Olimpiade Sains Kota (OSK) 2017 - Komputer, Nomor 2

AB + C + D = 3...(i)
BC + A + D = 5...(ii)
CD + A + B = 2...(iii)
DA + B + C = 6...(iv)

Step 1 : (i) - (ii)
C - A + B(A - C) = -2 →
(A - C)(1 - B) = 2 →
A - C = 2/(1 - B)...(v)

Step 2 : (iii) - (iv)
A - C + D(C - A) = -4 →
(A - C)(1 - D) = -4 →
A - C = -4/(1 - D)…(vi)

Step 3 : (v) = (vi)
2/(1 - B) = -4/(1 - D) →
2 - 2D = -4 + 4B →
3B + D = 3…(vii)

Step 4 : (i) - (iv)
D - B + A(B - D) = -3 →
(B - D)(1 - A) = 3 →
B - D = 3/(1 - A)...(viii)

Step 5 : (ii) - (iii)
D - B + C(B - D) = 3 →
(B - D)(1 - C) = -3 →
B - D = -3/(1 - C)...(ix)

Step 6 : (viii) = (ix)
3/(1 - A) = -3/(1 - C) →
3 - 3C = -3 + 3A →
A + C = 2…(x)

 

Step 7 : (i) + (ii) + (iii) + (iv)
A + B + B + C + C + D + D + A + AB + BC + CD + DA = 16
→ 2(A + B + C + D) + (A + C)(B + D) = 16
Substitusikan persamaan (x) : A + C = 2
→ 2(2 + B + D) + 2(B + D) = 16 → 4 + 4B + 4D = 16 → B + D = 3…(xi)

Step 8 : Gunakan Eliminasi/Substitusi pada persamaan (vii) dan (xi)
3B + D = 3
B + D = 3
------------- -
B = 0, D = 3

Step 9 : Substitusikan nilai B dan D ke persamaan (ii)
0 + A + 3 = 5 → A = 2