My favourite problem ever
If a , b , c , d satisfy the equations
a
+
7
b
+
3
c
+
5
d
=
0
,
8
a
+
4
b
+
6
c
+
2
d
=
−
1
6
,
2
a
+
6
b
+
4
c
+
8
d
=
1
6
,
5
a
+
3
b
+
7
c
+
d
=
−
1
6
,
find the value of ( a + d ) ( b + c ) .
The answer is -16.
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2 solutions
Add the second and third equations and one gets 1 0 a + 1 0 b + 1 0 c + 1 0 d = 0 . Therefore, a + b + c + d = 0 .
In addition, adding the first and last equations gives 6 a + 1 0 b + 1 0 c + 6 d = − 1 6 . 6 a + 1 0 b + 1 0 c + 6 d − 6 ( a + b + c + d ) = − 1 6 − 6 ( 0 ) = 4 b + 4 c = − 1 6 . Therefore b + c = − 4 , and since a − 4 + d = 0 , a + d = 4 For our answer, we multiply ( a + d ) ( b + c ) = ( 4 ) ( − 4 ) = − 1 6
You can easily deduct that a+b+c+d=0 using the two equations in the middle. This means that a+d=-b-c From here, add all the equations up and simplify to get 4a+4d+5b+5c=-4 This simplifies further to 4(A+D) +5(B+C)= -4 We know that a+d= -b-c thus we can rewrite the equation as 4(-b-c)+5(b+c) =4 and if you simplify, you should end with b+c=-4 . Notice that (a+d)(b+c) can be rewritten as -[(b+c)^2] if we use the fact that a+d=-b-c Since b+c=-4 we know that the answer is -[(-4)^2] which is -(16)= -16