MO Z8-I-1 2018

Fero and David meet daily in the elevator. One morning they found that if they multiply their current age, they get 238. If they did the same after four years, this product would be 378. Determine the sum of the current ages of Fero and David.

Correct result:

s =  31

Solution:

a (sa)=238 (a+4) (sa+4)=378  asa2=238 asa2+4a+4s4a+16=378  asa2=238 asa2+4s+16=378   238+4s+16=378  238+4 s+16=378  4s=124  s=31a \cdot \ (s-a)=238 \ \\ (a+4) \cdot \ (s-a+4)=378 \ \\ \ \\ as-a^2=238 \ \\ as-a^2+4a +4s-4a+16=378 \ \\ \ \\ as-a^2=238 \ \\ as-a^2+4s+16=378 \ \\ \ \\ \ \\ 238 + 4s + 16=378 \ \\ \ \\ 238 + 4 \cdot \ s + 16=378 \ \\ \ \\ 4s=124 \ \\ \ \\ s=31

The equations have the following integer solutions:
a*(s-a)=238
(a+4)*(s-a+4) = 378


Number of solutions found: 2
a1=14, s1=31
a2=17, s2=31

Calculated by our Diofant problems and integer equations.



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