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Mala olimpijada 1998 zadatak 3

1998 alg hrv izborno nejednakost niz
Neka je \{ F_i \}, i=0,1, \ldots niz brojeva definiran na sljedeći način:
F_0=0, \ F_1=1,\ F_{i+2}=F_{i+1}+F_{i}, \ i=0,1, \ldots
Za prirodan broj n \geq 2 neka su a_0, a_1, \ldots a_n nenegativni brojevi koji zadovoljavaju uvjet
a_0=1, \ a_i \leq a_{i+1} + a_{i+2}, \ i=0,1, \ldots, n-2.
Dokažite da je a_0+a_1+\ldots+a_n \geq \frac{F_{n+2}-1}{F_{n}}. Da li se postiže jednakost?

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IMO Shortlist 2009 problem A2

2009 alg nejednakost shortlist
Let a, b, c be positive real numbers such that \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c. Prove that:
\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}


Proposed by Juhan Aru, Estonia

IMO Shortlist 2006 problem A4

2006 alg nejednakost shortlist
Prove the inequality:

\displaystyle \sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} + a_{j}}}\leq \frac {n}{2(a_{1} + a_{2} + ... + a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}

for positive reals a_{1}, a_{2}, ..., a_{n}.

IMO Shortlist 2001 problem A3

2001 alg nejednakost shortlist
Let x_1,x_2,\ldots,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} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.

IMO Shortlist 2001 problem A2

2001 alg nejednakost niz shortlist tb
Let a_0, a_1, a_2, \ldots be an arbitrary infinite sequence of positive numbers. Show that the inequality 1 + a_n > a_{n-1} \sqrt[n]{2} holds for infinitely many positive integers n.

IMO Shortlist 1998 problem A1

1998 alg nejednakost shortlist
Let a_{1},a_{2},\ldots ,a_{n} be positive real numbers such that a_{1}+a_{2}+\cdots +a_{n}<1. Prove that

\frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_1)(1 - a_2) \cdots (1 - a_n)} \leqslant \frac{1}{n^{n+1}}

IMO Shortlist 1995 problem A3

1995 alg nejednakost shortlist
Let n be an integer, n \geq 3. Let a_1, a_2, \ldots, a_n be real numbers such that 2 \leq a_i \leq 3 for i = 1, 2, \ldots, n. If s = a_1 + a_2 + \ldots + a_n, prove that \frac{a^{2}_{1}+a^{2}_{2}-a^{2}_{3}}{a_{1}+a_{2}-a_{3}}+\frac{a^{2}_{2}+a^{2}_{3}-a^{2}_{4}}{a_{2}+a_{3}-a_{4}}+\ldots+\frac{a^{2}_{n}+a^{2}_{1}-a^{2}_{2}}{a_{n}+a_{1}-a_{2}}\leq 2s-2n.