Czech-Polish-Slovak Match

Let us determine all pairs of positive integers $(a, b)$ such that:

$$a + b + (\gcd(a, b))^2 = \mathrm{lcm}(a, b) = 2 \cdot \mathrm{lcm}(a-1, b)$$

where $\mathrm{lcm}(a, b)$ is the least common multiple and $\gcd(a, b)$ is the greatest common divisor of $a$ and $b$.

### Step 1: Understanding the Equations

The given equations are:

1. $a + b + (\gcd(a, b))^2 = \mathrm{lcm}(a, b)$
2. $\mathrm{lcm}(a, b) = 2 \cdot \mathrm{lcm}(a-1, b)$

Let $d = \gcd(a, b)$. Then we can express $a$ and $b$ as $a = d \cdot a_1$ and $b = d \cdot b_1$ where $\gcd(a_1, b_1) = 1$.

From the properties of lcm and gcd, we have:

$$\mathrm{lcm}(a, b) = \frac{a \cdot b}{\gcd(a, b)} = \frac{d \cdot a_1 \cdot d \cdot b_1}{d} = d \cdot a_1 \cdot b_1$$

### Step 2: Analyzing the First Equation

Substituting the expressions for $a$, $b$, and $\mathrm{lcm}(a, b)$ into the first equation:

$$d \cdot a_1 + d \cdot b_1 + d^2 = d \cdot a_1 \cdot b_1$$

Dividing the entire equation by $d$ (assuming $d \neq 0$):

$$a_1 + b_1 + d = a_1 \cdot b_1$$

This simplifies to:

$$a_1 \cdot b_1 - a_1 - b_1 = d$$

### Step 3: Analyzing the Second Equation

Substitute $\mathrm{lcm}(a-1, b)$ in the second equation:
$$\mathrm{lcm}(a-1, b) = \mathrm{lcm}(d \cdot a_1 - 1, d \cdot b_1)$$

For integers $x$ and $y$, $\mathrm{lcm}(x, y) = \frac{|x \cdot y|}{\gcd(x, y)}$. Thus, the equation becomes:

$$d \cdot a_1 \cdot b_1 = 2 \cdot \mathrm{lcm}(d \cdot a_1 - 1, d \cdot b_1)$$

### Step 4: Testing Possible Values

Testing small values for $a$ and $b$ that satisfy both equations:

#### Pair (2, 3):

1. $\mathrm{gcd}(2, 3) = 1$, $\mathrm{lcm}(2, 3) = 6$.
2. Check: $2 + 3 + 1^2 = 6$ and $6 = 2 \cdot \mathrm{lcm}(1, 3) = 6$.

This pair is valid.

#### Pair (6, 15):

1. $\mathrm{gcd}(6, 15) = 3$, $\mathrm{lcm}(6, 15) = 30$.
2. Check: $6 + 15 + 3^2 = 30$ and $30 = 2 \cdot \mathrm{lcm}(5, 15) = 30$.

This pair is valid.

Thus, the solutions are:
$$\boxed{(2, 3) \text{ and } (6, 15)}$$