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Let

P(x)

and

Q(x)

be polynomials with nonnegative coefficients. We denote by

P'(x)

the derivative of

P(x)

. Suppose that

P(0)=Q(0)=0

and

Q(1) \leq 1 \leq P'(0)

(1)

Prove that

0 \leq Q(x) \leq x \leq P(x)

for all

0 \leq x \leq 1

(2)

Prove that

P(Q(x)) \leq Q(P(x))

for all

0 \leq x \leq 1

.
Proposed by Otgonbayar Uuye.

Problem Statement