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IMO Shortlist 1996 problem A6

Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that

$k = f(x) \cdot (x+1)^n + g(x) \cdot (x^n + 1)$

for some polynomials $f(x), g(x)$ having integer coefficients. If $k_0$ denotes the least such $k,$ determine $k_0$ as a function of $n,$ i.e. show that $k_0 = 2^q$ where $q$ is the odd integer determined by $n = q \cdot 2^r, r \in \mathbb{N}.$

Note: This is variant A6' of the three variants given for this problem.

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IMO Shortlist 1996 problem A4

Let $a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that
(a) The polynomial $p(x) = x^{n} - a_{1}x^{n - 1} + ... - a_{n - 1}x - a_{n}$ has preceisely 1 positive real root $R$ .
(b) let $A = \sum_{i = 1}^n a_{i}$ and $B = \sum_{i = 1}^n ia_{i}$ . Show that $A^{A} \leq R^{B}$ .

IMO Shortlist 1996 problem A7

Let $f$ be a function from the set of real numbers $\mathbb{R}$ into itself such for all $x \in \mathbb{R},$ we have $|f(x)| \leq 1$ and

$f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac{1}{6} \right) + f \left( x + \frac{1}{7} \right).$

Prove that $f$ is a periodic function (that is, there exists a non-zero real number $c$ such $f(x+c) = f(x)$ for all $x \in \mathbb{R}$ ).

IMO Shortlist 2003 problem A5

Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+ \longrightarrow \mathbb{R}^+$ that satisfy the following conditions:

- $f(xyz)+f(x)+f(y)+f(z)=f(\sqrt{xy})f(\sqrt{yz})f(\sqrt{zx})$ for all $x,y,z\in\mathbb{R}^+$ ;

- $f(x)<f(y)$ for all $1\le x<y$ .

IMO Shortlist 2004 problem A6

Find all functions $f\colon\mathbb{R} \rightarrow\mathbb{R}$ satisfying the equation $f\left(x^2 + y^2 + 2f\left(xy\right)\right) = \left(f\left(x + y\right)\right)^2$ for all $x,y\in \mathbb{R}$ .

IMO Shortlist 2008 problem A6

Let $f: \mathbb{R}\to\mathbb{N}$ be a function which satisfies $f\left(x + \dfrac{1}{f(y)}\right) = f\left(y + \dfrac{1}{f(x)}\right)$ for all $x$ , $y\in\mathbb{R}$ . Prove that there is a positive integer which is not a value of $f$ .

Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithania

IMO Shortlist 2009 problem A5

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that $f\left(x-f(y)\right)>yf(x)+x$

Proposed by Igor Voronovich, Belarus