Olympiad problems

1990 IMO Shortlist, 24

Let $ w, x, y, z$ are non-negative reals such that $ wx \plus{} xy \plus{} yz \plus{} zw \equal{} 1$. Show that $ \frac {w^3}{x \plus{} y \plus{} z} \plus{} \frac {x^3}{w \plus{} y \plus{} z} \plus{} \frac {y^3}{w \plus{} x \plus{} z} \plus{} \frac {z^3}{w \plus{} x \plus{} y}\geq \frac {1}{3}$.

2016 Croatia Team Selection Test, Problem 2

Tags: linear combination , coloring , cauchy schwarz , combinatorics , matrix

Let $N$ be a positive integer. Consider a $N \times N$ array of square unit cells. Two corner cells that lie on the same longest diagonal are colored black, and the rest of the array is white. A [i]move[/i] consists of choosing a row or a column and changing the color of every cell in the chosen row or column. What is the minimal number of additional cells that one has to color black such that, after a finite number of moves, a completely black board can be reached?

2016 Croatia Team Selection Test, Problem 2

Tags: combinatorics , linear combination , coloring , cauchy schwarz , matrix

Let $N$ be a positive integer. Consider a $N \times N$ array of square unit cells. Two corner cells that lie on the same longest diagonal are colored black, and the rest of the array is white. A [i]move[/i] consists of choosing a row or a column and changing the color of every cell in the chosen row or column. What is the minimal number of additional cells that one has to color black such that, after a finite number of moves, a completely black board can be reached?

2011 Laurențiu Duican, 3

Tags: cauchy schwarz , polynom , algebra

Let $ n\ge 2 $ be a perfect square and let be $ n $ natural numbers $ m_1,m_2,\ldots ,m_n. $ Prove that if the polynom $$ X^2-\left( 1+ m_1^2+m_2^2+\cdots +m_n^2 \right) X+m_1m_2+m_2m_3+\cdots +m_{n-1}m_n +m_nm_1\in \mathbb{N} [X] $$ is reducible, then its two roots are perfect squares.

2011 Bogdan Stan, 3

Tags: inequalities , cauchy schwarz

Prove that $$ a+b+c>\left( \sqrt\alpha +\sqrt\beta +\sqrt\gamma \right)^2, $$ for all positive real numbers $ a,b,c,\alpha ,\beta ,\gamma $ that are under the condition $$ abc>\alpha bc+\beta ac+\gamma ab. $$ [i]Țuțescu Lucian[/i] and [i]Chiriță Aurel[/i]

2022 Bulgaria JBMO TST, 2

Tags: am-gm , titu's lemma , cauchy schwarz , inequalities

Let $a$, $b$ and $c$ be positive real numbers with $abc = 1$. Determine the minimum possible value of $$ \left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \cdot \left(\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}\right) $$ as well as all triples $(a,b,c)$ which attain the minimum.

2005 Iran MO (3rd Round), 1

Tags: inequalities , 3-variable inequality , cyclic inequality , cauchy schwarz

Suppose $a,b,c\in \mathbb R^+$. Prove that :\[\left(\frac ab+\frac bc+\frac ca\right)^2\geq (a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\]

1990 IMO Longlists, 88

Tags: cauchy schwarz , holder , inequality , 4-variable inequality , algebra

Let $ w, x, y, z$ are non-negative reals such that $ wx \plus{} xy \plus{} yz \plus{} zw \equal{} 1$. Show that $ \frac {w^3}{x \plus{} y \plus{} z} \plus{} \frac {x^3}{w \plus{} y \plus{} z} \plus{} \frac {y^3}{w \plus{} x \plus{} z} \plus{} \frac {z^3}{w \plus{} x \plus{} y}\geq \frac {1}{3}$.