Problems

Problem 1
 

If \( a,b,c\geq 1 \), prove that \[ \sqrt{a-1}+\sqrt{b-1}+ \sqrt{c-1} \leq \sqrt{c(ab+1)}.\]

Problem 2 (Rade TodoroviŠ)
 

Let \( a_1,a_2,\dots,a_n,b_1,b_2,\dots,b_n \) be positive real numbers. Prove that \[ \left(\sum_{i\neq j}a_ib_j\right)^2\geq\left(\sum _{i\neq j}a_ia_j\right)\left(\sum_{i\neq j}b_ib_j\right).\]

Problem 3
 

If \( \frac1x+ \frac1y+\frac1z=1 \) for \( x,y,z> 0 \), prove that \[ (x-1)(y-1)(z-1)\geq 8.\]

Problem 4 (Rade StanojeviŠ)
 

Let \( a,b,c> 0 \) satisfy \( abc=1 \). Prove that \[ \frac1{\sqrt{b+\frac1a+\frac12}} + \frac1{\sqrt{c+\frac1b+\frac12}}+ \frac1{\sqrt{a+\frac1c+\frac12}}\geq \sqrt2.\]

Problem 5
 

Given positive numbers \( a,b,c,x,y,z \) such that \( a+x=b+y=c+z=S \), prove that \( ay+bz+cx< S^2 \).

Problem
 

Determine the maximal real number \( a \) for which the inequality \[ x_1^2+x_2^2+x_3^2+x_4^2+x_5^2\geq a(x_1x_2+x_2x_3+x_3x_4+ x_4x_5)\] holds for any five real numbers \( x_1,x_2,x_3,x_4,x_5 \).

Problem 7
 

If \( x,y,z \geq 0 \) and \( x+y+z=1 \), prove that \[ 0\leq xy+yz+zx - 2xyz \leq \frac7{27}.\]

Problem 8
 

Let \( a_1, \dots, a_n \) be positive real numbers. Prove that \[ \frac{a_1^3}{a_2}+\frac{a_2^3}{a_3}+ \cdots + \frac{a_n^3}{a_1}\geq a_1^2+a_2^2+\cdots + a_n^2.\]

Problem 9
 

Let \( a_1, \dots, a_n \) be positive real numbers. Prove that \[ (1+a_1)(1+a_2)\cdots (1+a_n) \leq \left(1+\frac{a_1^2}{a_2}\right)\cdot \left(1+\frac{a_2^2}{a_3}\right)\cdot \cdots \cdot \left(1+\frac{a_n^2}{a_1}\right).\]

Problem 10
 

Let \( 0< x_1 \leq x_2 \leq \cdots \leq x_n \) (\( n\geq 2 \)) and \[ \frac1{1+x_1} + \frac1{1+x_2}+ \cdots + \frac1{1+x_n} = 1.\] Prove that \[ \sqrt{x_1} + \sqrt{x_2} + \cdots + \sqrt{x_n} \geq (n-1) \left( \frac1{\sqrt{x_1}}+ \frac1{\sqrt{x_2}} + \cdots + \frac1{\sqrt{x_n}}\right).\]

Problem 11
 

Suppose that any two members of certain society are either friends or enemies. Suppose that there is total of \( n \) members, that there is total of \( q \) pairs of friends, and that in any set of three persons there are two who are enemies to each other. Prove that there exists at least one member among whose enemies we can find at most \( q\cdot \left(1-\frac{4q}{n^2}\right) \) pairs of friends.

Problem 12
 

Let \( a \), \( b \), \( c \) be positive real numbers such that \( \frac1{1+a}+\frac1{1+b}+\frac1{1+c}=2 \). Prove that \[ \sqrt{3+a+b+c}\geq\sqrt a+\sqrt b+\sqrt c.\]

Problem 13
 

Let \( a \), \( b \), \( c \) be positive real numbers. Prove that \[ \frac{2a^3}{a^2+b^2}+\frac{2b^3}{b^2+c^2}+\frac{2c^3}{c^2+a^2}\geq a+b+c.\]

Problem 14
 

If \( a \), \( b \), \( c \) are positive real numbers such that \( a+b+c=1 \) prove that \[ \frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\geq \frac12.\]

Problem 15
 

Let \( x_1 \), \( x_2 \), \( \dots \), \( x_{2009} \) be positive real numbers such that \[ x_i^2 \geq x_1^2+\frac{x_2^2}{2^3}+\frac{x_3^2}{3^3}+\cdots+ \frac{x_{i-1}^2}{(i-1)^3},\;\; 2\leq i\leq 2009.\] Prove that \[ \sum_{i=2}^{2009}\frac{x_i}{x_1+x_2+\cdots+ x_{i-1}}> 1.999.\]

Problem 16
 

Let \( a \), \( b \), \( c \) be positive real numbers. Assume that \( n \) and \( k \) are positive integers. Prove that \[ \frac{a^{n+k}}{b^n}+\frac{b^{n+k}}{c^n}+\frac{c^{n+k}}{a^n}\geq a^k+b^k+c^k.\]

Problem 17
 

If \( a \), \( b \), \( c \) are positive real numbers prove that \[ \frac{a+b}{2b+c}+\frac{b+c}{2c+a}+\frac{c+a}{2a+b}\geq 2.\]

Problem 18
 

If \( x \), \( y \), \( z \) are real numbers from the interval \( (0,\sqrt 5) \) such that \( xyz=1 \), prove that \[ \frac x{5-y^2}+\frac y{5-z^2}+\frac z{5-x^2}\geq \frac34.\]

Problem 19
 

Let \( m \), \( n \) be positive integers. Prove that for all real numbers \( x_1, \dots, x_n, y_1, \dots, y_n\in[0,1] \) such that \( x_i+y_i=1 \) for \( i=1,\dots, n \) the following inequality holds: \[ (1-x_1\cdots x_n)^m+(1-y_1^m)\cdots (1-y_n^m)\geq 1.\]


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