5. Zadatak

Prvo rješenje.

Zbrajanjem sve tri jednadžbe dobivamo

$$2\cdot\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=2025+2026+2027$$ $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3039.$$

Kako je

$$\frac{1}{a}+\frac{1}{b}=2025,$$

slijedi

$$\frac{1}{c}=3039-2025=1014.$$

Na sličan način dobivamo

$$\frac{1}{a}=3039-2026=1013.$$ $$\frac{1}{b}=3039-2027=1012.$$

Slijedi

$$a=\frac{1}{1013},\qquad b=\frac{1}{1012},\qquad c=\frac{1}{1014}.$$

Drugo rješenje.

Zbrajanjem prve dvije jednadžbe imamo

$$\frac{1}{a}+\frac{1}{b}+\frac{1}{b}+\frac{1}{c}=2025+2026$$ $$\frac{1}{a}+2\cdot\frac{1}{b}+\frac{1}{c}=4051.$$

Kako je

$$\frac{1}{c}+\frac{1}{a}=2027,$$

slijedi

$$2\cdot\frac{1}{b}+2027=4051$$ $$2\cdot\frac{1}{b}=2024$$ $$\frac{1}{b}=1012.$$

Tada je

$$\frac{1}{a}=2025-\frac{1}{b}=2025-1012=1013$$ $$\frac{1}{c}=2026-\frac{1}{b}=2026-1012=1014.$$

Slijedi

$$a=\frac{1}{1013},\qquad b=\frac{1}{1012},\qquad c=\frac{1}{1014}.$$