Found problems: 9
On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set.
[i]Proposed by Tonći Kokan, Croatia[/i]
A holey triangle is an upward equilateral triangle of side length $n$ with $n$ upward unit triangular holes cut out. A diamond is a $60^\circ-120^\circ$ unit rhombus.
Prove that a holey triangle $T$ can be tiled with diamonds if and only if the following condition holds: Every upward equilateral triangle of side length $k$ in $T$ contains at most $k$ holes, for $1\leq k\leq n$.
[i]Proposed by Federico Ardila, Colombia [/i]
Let $m$ be a positive integer, and $A_1, A_2, \ldots, A_m$ (not necessarily different) be $m$ subsets of a finite set $A$. It is known that for any nonempty subset $I$ of $\{1, 2 \ldots, m \}$,
\[ \Big| \bigcup_{i \in I} A_i \Big| \ge |I|+1. \]
Show that the elements of $A$ can be colored black and white, so that each of $A_1,A_2,\ldots,A_m$ contains both black and white elements.
An airline operates flights between any two capital cities in the European Union. Each flight has a fixed price which is the same in both directions. Furthermore, the flight prices from any given city are pairwise distinct. Anna and Bella wish to visit each city exactly once, not necessarily starting from the same city. While Anna always takes the cheapest flight from her current city to some city she hasn't visited yet, Bella always continues her tour with the most expensive flight available. Is it true that Bella's tour will surely cost at least as much as Anna's tour?
[i](Based on a Soviet problem)[/i]
Santa Claus has at least $n$ gifts for $n$ children. For $i\in\{1,2, ... , n\}$, the $i$-th child considers $x_i > 0$ of these items to be desirable. Assume that
\[\dfrac{1}{x_1}+\cdots+\dfrac{1}{x_n}\le1.\]
Prove that Santa Claus can give each child a gift that this child likes.
There are $100$ students who praticipate at exam.Also there are $25$ members of jury.Each student is checked by one jury.Known that every student likes $10$ jury
$a)$ Prove that we can select $7$ jury such that any student likes at least one jury.
$b)$ Prove that we can make this every student will be checked by the jury that he likes and every jury will check at most $10$ students.
On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set.
[i]Proposed by Tonći Kokan, Croatia[/i]
On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set.
[i]Proposed by Tonći Kokan, Croatia[/i]
There are $100$ students who praticipate at exam.Also there are $25$ members of jury.Each student is checked by one jury.Known that every student likes $10$ jury
$a)$ Prove that we can select $7$ jury such that any student likes at least one jury.
$b)$ Prove that we can make this every student will be checked by the jury that he likes and every jury will check at most $10$ students.