    ) {
        return a1 +
               (b1 - a1) * ((a2 - a1) ^ (b2 - a2)) / ((b1 - a1) ^ (b2 - a2));
    }

    // Check whether two vectors are collinear.
    friend bool collinear(const Point& a, const Point& b) {
        return abs(a ^ b) < eps;
    }

    // Compute circumcenter of triangle ABC.
    friend Point circumcenter(const Point& a, const Point& b, const Point& c) {
        Point mid_ab = (a + b) / 2.0;
        Point mid_ac = (a + c) / 2.0;
        Point perp_ab = (b - a).perp();
        Point perp_ac = (c - a).perp();

        return line_line_intersection(
            mid_ab, mid_ab + perp_ab, mid_ac, mid_ac + perp_ac
        );
    }

    // Compute signed area contribution of a circular arc.
    friend coord_t arc_area(
        const Point& center, coord_t r, const Point& p1, const Point& p2
    ) {
        coord_t theta1 = (p1 - center).angle();
        coord_t theta2 = (p2 - center).angle();

        if(theta2 < theta1 - eps) {
            theta2 += 2 * PI;
        }

        coord_t d_theta = theta2 - theta1;
        coord_t cx = center.x, cy = center.y;

        coord_t area = r * cx * (sin(theta2) - sin(theta1)) -
                       r * cy * (cos(theta2) - cos(theta1)) + r * r * d_theta;

        return area / 2.0;
    }

    // Find intersection points of two circles.
    friend vector<Point> intersect_circles(
        const Point& c1, coord_t r1, const Point& c2, coord_t r2
    ) {
        Point d = c2 - c1;
        coord_t dist = d.norm();

        if(dist > r1 + r2 + eps || dist < abs(r1 - r2) - eps || dist < eps) {
            return {};
        }

        coord_t a = (r1 * r1 - r2 * r2 + dist * dist) / (2 * dist);
        coord_t h_sq = r1 * r1 - a * a;

        if(h_sq < -eps) {
            return {};
        }

        if(h_sq < 0) {
            h_sq = 0;
        }

        coord_t h = sqrt(h_sq);

        Point mid = c1 + d.unit() * a;
        Point perp_dir = d.perp().unit();

        if(h < eps) {
            return {mid};
        }

        return {mid + perp_dir * h, mid - perp_dir * h};
    }

    // Find intersection between a ray and a segment.
    friend optional<Point> intersect_ray_segment(
        const Point& ray_start, const Point& ray_through, const Point& seg_a,
        const Point& seg_b
    ) {
        Point ray_dir = ray_through - ray_start;

        if(ray_dir.norm2() < Point::eps) {
            return {};
        }

        Point seg_dir = seg_b - seg_a;
        coord_t denom = ray_dir ^ seg_dir;

        if(fabs(denom) < eps) {
            return {};
        }

        coord_t t = ((seg_a - ray_start) ^ seg_dir) / denom;

        if(t < eps) {
            return {};
        }

        coord_t s = ((seg_a - ray_start) ^ ray_dir) / denom;

        if(s < eps || s > 1 - eps) {
            return {};
        }

        return ray_start + ray_dir * t;
    }
};

// Global variables for positions, velocities, radii, masses, and time.
Point pos_a, vel_a, pos_b, vel_b;
coord_t r_a, m_a, r_b, m_b, t;

// Read input.
void read() {
    // Read disk A: position, velocity, radius, mass.
    cin >> pos_a >> vel_a >> r_a >> m_a;

    // Read disk B: position, velocity, radius, mass.
    cin >> pos_b >> vel_b >> r_b >> m_b;

    // Read elapsed time.
    cin >> t;
}

// Solve the problem.
void solve() {
    // Sum of radii; collision happens when distance between centers equals R.
    coord_t R = r_a + r_b;

    // Relative position of A with respect to B.
    Point p = pos_a - pos_b;

    // Relative velocity of A with respect to B.
    Point v = vel_a - vel_b;

    // Coefficient of s^2 in |p + v*s|^2 = R^2.
    coord_t a = v * v;

    // Coefficient of s.
    coord_t b = 2 * (p * v);

    // Constant term.
    coord_t c = p * p - R * R;

    // Collision time; -1 means no collision found.
    coord_t hit = -1;

