Question

Determinati a,b,c,d daca a/5=b/6=c/10=d/13 si a la a2 +b la a doua+c la a doua+d la a doua< sau = cu 1997 cat face??

Answer 1

a/5 = b/6 =c/10 = d/13 = k/34    a = 5k/34  b = 6k/34  c = 10k/34   d = 13k/34
a² + b² + c² + d² ≤ 1997
(25 + 36 + 100 + 169)·k² ≤ 34²·1997
330k² ≤ 34²·1997    k² = 34²·(n/330)²   n/330 ≤ 6  (1997 = 6·330 + 17)·
 n/330  ∈ D6   n/330 ∈ { 1,2 , 3, 6}
ptr. n/330 = 1   k = 34      a = 5     b = 6  c = 10  d = 13 
25 + 36 + 100 + 169 = 330 < 1997
n/330 = 2     a = 10   b = 12   c = 20    d = 26
100 + 144 + 400 + 676 = 1020 < 1997
n/330 = 3    a = 15    b = 18    c = 30    d = 39
225 + 324 + 900 + 1521 = 2970 > 1997
raspuns :  {a,b,c,d} = 5,6,10,13 }  sau {a,b,c,d} = {10, 12 , 20,26}