Problem (Number Theory):
A number is called Norwegian if it has three distinct positive divisors whose sum is equal to 2022. Determine the smallest Norwegian number.

(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than 3.)

Solution:
Observe that 1344 is a Norwegian number as 6, 672 and 1344 are three distinct divisors of 1344 and $6+672+1344=2022$. It remains to show that this is the smallest such number.

Assume for contradiction that $N<1344$ is Norwegian and let $N / a, N / b$ and $N / c$ be the three distinct divisors of $N$, with $a<b<c$. Then

$$
2022=N\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)<1344\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)
$$

and so

$$
\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)>\frac{2022}{1344}=\frac{337}{224}=\frac{3}{2}+\frac{1}{224} .
$$

If $a>1$ then

$$
\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \leqslant \frac{1}{2}+\frac{1}{3}+\frac{1}{4}=\frac{13}{12}<\frac{3}{2}
$$

so it must be the case that $a=1$. Similarly, it must hold that $b<4$ since otherwise

$$
1+\frac{1}{b}+\frac{1}{c} \leqslant 1+\frac{1}{4}+\frac{1}{5}<\frac{3}{2}
$$

This leaves two cases to check, $b=2$ and $b=3$.

Case $b=3$. Then

$$
\frac{1}{c}>\frac{3}{2}+\frac{1}{224}-1-\frac{1}{3}>\frac{1}{6}
$$

so $c=4$ or $c=5$. If $c=4$ then

$$
2022=N\left(1+\frac{1}{3}+\frac{1}{4}\right)=\frac{19}{12} N
$$

but this is impossible as $19 \nmid 2022$. If $c=5$ then

$$
2022=N\left(1+\frac{1}{3}+\frac{1}{5}\right)=\frac{23}{15} N
$$

which again is impossible, as $23 \nmid 2022$.

Case $b=2$. Note that $c<224$ since

$$
\frac{1}{c}>\frac{3}{2}+\frac{1}{224}-1-\frac{1}{2}=\frac{1}{224}
$$

It holds that

$$
2022=N\left(1+\frac{1}{2}+\frac{1}{c}\right)=\frac{3 c+2}{2 c} N \Rightarrow(3 c+2) N=4044 c
$$

Since $(c, 3 c-2)=(c, 2) \in\{1,2\}$, then $3 c+2 \mid 8088=2^{3} \cdot 3 \cdot 337$ which implies that $3 c+2 \mid 2^{3} \cdot 337$. But since $3 c+2 \geqslant 3 \cdot 3+2>8=2^{3}$ and $3 c+2 \neq 337$, then it must hold that $3 c+2 \geqslant 2 \cdot 337$, contradicting $c<224$.

Final answer: 1344