    // If relative velocity is nonzero, solve the quadratic equation.
    if(a > Point::eps) {
        // Discriminant of the quadratic equation.
        coord_t disc = b * b - 4 * a * c;

        // Real roots exist only when discriminant is nonnegative.
        if(disc >= 0) {
            // Earlier root: first time the disks touch.
            coord_t s = (-b - sqrt(disc)) / (2 * a);

            // Collision must happen after time 0 and not after target time t.
            if(s > Point::eps && s <= t + Point::eps) {
                // Store collision time.
                hit = min(s, t);
            }
        }
    }

    // If there is no collision before time t.
    if(hit < 0) {
        // Move disk A uniformly for the whole time.
        pos_a = pos_a + vel_a * t;

        // Move disk B uniformly for the whole time.
        pos_b = pos_b + vel_b * t;
    } else {
        // Move disk A to the collision point.
        pos_a = pos_a + vel_a * hit;

        // Move disk B to the collision point.
        pos_b = pos_b + vel_b * hit;

        // Unit normal vector along the line from B to A.
        Point n = (pos_a - pos_b).unit();

        // Relative velocity component along the collision normal.
        coord_t rel = (vel_a - vel_b) * n;

        // Update velocity of A using elastic collision formula.
        vel_a = vel_a - n * (2 * m_b / (m_a + m_b) * rel);

        // Update velocity of B using elastic collision formula.
        vel_b = vel_b + n * (2 * m_a / (m_a + m_b) * rel);

        // Remaining time after the collision.
        coord_t rest = t - hit;

        // Move disk A for the remaining time using new velocity.
        pos_a = pos_a + vel_a * rest;

        // Move disk B for the remaining time using new velocity.
        pos_b = pos_b + vel_b * rest;
    }

    // Print fixed-point decimal numbers.
    cout << fixed << setprecision(6);

    // Output final position and velocity of disk A.
    cout << pos_a << ' ' << vel_a << '\n';

    // Output final position and velocity of disk B.
    cout << pos_b << ' ' << vel_b << '\n';
}

// Main function.
int main() {
    // Speed up input/output.
    ios_base::sync_with_stdio(false);

    // Untie cin from cout.
    cin.tie(nullptr);

    // Number of test cases. The original problem has exactly one.
    int T = 1;

    // Process all test cases.
    for(int test = 1; test <= T; test++) {
        // Read input.
        read();

        // Solve and output answer.
        solve();
    }

    // Successful program termination.
    return 0;
}
```

---

## 4. Python Solution with Detailed Comments

```python
import math
import sys


# Small value used to handle floating-point comparison errors.
EPS = 1e-9


class Point:
    def __init__(self, x=0.0, y=0.0):
        # Store x-coordinate as float.
        self.x = float(x)

        # Store y-coordinate as float.
        self.y = float(y)

    def __add__(self, other):
        # Vector addition.
        return Point(self.x + other.x, self.y + other.y)

    def __sub__(self, other):
        # Vector subtraction.
        return Point(self.x - other.x, self.y - other.y)

    def __mul__(self, value):
        # If multiplying by another Point, return dot product.
        if isinstance(value, Point):
            return self.x * value.x + self.y * value.y

        # Otherwise multiply vector by scalar.
        return Point(self.x * value, self.y * value)

    def __rmul__(self, value):
        # Support scalar * Point.
        return self.__mul__(value)

    def __truediv__(self, value):
        # Divide vector by scalar.
        return Point(self.x / value, self.y / value)

    def norm2(self):
        # Squared vector length.
        return self.x * self.x + self.y * self.y

    def norm(self):
        # Vector length.
        return math.sqrt(self.norm2())

    def unit(self):
        # Unit vector in the same direction.
        length = self.norm()
        return self / length

    def __repr__(self):
        # String representation for debugging.
        return f"{self.x} {self.y}"


def main():
    # Read all input tokens.
    data = sys.stdin.read().strip().split()

