240 4 Geometry and Trigonometry
677.Find the maximum value of
S=( 1 −x 1 )( 1 −y 1 )+( 1 −x 2 )( 1 −y 2 )ifx 12 +x 22 =y 12 +y 22 =c^2 , wherecis some positive number.678.Prove for all real numbersa, b, cthe inequality
|a−b|
√
1 +a^2√
1 +b^2≤
|a−c|
√
1 +a^2√
1 +c^2+
|b−c|
√
1 +b^2√
1 +c^2.
679.Leta, b, cbe real numbers. Prove that
(ab+bc+ca− 1 )^2 ≤(a^2 + 1 )(b^2 + 1 )(c^2 + 1 ).680.Prove that
x
√
1 +x^2+
y
√
1 +y^2+
z
√
1 +z^2≤
3
√
3
2
if the positive real numbersx, y, zsatisfyx+y+z=xyz.681.Prove that
x
1 −x^2+
y
1 −y^2+
z
1 −z^2≥
3
√
3
2
if 0< x,y,z <1 andxy+yz+xz=1.682.Solve the following system of equations in real numbers:
3 x−y
x− 3 y=x^2 ,3 y−z
y− 3 z=y^2 ,3 z−x
z− 3 x=z^2.683.Leta 0 =
√
2,b 0 =2, andan+ 1 =√
2 −
√
4 −a^2 n,bn+ 1 =
2 bn
2 +√
4 +bn^2,n≥ 0.(a) Prove that the sequences(an)nand(bn)nare decreasing and converge to zero.
(b) Prove that the sequence( 2 nan)nis increasing, the sequence( 2 nbn)nis decreas-
ing, and these two sequences converge to the same limit.