MathDB

4

Part of 1997 May Olympiad

Problems(2)

graceful distribution of numbers in circles

Source: III May Olympiad (Olimpiada de Mayo) 1997 L2 P4

9/17/2022
In the figures, the vertices are marked with a circle. The segments that join vertices are called paths. Non-negative integers are distributed to the vertices and, to the paths, the differences between the numbers at their ends. https://cdn.artofproblemsolving.com/attachments/d/6/e6fce93719a5b35dbf34d58652b01a8631de57.gif
We will say that a distribution of numbers is graceful if all the numbers from 11 to nn appear in the paths, where nn is the number of paths.
The following is an example of graceful distribution: https://cdn.artofproblemsolving.com/attachments/1/1/a8c2b4fde673ca902b655804c4f5321f9666e9.gif Give -if possible- a graceful distribution for the following figures. If you can't do it, show why.
combinatorics
5-digits numbers of tickets

Source: III May Olympiad (Olimpiada de Mayo) 1997 L1 P4

9/17/2022
Joaquín and his brother Andrés go to class every day on the 6262 bus. Joaquín always pays for the tickets. Each ticket has a 55-digit number printed on it. One day, Joaquín observes that the numbers on his tickets - his and his brother's - as well as being consecutive, are such that the sum of the ten digits is precisely 6262. Andrés asks him if the sum of the digits of any of the tickets is 3535 and, knowing the answer, he can directly say the number of each ticket. What were those numbers?
number theory