Advanced book on Mathematics Olympiad

570 Real Analysis

and this is equivalent to(a+b−c)^2 >0. Hence the conclusion.
(Kvant(Quantum), proposed by R.P. Ushakov)

514.The domain is bounded by the hyperbolasxy=1,xy=2 and the linesy=xand
y= 2 x. This domain can mapped into a rectangle by the transformation

T: u=xy, v=

y

x

.

Thus it is natural to consider the change of coordinates

T−^1 : x=

√

u

v

,y=

√

uv.

The domain becomes the rectangleD∗={(u, v)∈R^2 | 1 ≤u≤ 2 , 1 ≤v≤ 2 }. The
Jacobian ofT−^1 is 21 v =0. The integral becomes

∫ 2

1

∫ 2

1

√

u

v

1

2 v

dudv=

1

2

∫ 2

1

u^1 /^2 du

∫ 2

1

v−^3 /^2 dv=

1

3

( 5

√

2 − 6 ).

(Gh. Bucur, E. Câmpu, S. G ̆aina, ̆ Culegere de Probleme de Calcul Diferen ̧tial ̧si
Integral(Collection of Problems in Differential and Integral Calculus), Editura Tehnic ̆a,
Bucharest, 1967)

515.Denote the integral byI. The change of variable(x,y,z)→(z,y,x)transforms
the integral into

∫∫∫

B

z^4 + 2 y^4

x^4 + 4 y^4 +z^4

dxdydz.

Hence

2 I=

∫∫∫

B

x^4 + 2 y^4

x^4 + 4 y^4 +z^4

dxdydz+

∫∫∫

B

2 y^4 +z^4

x^4 + 4 y^4 +z^4

dxdydz

=

∫∫∫

B

x^4 + 4 y^4 +z^4

x^4 + 4 y^4 +z^4

dxdydz=

4 π

3

.

It follows thatI=^23 π.

516.The domainDis depicted in Figure 71. We transform it into the rectangleD 1 =
[^14 ,^12 ]×[^16 ,^12 ]by the change of coordinates

x=

u

u^2 +v^2

,y=

v

u^2 +v^2

.

The Jacobian is