IZhO 2023

Peter has a deck of $1001$ cards, and with a blue pen he has written the numbers $1,2,\ldots,1001$ on the cards (one number on each card). He replaced cards in a circle so that blue numbers were on the bottom side of the card. Then, for each card $C$ , he took $500$ consecutive cards following $C$ (clockwise order), and denoted by $f(C)$ the number of blue numbers written on those $500$ cards that are greater than the blue number written on $C$ itself. After all, he wrote this $f(C)$ number on the top side of the card $C$ with a red pen. Prove that Peter's friend Basil, who sees all the red numbers on these cards, can determine the blue number on each card.

The tangent at $C$ to $\Omega$ , the circumcircle of scalene triangle $ABC$ intersects $AB$ at $D$ . Through point $D$ , a line is drawn that intersects segments $AC$ and $BC$ at $K$ and $L$ respectively. On the segment $AB$ points $M$ and $N$ are marked such that $AC \parallel NL$ and $BC \parallel KM$ . Lines $NL$ and $KM$ intersect at point $P$ lying inside the triangle $ABC$ . Let $\omega$ be the circumcircle of $MNP$ . Suppose $CP$ intersects $\omega$ again at $Q$ . Show that $DQ$ is tangent to $\omega$ .

Let $a_1, a_2, \cdots, a_k$ be natural numbers. Let $S(n)$ be the number of solutions in nonnegative integers to $a_1x_1 + a_2x_2 + \cdots + a_kx_k = n$ . Suppose $S(n) \neq 0$ for all big enough $n$ . Show that for all sufficiently large $n$ , we have $S(n+1) < 2S(n)$ .

The sum of $n > 2$ nonzero real numbers (not necessarily distinct) equals zero. For each of the $2^n - 1$ ways to choose one or more of these numbers, their sums are written in non-increasing order in a row. The first number in the row is $S$ . Find the smallest possible value of the second number.

We call a positive integer $n$ is $good$ , if there exist integers $a,b,c,x,y$ such that $n=ax^2+bxy+cy^2$ and $b^2-4ac=-20$ . Prove that the product of any two good numbers is also a good number.

Several blue and green rectangular napkins (perhaps of different sizes) with vertical and horizontal sides were placed on the plane. It turned out that any two napkins of different colors can be crossed by a vertical or horizontal line (perhaps along the border). Prove that it is possible to choose a color, two horizontal lines and one vertical line, so that each napkin of the chosen color is intersected by at least one of the chosen lines.