MathDB

5

Part of 2004 May Olympiad

Problems(2)

3 square pieces on 9x9

Source: X May Olympiad (Olimpiada de Mayo) 2004 L2 P5

9/19/2022
On a 9×9 9\times 9 board, divided into 1×11\times 1 squares, pieces of the form Each piece covers exactly 33 squares. (a) Starting from the empty board, what is the maximum number of pieces that can be placed? (b) Starting from the board with 33 pieces already placed as shown in the diagram below, what is the maximum number of pieces that can be placed? https://cdn.artofproblemsolving.com/attachments/d/4/3bd010828accb2d1811d49eb17fa69662ff60d.gif
combinatoricscombinatorial geometry
2 digits written on 90 cards

Source: X May Olympiad (Olimpiada de Mayo) 2004 L1 P5

9/22/2022
There are 9090 cards and two different digits are written on each one: 0101, 0202, 0303, 0404, 0505, 0606, 0707, 0808, 0909, 1010, 1212, and so on up to 9898. A set of cards is correct if it does not contain any cards whose first digit is the same as the second digit of another card in the set. We call the value of a set of cards the sum of the numbers written on each card. For example, the four cards 0404, 3535, 7878 and 9898 form a correct set and their value is 215215, since04+35+78+98=215 04+35+78+98=215. Find a correct set that has the largest possible value. Explain why it is impossible to achieve a correct set of higher value.
combinatorics