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IMO Shortlist 2007 problem A1

Real numbers $a_{1}$ , $a_{2}$ , $\ldots$ , $a_{n}$ are given. For each $i$ , $(1 \leq i \leq n )$ , define
$d_{i} = \max \{ a_{j}\mid 1 \leq j \leq i \} - \min \{ a_{j}\mid i \leq j \leq n \}$
and let $d = \max \{d_{i}\mid 1 \leq i \leq n \}$ .

(a) Prove that, for any real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ ,
$\max \{ |x_{i} - a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*)$
(b) Show that there are real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*).

Author: Michael Albert, New Zealand

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IMO Shortlist 1994 problem A2

Let $m$ and $n$ be two positive integers. Let $a_1$ , $a_2$ , $\ldots$ , $a_m$ be $m$ different numbers from the set $\{1, 2,\ldots, n\}$ such that for any two indices $i$ and $j$ with $1\leq i \leq j \leq m$ and $a_i + a_j \leq n$ , there exists an index $k$ such that $a_i + a_j = a_k$ . Show that
$\frac {a_1 + a_2 + ... + a_m}{m} \geq \frac {n + 1}{2}.$

IMO Shortlist 1995 problem A1

Let $a$ , $b$ , $c$ be positive real numbers such that $abc = 1$ . Prove that $\frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}.$

IMO Shortlist 1995 problem S2

Find the maximum value of $x_{0}$ for which there exists a sequence $x_{0},x_{1}\cdots ,x_{1995}$ of positive reals with $x_{0} = x_{1995}$ , such that
$x_{i - 1} + \frac {2}{x_{i - 1}} = 2x_{i} + \frac {1}{x_{i}},$
for all $i = 1,\cdots ,1995$ .

IMO Shortlist 2000 problem A1

Let $a, b, c$ be positive real numbers so that $abc = 1$ . Prove that
$\left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.$

IMO Shortlist 2007 problem A2

Consider those functions $f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition
$f(m + n) \geq f(m) + f(f(n)) - 1$
for all $m,n \in \mathbb{N}.$ Find all possible values of $f(2007).$

Author: unknown author, Bulgaria

IMO Shortlist 2008 problem A1

Find all functions $f: (0, \infty) \mapsto (0, \infty)$ (so $f$ is a function from the positive real numbers) such that
$\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}$
for all positive real numbes $w,x,y,z,$ satisfying $wx = yz.$

Author: Hojoo Lee, South Korea