Problems of the weekend

1) Show that among any nine distinct real numbers, there are two, say a and b, such that

0 < (a-b)/(1+ab) < root(2) - 1

2) Find all the functions f:R to R, with the property :

f(xf(y)+x) = xy + f(x)

for any real numbers x and y.

3) Solve the equation

cos(cos(cos(cosx))) = sin(sin(sin(sinx)))

川の流れのように

長い間ブロックを更新しなかった。申し訳ござらん。