Question

[Ngọc Diệu]

a,Cho các số a,b,c,d thỏa mãn a/b+c+d=b/c+d+a=c/d+a+b=d/a+b+c

Tính giá trị của biểu thức:

P=a+b/c+d=b+c/d+a=c+d/b+a=d+a/b+c
Giúp mình vs ạ

Answer 1

[No name]

⇒ab+c+d+1=bc+d+a+1=cd+a+b+1=da+b+c+1⇒ab+c+d+1=bc+d+a+1=cd+a+b+1=da+b+c+1

⇒a+b+c+db+c+d=b+c+d+ac+d+a=c+d+a+bd+a+b=d+a+b+ca+b+c⇒a+b+c+db+c+d=b+c+d+ac+d+a=c+d+a+bd+a+b=d+a+b+ca+b+c

Xét a+b+c+d≠0a+b+c+d≠0 suy ra a=b=c=da=b=c=d. Khi đó:

P=a+bc+d+b+cd+a+c+db+a+d+ab+c=1+1+1+1=4P=a+bc+d+b+cd+a+c+db+a+d+ab+c=1+1+1+1=4

Xét a+b+c+d=0a+b+c+d=0 suy ra ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩a+b=−(c+d)b+c=−(d+a)c+d=−(b+a)d+a=−(b+c){a+b=−(c+d)b+c=−(d+a)c+d=−(b+a)d+a=−(b+c). Khi đó:

P=a+bc+d+b+cd+a+c+db+a+d+ab+c=(−1)+(−1)+(−1)+(−1)=−4