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Part of 2015 Middle European Mathematical Olympiad

Problems(2)

Surjective functions on N satisfying special condition

Source: MEMO 2015, problem I-1

Find all surjective functions

f:\mathbb{N}\to\mathbb{N}

such that for all positive integers

a

b

, exactly one of the following equations is true: \begin{align*} f(a)&=f(b),
\\ f(a+b)&=\min\{f(a),f(b)\}. \end{align*} Remarks:

\mathbb{N}

denotes the set of all positive integers. A function

f:X\to Y

is said to be surjective if for every

y\in Y

x\in X

f(x)=y

functionalgebra

a/(2b+c^2) cyclic sum is at most (a^2+b^2+c^2)/3

Source: MEMO 2015, problem T-1

Prove that for all positive real numbers

a

b

c

abc=1

the following inequality holds:

\frac{a}{2b+c^2}+\frac{b}{2c+a^2}+\frac{c}{2a+b^2}\le \frac{a^2+b^2+c^2}3.

inequalitiesFractionmuirhed inequality