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\begin{document}

\begin{enumerate}

\item (Nesbitt)
Prove that for all positive real numbers \( a \), \( b \), and \( c \),  the following inequality holds:
\[
\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} \geq \frac{3}{2}
\]

\item (1996 \ \text{IMO Shortlist})\ \text{Let } $a$, $b$, and $c$ \text{be positive real numbers satisfying} $abc = 1$. \text{Prove that:}
\[
\frac{ab}{a^5 + b^5 + ab} + \frac{bc}{b^5 + c^5 + bc} + \frac{ca}{c^5 + a^5 + ca} \leq 1
\]

\item ((1995 \ \text{IMO Q2}))\ \text{Let} $a$, $b$, $c$ \text{be positive real numbers such that} $abc = 1$. \text{Prove that: }
\[
\frac{1}{a^3(b + c)} + \frac{1}{b^3(c + a)} + \frac{1}{c^3(a + b)} \geq \frac{3}{2}
\]

\item ((1998 \ \text{APMO Q3}))\ Let\ $a$, $b$, $c$\ be\ positive\ real\ numbers. Prove\ that:
\[
\left( 1 + \frac{a}{b} \right) \left( 1 + \frac{b}{c} \right) \left( 1 + \frac{c}{a} \right) \geq 2 \left( 1 + \frac{a + b + c}{abc} \right)
\]

\item 
((2000 \ \text{IMO Q2}))\ Let\ $a$, $b$, $c$\ be\ positive\ real\ numbers\ such\ that\ $abc = 1$. \ Prove\ that:
\[
\left( a - 1 + \frac{1}{b} \right) \left( b - 1 + \frac{1}{c} \right) \left( c - 1 + \frac{1}{a} \right) \leq 1
\]

\item ((2001 \ \text{IMO Q2}))\ Prove\ that\ for\ all\ positive\ real\ numbers\ $a$, $b$, $c$:
\[
\frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1
\]

\item Let $x$, $y$, $z$ be positive reals satisfying $xyz = 1$. Show that:
\[
\frac{x^3}{(1 + y)(1 + z)} + \frac{y^3}{(1 + z)(1 + x)} + \frac{z^3}{(1 + x)(1 + y)} \geq \frac{3}{4}
\]

\item Let $x$, $y$, $z$ be positive reals. Show that:
\[
\frac{3(x + y)(y + z)(z + x)}{8xyz} \geq (x + y + z)\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)
\]

\item Let $x$, $y$, $z$ be positive reals. Show that:
\[
(xy + yz + zx) \left( \frac{1}{(x + y)^2} + \frac{1}{(y + z)^2} + \frac{1}{(z + x)^2} \right) \geq \frac{9}{4}
\]

\item Let $k$, $a_i > 0$ and $s = \sum_{i=1}^n a_i$. Show that:
\[
\sum_{i=1}^n \left( \frac{a_i}{s - a_i} \right)^k \geq \frac{n}{(n - 1)^k}
\]

\item Let $r_i \geq 1$ for $i = 1, 2, \dots, n$. Show that:
\[
\sum_{i=1}^n \frac{1}{r_i + 1} \geq \frac{n}{\left( \prod_{i=1}^n r_i \right)^{1/n} + 1}
\]

\item Let $a_i$ be positive reals such that $\sum_{i=1}^n a_i < 1$. Show that:
\[
\frac{a_1 a_2 \dots a_n \left( 1 - \left( a_1 + a_2 + \dots + a_n \right) \right)}{  \left( a_1 + a_2 + \dots + a_n \right) \prod_{i=1}^n (1 - a_i) } \leq \frac{1}{n^{n+1}}
\]

\item (2001 \ \text{IMO Shortlist}) Let $x_1, x_2, \dots, x_n$ be arbitrary real numbers. Prove the inequality:
\[
\frac{x_1}{1 + x_1^2} + \frac{x_2}{1 + x_1^2 + x_2^2} + \dots + \frac{x_n}{1 + x_1^2 + \dots + x_n^2} < n
\]

\item (2000\ \text{AMO Q3}) Let $x_1, x_2, \dots, x_n, y_1, y_2, \dots, y_n$ be real numbers such that

(i) $0 < x_1 y_1 < x_2 y_2 < \dots < x_n y_n$ and

(ii) $x_1 + x_2 + \dots + x_i \geq y_1 + y_2 + \dots + y_i$ for $1 \leq i \leq n$.

Prove that:
\[
\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n} \leq \frac{1}{y_1} + \frac{1}{y_2} + \dots + \frac{1}{y_n}.
\]

When does equality occur?

\item All coefficients of a quadratic polynomial $f(x) = ax^2 + bx + c$ are positive, and $a + b + c = 1$. Prove that the inequality
\[
f(x_1) f(x_2)  \cdots  f(x_n) \geq 1
\]
holds for all positive numbers $x_1, x_2, \dots, x_n$ satisfying $x_1 x_2 \dots x_n = 1$.

\item If $x_i > 0$ for $i = 1, 2, \dots, n$, show that,
\[
\sum_{i=1}^n \frac{x_i^2}{x_{i+1}} \geq \sum_{i=1}^n x_i
\]
where the indices are read modulo $n$.

\item If $x_i > 0$ for $i = 1, 2, \dots, n$, show that:
\[
\sum_{i=1}^n \frac{x_i^2}{x_i^2 + x_{i+1} x_{i+2}} \leq n - 1,
\]
where the indices are read modulo $n$.

