MathDB

5

Part of 1998 Estonia National Olympiad

Problems(4)

13 children are sitting at a round table each holding 2 cards, no 1-13

Source: 1998 Estonia National Olympiad Final Round grade 9 p5

11/4/2020
Thirteen children are sitting at a round table, each holding two cards. Each card has one of the numbers 1,2,...,131, 2, ..., 13 written on it, and each number is written on exactly two cards. On a signal, each child gives the card with the lower number to his neighbor on the right (and at the same time receives his card with the lower number from the neighbor on the left). Prove that after a finite number of such exchanges, a situation arises when at least one of the children will have two cards with the same number.
combinatorics
game with blue and red dots, special points

Source: 1998 Estonia National Olympiad Final Round grade 10 p5

3/14/2020
The paper is marked with the finite number of blue and red dots and some these points are connected by lines. Let's name a point PP special if more than half of the points connected with PP has a color other than point PP. Juku selects one special point and reverses its color. Then Juku selects a new special point and changes its color, etc. Prove that by a finite number of integers Juku ends up in a situation where the paper has not made a special point.
combinatoricsColoring
circle is divided into n equal arcs by n points, color points in 2 colors

Source: 1998 Estonia National Olympiad Final Round grade 11 p5

3/11/2020
A circle is divided into nn equal arcs by nn points. Assume that, no matter how we color the nn points in two colors, there always exists an axis of symmetry of the set of points such that any two of the nn points which are symmetric with respect to that axis have the same color. Find all possible values of nn.
circlespointsColoringcombinatoricscombinatorial geometry
L- trominos in a nxn square

Source: 1998 Estonia National Olympiad Final Round grade 12 p5

3/11/2020
From an n×nn\times n square divided into n2n^2 unit squares, one corner unit square is cut off. Find all positive integers nn for which it is possible to tile the remaining part of the square with LL-trominos. https://cdn.artofproblemsolving.com/attachments/0/4/d13e6e7016d943b867f44375a2205b10ccf552.png
combinatoricscombinatorial geometrysquare