67th Romanian Mathematical Olympiad

Question

Let I R be an open interval and f, g : I R two functions that satisfy f(x) - g(y)x - y + |x - y| 0, for any x, y I, x y. i) Deduce that f and g are non-decreasing. ii) Find f, g : R R, f g, having the above property.

Accepted Answer

i) The given relation can be rewritten as

f (x) + (x - z)^{2} \geq g (z) \geq f (y) - (z - y)^{2}

, for all triple

x > z > y

,

x, y, z \in I

; in particular,

f (x) + (x - y)^{2} \geq f (y)

, for all triplets

x > y

,

x, y \in I

.

For

a, b \in I

,

a > b

, taking

n \in N^{*}

and

x_{k} = a - \frac{k}{n} (a - b)

,

k \in \overline{0, n}

we obtain

f (x_{k}) + (x_{k} - x_{k + 1})^{2} \geq f (x_{k + 1})

,

k = \overline{0, n - 1}

.

Summing up, we get

f (a) + \frac{1}{n} (a - b)^{2} \geq f (b)

. As this inequality is true for all

n \in N^{*}

, we obtain at the limit

f (a) \geq f (b)

. In the same way we prove that

g

is non-decreasing.

ii) One example with

I = R

is

f (x) = 0

for

x < 0

, and

f (x) = 1

for

x \geq 0

and

g (x) = 0

for

x \leq 0

,

g (x) = 1

for

x > 0

.