Advanced book on Mathematics Olympiad

74 2 Algebra

cαx+dαy+cγ z+dγt=p,

cβ x+dβy+cδz+dδt=q.

We saw above that this system has a unique solution, which implies that its coefficient
matrix is invertible. This coefficient matrix isC. 

The second problem we found in an old textbook on differential and integral calculus.

Example.Given the distinct real numbersa 1 ,a 2 ,a 3 , letx 1 ,x 2 ,x 3 be the three roots of
the equation

u 1

a 1 +t

+

u 2

a 2 +t

+

u 3

a 3 +t

= 1 ,

whereu 1 ,u 2 ,u 3 are real parameters. Prove thatu 1 ,u 2 ,u 3 are smooth functions of
x 1 ,x 2 ,x 3 and that

det

(

∂ui

∂xj

)

=−

(x 1 −x 2 )(x 2 −x 3 )(x 3 −x 1 )

(a 1 −a 2 )(a 2 −a 3 )(a 3 −a 1 )

.

Solution.After eliminating the denominators, the equation from the statement becomes
a cubic equation int,sox 1 ,x 2 ,x 3 are well defined. The parametersu 1 ,u 2 ,u 3 satisfy the
system of equations

1

a 1 +x 1

u 1 +

1

a 2 +x 1

u 2 +

1

a 3 +x 1

u 3 = 1 ,

1

a 1 +x 2

u 1 +

1

a 2 +x 2

u 2 +

1

a 3 +x 2

u 3 = 1 ,

1

a 1 +x 3

u 1 +

1

a 2 +x 3

u 2 +

1

a 3 +x 3

u 3 = 1.

When solving this system, we might end up entangled in algebraic computations. Thus
it is better instead to take a look at the two-variable situation. Solving the system

1

a 1 +x 1

u 1 +

1

a 2 +x 1

u 2 = 1 ,

1

a 1 +x 2

u 1 +

1

a 2 +x 2

u 2 = 1 ,

with Cramer’s rule we obtain

u 1 =

(a 1 +x 1 )(a 1 +x 2 )

(a 1 −a 2 )

and u 2 =

(a 2 +x 1 )(a 2 +x 2 )

(a 2 −a 1 )

.

Now we can extrapolate to the three-dimensional situation and guess that