\item Let $a_0, a_1, \dots, a_{n-1}$ ($n \geq 3$) be positive integers written on the circle such that each of them divides the sum of its neighbours. Let us denote:
\[
S_n = \sum_{i=1}^n \frac{a_{i-1} + a_{i+1}}{a_i}
\]
For each $n$, determine the maximum and minimum possible values for $S_n$.

\item Let $f : [0, 1] \to \mathbb{R}$ satisfy $f(0) = f(1)$ and
\[
|f(a) - f(b)| \leq |a - b| \quad \text{for all } a, b \in [0, 1].
\]
Show that
\[
|f(a) - f(b)| \leq \frac{1}{2} \quad \text{for all } a, b \in [0, 1].
\]

\item Let $a_1, a_2, \dots, a_n$ be $n$ real numbers. Show that $a_i + a_j \geq 0$ for all $i \neq j$ if and only if
\[
\sum_{i=1}^n a_i x_i \geq \sum_{i=1}^n a_i x_i^2 \quad \text{for all } x_i \geq 0 \text{ such that } \sum_{i=1}^n x_i = 1.
\]

\item Let $n \geq 3$ be an integer. Suppose that the inequality
\[
\left( a_1^2 + a_2^2 + \dots + a_n^2 \right)^2 > (n - 1) \left( a_1^4 + a_2^4 + \dots + a_n^4 \right)
\]
holds for some positive real numbers $a_1, a_2, \dots, a_n$. Prove that $a_i, a_j, a_k$ are the sides of some triangle for all $i, j, k$.

\item (1978 \ \text{IMO Q5}) Let $\{a_k\}$ be a sequence of pairwise distinct positive integers. Prove that:
\[
\sum_{k=1}^n \frac{a_k}{k^2} \geq \sum_{k=1}^n \frac{1}{k}
\]

\item  Let $z_1, z_2, \dots, z_n$ be complex numbers such that $\sum_{i=1}^n |z_i| = 1$. Show that
\[
\left| \sum_{i \in S} z_i \right| \geq \frac{1}{6}
\]
for some subset $S$ of $\{z_1, z_2, \dots, z_n\}$.

\item Consider $n$ complex numbers $z_k$ such that $|z_k| \leq 1$ for $k = 1, 2, \dots, n$. Prove that there exist $e_1, e_2, \dots, e_n \in \{-1, 1\}$ such that for any $m \leq n$, 
\[
\left| e_1 z_1 + e_2 z_2 + \dots + e_n z_n \right| \leq 2.
\]

\item Find all non-negative real numbers $a_1 \leq a_2 \leq \dots \leq a_n$ satisfying:
\[
\sum_{i=1}^n a_i = 96, \quad \sum_{i=1}^n a_i^2 = 144, \quad \sum_{i=1}^n a_i^3 = 216.
\]

\item (1999 \ \text{APMO Q2}) Let $a_1, a_2, \dots$ be a sequence of real numbers satisfying $a_{i+j} \leq a_i + a_j$ for all $i, j = 1, 2, \dots$. Prove that:
\[
a_1 + \frac{a_2}{2} + \frac{a_3}{3} + \dots + \frac{a_n}{n} \geq a_n
\]
for each positive integer $n$.

\item Let $a_1, a_2, \dots$ be non-negative numbers satisfying $a_{n+m} \leq a_n + a_m$ for all $m, n \in \mathbb{N}$. Prove that:
\[
a_n \leq m a_1 + \left(\frac{n}{m} - 1\right) a_m
\]
for all $n \geq m$.


\item (2001 \ \text{IMO Shortlist}) Let $a_0, a_1, \dots$ be an arbitrary infinite sequence of positive numbers. Show that:
\[
1 + a_n > a_{n-1} \sqrt[n]{2} \quad \text{for infinitely many positive integers } n.
\]

\item Let $n \geq 3$, $a_i \in [2, 3]$, and $s = \sum_{i=1}^n a_i$. Show that:
\[
\sum_{i \mod n} \frac{a_i^2 + a_{i+1}^2 - a_{i+2}^2}{a_i + a_{i+1} - a_{i+2}} \leq 2s - 2n.
\]

\item Let \( p(x) \) be a polynomial with real coefficients such that \( p(x) > 0 \) whenever \( x > 0 \). Show that there exists a positive integer \( n \) such that \( (1 + x)^n p(x) \) has all its coefficients positive.

\item (1997\ IMO\ Q3) Let \( x_1, x_2, \dots, x_n \) be real numbers satisfying the conditions:
\[
|x_1 + x_2 + \dots + x_n| = 1 \quad \text{and} \quad |x_i| \leq \frac{n+1}{2} \quad \text{for} \quad i = 1, 2, \dots, n.
\]
Show that there exists a permutation \( y_1, y_2, \dots, y_n \) of \( x_1, x_2, \dots, x_n \) such that
\[
|y_1 + 2 y_2 + \dots + n y_n| \leq \frac{n+1}{2}.
\]

\item (Klamkin) Let
\[
x_n = \sqrt[2]{2+\sqrt[3]{3+\sqrt[4]{\cdots +\sqrt[n]{n}}}}
\]
 Show that
\[
x_{n+1} - x_n < \frac{1}{n!}.
\]

\item 
Prove that for all positive integers \(n\),

\[
\prod_{i=1}^{n} \frac{2i - 1}{2i} < \frac{1}{3n}
\]

\item 

Prove that for natural numbers \(k < l < m < n\) satisfying \(kn = lm\), the following inequality holds:

\[
\left(\frac{n - k}{2}\right)^2 \geq k + 2.
\]

\end{enumerate}
\end{document}
