Estonian Math Competitions

Solution 1: Let $d = \gcd(a,b)$, $a = da'$, $b = db'$. The given equation reduces to $\frac{1}{da'} + \frac{1}{db'} = \frac{1}{2021}$ which is equivalent to
$$2021(a' + b') = da'b'. \qquad (4)$$
As $a'$ and $b'$ are relatively prime to each other, they both are relatively prime to $a' + b'$, implying that $a'b'$ and $a' + b'$ are relatively prime. By (4), $a'b' \mid 2021(a' + b')$, implying $a'b' \mid 2021$. As $2021 = 43 \cdot 47$ where the factors are prime, the number $2021$ has exactly four positive factors $1$, $43$, $47$ and $2021$. Taking into account that $a \ge b$ holds if and only if $a' \ge b'$, consider all cases:
* If $a' = 2021$ and $b' = 1$ then (4) implies $d = 2022$. Consequently, $(a,b) = (2021 \cdot 2022, 2022)$.
* If $a' = 47$ and $b' = 43$ then (4) implies $d = 90$. Consequently, $(a,b) = (47 \cdot 90, 43 \cdot 90)$.
* If $a' = 47$ and $b' = 1$ then (4) implies $d = 43 \cdot 48$. Consequently, $(a,b) = (2021 \cdot 48, 43 \cdot 48)$.
* If $a' = 43$ and $b' = 1$ then (4) implies $d = 47 \cdot 44$. Consequently, $(a,b) = (2021 \cdot 44, 47 \cdot 44)$.
* If $a' = 1$ and $b' = 1$ then (4) implies $d = 2021 \cdot 2$. Thus $(a,b) = (2021 \cdot 2, 2021 \cdot 2)$.

Solution 2: The given equation is equivalent to $\frac{a+b}{ab} = \frac{1}{2021}$ which reduces to $ab - 2021a - 2021b = 0$. After adding $2021^2$ to both sides and factorizing in the l.h.s, we get
$$(a - 2021)(b - 2021) = 2021^2. \qquad (5)$$
As both $a$ and $b$ are positive, the factors in the l.h.s. of (5) are greater than $-2021$. Thus if both factors were negative then the absolute value of their product would be less than $2021^2$ and (5) could not hold. Hence both factors are positive. Since $a \ge b$, we must have $a - 2021 \ge b - 2021$. As $2021 = 43 \cdot 47$ with factors being prime, we obtain variants $(a - 2021, b - 2021) = (2021^2, 1)$, $(a - 2021, b - 2021) = (2021 \cdot 47, 43)$, $(a - 2021, b - 2021) = (2021 \cdot 43, 47)$, $(a - 2021, b - 2021) = (47^2, 43^2)$ and $(a - 2021, b - 2021) = (2021, 2021)$. Consequently, $(a,b) = (2021 \cdot 2022, 2022)$, $(a,b) = (2021 \cdot 48, 43 \cdot 48)$, $(a,b) = (2021 \cdot 44, 47 \cdot 44)$, $(a,b) = (47 \cdot 90, 43 \cdot 90)$, or $(a,b) = (2021 \cdot 2, 2021 \cdot 2)$.