Marinada '24

[lang=hr] "TO SAM JA VELIKI ČAROBNJAK FEEENIKS!!! Princ tame je oteo mog prijatelja Marka i moju sovu Sandstorm. Moja magija je dovoljno jaka da ih oslobodim, ali moram prvo proći kroz ove skibidifantske jednadžbe kako bih oslobodio puni potencijal svojih moći."\\ Odredite sve cjelobrojne parove $(n,m)$ koji zadovoljavaju: $$\frac{1}{n} + \frac{4}{m} + \frac{7}{nm} = 1$$ Rješenja u ovom lancu zapišite kao sumu kvadrata svih $n,m$ koji zadovoljavaju danu skibidifantsku jednadžbu npr. ako su rješenja $(n,m) = (5,5)$ i $(n,m) = (10,10)$ ona u polje za rješenje upišite $5^2 + 5^2 + 10^2 + 10 ^2 = 250$. [/lang] [lang=en] "I AM THE GREAT WIZARD PHOENIX!!! The Prince of Darkness has kidnapped my friend Marko and my owl, Sandstorm. My magic is strong enough to free them, but I must first solve these skibidifant equations to unlock the full potential of my powers." Determine all integer pairs \((n, m)\) that satisfy the equation: \[ \frac{1}{n} + \frac{4}{m} + \frac{7}{nm} = 1 \] For this chain, write the solution as the sum of the squares of all \(n, m\) that satisfy the given skibidifant equation. For example, if the solutions are \((n,m) = (5,5)\) and \((n,m) = (10,10)\), then in the solution field, you would write \(5^2 + 5^2 + 10^2 + 10^2 = 250\). [/lang]