    # If input is empty, do nothing.
    if not data:
        return

    # Convert all tokens to float.
    data = list(map(float, data))

    # Input format:
    # A: x y vx vy radius mass
    # B: x y vx vy radius mass
    # t
    idx = 0

    # Read disk A position.
    pos_a = Point(data[idx], data[idx + 1])
    idx += 2

    # Read disk A velocity.
    vel_a = Point(data[idx], data[idx + 1])
    idx += 2

    # Read disk A radius.
    r_a = data[idx]
    idx += 1

    # Read disk A mass.
    m_a = data[idx]
    idx += 1

    # Read disk B position.
    pos_b = Point(data[idx], data[idx + 1])
    idx += 2

    # Read disk B velocity.
    vel_b = Point(data[idx], data[idx + 1])
    idx += 2

    # Read disk B radius.
    r_b = data[idx]
    idx += 1

    # Read disk B mass.
    m_b = data[idx]
    idx += 1

    # Read total elapsed time.
    total_time = data[idx]

    # Collision happens when distance between centers equals sum of radii.
    R = r_a + r_b

    # Relative position of A with respect to B.
    p = pos_a - pos_b

    # Relative velocity of A with respect to B.
    v = vel_a - vel_b

    # Quadratic coefficient a in |p + v*s|^2 = R^2.
    a = v * v

    # Quadratic coefficient b.
    b = 2.0 * (p * v)

    # Quadratic coefficient c.
    c = (p * p) - R * R

    # Collision time. None means no collision before total_time.
    hit = None

    # If relative velocity is zero, the distance never changes.
    if a > EPS:
        # Discriminant of the quadratic equation.
        disc = b * b - 4.0 * a * c

        # If discriminant is nonnegative, the relative path reaches the circle.
        if disc >= -EPS:
            # Avoid tiny negative values caused by floating-point errors.
            if disc < 0.0:
                disc = 0.0

            # The earlier root is the first contact time.
            s = (-b - math.sqrt(disc)) / (2.0 * a)

            # Collision is relevant only if it occurs during the interval.
            if s > EPS and s <= total_time + EPS:
                hit = min(s, total_time)

    # Case 1: no collision happens before the target time.
    if hit is None:
        # Move A uniformly.
        pos_a = pos_a + vel_a * total_time

        # Move B uniformly.
        pos_b = pos_b + vel_b * total_time

    # Case 2: collision happens.
    else:
        # Move A to the collision instant.
        pos_a = pos_a + vel_a * hit

        # Move B to the collision instant.
        pos_b = pos_b + vel_b * hit

        # Unit normal vector from B to A at collision.
        n = (pos_a - pos_b).unit()

        # Relative velocity projected onto the collision normal.
        rel = (vel_a - vel_b) * n

        # Elastic collision formula for A.
        vel_a = vel_a - n * (2.0 * m_b / (m_a + m_b) * rel)

        # Elastic collision formula for B.
        vel_b = vel_b + n * (2.0 * m_a / (m_a + m_b) * rel)

        # Remaining time after collision.
        rest = total_time - hit

        # Move A with its new velocity.
        pos_a = pos_a + vel_a * rest

        # Move B with its new velocity.
        pos_b = pos_b + vel_b * rest

    # Print final position and velocity of A.
    print(f"{pos_a.x:.6f} {pos_a.y:.6f} {vel_a.x:.6f} {vel_a.y:.6f}")

    # Print final position and velocity of B.
    print(f"{pos_b.x:.6f} {pos_b.y:.6f} {vel_b.x:.6f} {vel_b.y:.6f}")


if __name__ == "__main__":
    main()
```

---

## 5. Compressed Editorial

Use relative motion. Let:

```text
p = pos_a - pos_b
v = vel_a - vel_b
R = r_a + r_b
```

The disks collide when:

```text
|p + v*s| = R
```

Squaring gives:

```text
(v·v)s² + 2(p·v)s + (p·p - R²) = 0
```

Solve this quadratic. The earlier positive root is the first collision time. If no such root exists in `(0, t]`, move both disks normally for time `t`.

If a collision happens at time `hit`, move both disks to that time. Let:

```text
n = unit(pos_a - pos_b)
rel = (vel_a - vel_b) · n
```

Then apply the elastic collision formulas:

```text
vel_a -= n * (2m_b / (m_a + m_b) * rel)
vel_b += n * (2m_a / (m_a + m_b) * rel)
```

Finally, move both disks for the remaining time `t - hit`.

Only one collision is possible because after the elastic impact the disks separate and there are no walls or other forces.

Complexity:

```text
O(1)
```